Editing List of unsolved problems in mathematics

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Many [[mathematical problems]] have been stated but not yet solved. These problems come from many [[areas of mathematics]], such as [[theoretical physics]], [[computer science]], [[algebra]], [[Mathematical analysis|analysis]], [[combinatorics]], [[Algebraic geometry|algebraic]], [[Differential geometry|differential]], [[Discrete geometry|discrete]] and [[Euclidean geometry|Euclidean geometries]], [[graph theory]], [[group theory]], [[model theory]], [[number theory]], [[set theory]], [[Ramsey theory]], [[dynamical system]]s, and [[partial differential equation]]s. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the [[Millennium Prize Problems]], receive considerable attention.

This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative, and the problems listed here vary widely in both difficulty and importance.

== Lists of unsolved problems in mathematics ==
Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions.

{| class="wikitable sortable"
|-
! List !! Number of<br />problems !! Number unsolved <br /> or incompletely solved !! Proposed by !! Proposed<br />in
|-
| [[Hilbert's problems]]<ref>{{citation|last=Thiele|first=Rüdiger|chapter=On Hilbert and his twenty-four problems|title=Mathematics and the historian's craft. The Kenneth O. May Lectures|pages=243–295|isbn=978-0-387-25284-1|editor-last=Van Brummelen|editor-first=Glen|year=2005|series=[[Canadian Mathematical Society|CMS]] Books in Mathematics/Ouvrages de Mathématiques de la SMC |volume=21|title-link=Kenneth May}}</ref> || 23 || 15 || [[David Hilbert]] || 1900
|-
| [[Landau's problems]]<ref>{{citation|title=Unsolved Problems in Number Theory|first=Richard|last=Guy|author-link=Richard K. Guy|edition=2nd|publisher=Springer|year=1994|page=vii|url=https://books.google.com/books?id=EbLzBwAAQBAJ&pg=PR7|isbn=978-1-4899-3585-4|access-date=2016-09-22|archive-url=https://web.archive.org/web/20190323220345/https://books.google.com/books?id=EbLzBwAAQBAJ&pg=PR7|archive-date=2019-03-23|url-status=live}}.</ref> || 4 || 4 || [[Edmund Landau]] || 1912
|-
| [[Taniyama's problems]]<ref>{{cite journal | last = Shimura | first = G. | author-link = Goro Shimura | title = Yutaka Taniyama and his time | journal = Bulletin of the London Mathematical Society | volume = 21 | issue = 2 | pages = 186–196 | year = 1989 | doi = 10.1112/blms/21.2.186 }}</ref> || 36 || – || [[Yutaka Taniyama]] || 1955
|-
| [[Thurston's 24 questions]]<ref>{{cite journal
 | last = Friedl | first = Stefan
 | doi = 10.1365/s13291-014-0102-x
 | issue = 4
 | journal = Jahresbericht der Deutschen Mathematiker-Vereinigung
 | mr = 3280572
 | pages = 223–241
 | title = Thurston's vision and the virtual fibering theorem for 3-manifolds
 | volume = 116
 | year = 2014| s2cid = 56322745
 }}</ref><ref>{{cite journal
 | last = Thurston | first = William P.
 | doi = 10.1090/S0273-0979-1982-15003-0
 | issue = 3
 | journal = Bulletin of the American Mathematical Society
 | mr = 648524
 | pages = 357–381
 | series = New Series
 | title = Three-dimensional manifolds, Kleinian groups and hyperbolic geometry
 | volume = 6
 | year = 1982}}</ref> || 24 || 2 || [[William Thurston]] || 1982
|-
| [[Smale's problems]] || 18 || 14 || [[Stephen Smale]] || 1998
|-
| [[Millennium Prize Problems]] || 7 || 6<ref name="auto1">{{cite web |title=Millennium Problems |url=http://claymath.org/millennium-problems |archive-url=https://web.archive.org/web/20170606121331/http://claymath.org/millennium-problems |archive-date=2017-06-06 |access-date=2015-01-20 |website=claymath.org}}</ref>|| [[Clay Mathematics Institute]] || 2000
|-
| [[Simon problems]] || 15 || <&nbsp;12<ref>{{cite web |url=http://www2.cnrs.fr/en/2435.htm?debut=8&theme1=12 |title=Fields Medal awarded to Artur Avila |website=[[Centre national de la recherche scientifique]] |date=2014-08-13 |access-date=2018-07-07 |archive-url=https://web.archive.org/web/20180710010437/http://www2.cnrs.fr/en/2435.htm?debut=8&theme1=12 |archive-date=2018-07-10 }}</ref><ref name="guardian">{{cite web |url=https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/aug/13/fields-medals-2014-maths-avila-bhargava-hairer-mirzakhani |title=Fields Medals 2014: the maths of Avila, Bhargava, Hairer and Mirzakhani explained |website=[[The Guardian]] |last=Bellos |first=Alex |date=2014-08-13 |access-date=2018-07-07 |archive-url=https://web.archive.org/web/20161021115900/https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/aug/13/fields-medals-2014-maths-avila-bhargava-hairer-mirzakhani |archive-date=2016-10-21 |url-status=live }}</ref> || [[Barry Simon]] || 2000
|-
| [[Unsolved Problems on Mathematics for the 21st Century]]<ref>{{cite book | last1 = Abe | first1 = Jair Minoro | last2 = Tanaka | first2 = Shotaro | title = Unsolved Problems on Mathematics for the 21st Century | publisher = IOS Press | year = 2001 | url = https://books.google.com/books?id=yHzfbqtVGLIC&q=unsolved+problems+in+mathematics | isbn = 978-90-5199-490-2}}</ref> || 22 || – || Jair Minoro Abe, Shotaro Tanaka || 2001
|-
| [[DARPA]]'s math challenges<ref>{{cite web | title = DARPA invests in math | publisher = [[Cable News Network|CNN]] | date = 2008-10-14 | url = http://edition.cnn.com/2008/TECH/science/10/09/darpa.challenges/index.html | access-date = 2013-01-14 | archive-url = https://web.archive.org/web/20090304121240/http://edition.cnn.com/2008/TECH/science/10/09/darpa.challenges/index.html | archive-date = 2009-03-04}}</ref><ref>{{cite web | title = Broad Agency Announcement (BAA 07-68) for Defense Sciences Office (DSO) | publisher = DARPA | date = 2007-09-10 | url = http://www.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html | access-date = 2013-06-25 | archive-url = https://web.archive.org/web/20121001111057/http://www.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html
 | archive-date = 2012-10-01}}</ref> || 23 || – || [[DARPA]] || 2007
|-
| [[Erdős's problems]]<ref>{{cite web|url=https://www.erdosproblems.com/|title=Erdős Problems|first=Thomas|last=Bloom|author-link=Thomas Bloom|access-date=2024-08-25}}</ref> || >&nbsp;934 || 617 || [[Paul Erdős]] || Over six decades of Erdős' career, from the 1930s to 1990s
|}
[[File:Riemann-Zeta-Func.png|thumb|250px|The [[Riemann zeta function]], subject of the [[Riemann hypothesis]]<ref>{{Cite web |title=Math Problems Guide: From Simple to Hardest Math Problems Tips & Examples. |url=https://blendedlearningmath.com/math-word-problems-to-challenge-university-students/ |access-date=2024-11-28 |website=blendedlearningmath |language=en-US}}</ref>]]

=== Millennium Prize Problems ===
Of the original seven [[Millennium Prize Problems]] listed by the [[Clay Mathematics Institute]] in 2000, six remain unsolved to date:<ref name="auto1"/>

* [[Birch and Swinnerton-Dyer conjecture]]
* [[Hodge conjecture]]
* [[Navier–Stokes existence and smoothness]]
* [[P versus NP problem|P versus NP]]
* [[Riemann hypothesis]]
* [[Yang–Mills existence and mass gap]]

The seventh problem, the [[Poincaré conjecture]], was solved by [[Grigori Perelman]] in 2003.<ref>{{cite web |title=Poincaré Conjecture |url=http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture |archive-url=https://web.archive.org/web/20131215120130/http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture |archive-date=2013-12-15 |website=Clay Mathematics Institute}}</ref> However, a generalization called the [[Generalized Poincaré conjecture|smooth four-dimensional Poincaré conjecture]]—that is, whether a ''four''-dimensional [[topological sphere]] can have two or more inequivalent [[smooth structure]]s—is unsolved.<ref>{{cite web |last=rybu |date=November 7, 2009 |title=Smooth 4-dimensional Poincare conjecture |url=http://www.openproblemgarden.org/?q=op/smooth_4_dimensional_poincare_conjecture |url-status=live |archive-url=https://web.archive.org/web/20180125203721/http://www.openproblemgarden.org/?q=op%2Fsmooth_4_dimensional_poincare_conjecture |archive-date=2018-01-25 |access-date=2019-08-06 |website=Open Problem Garden}}</ref>

=== Notebooks ===
* The [[Kourovka, Sverdlovsk Oblast|Kourovka]] Notebook ({{Langx|ru|Коуровская тетрадь}}) is a collection of unsolved problems in [[group theory]], first published in 1965 and updated many times since.<ref>{{citation |last1=Khukhro |first1=Evgeny I. |title=Unsolved Problems in Group Theory. The Kourovka Notebook |year=2019 |arxiv=1401.0300v16 |last2=Mazurov |first2=Victor D. |author-link2=Victor Mazurov}}</ref>
* The [[Yekaterinburg|Sverdlovsk]] Notebook ({{Langx|ru|Свердловская тетрадь}}) is a collection of unsolved problems in [[semigroup theory]], first published in 1965 and updated every 2 to 4 years since.<ref>{{Cite book |last1=RSFSR |first1=MV i SSO |url=https://books.google.com/books?id=nKwgzgEACAAJ |title=Свердловская тетрадь: нерешенные задачи теории подгрупп |last2=Russie) |first2=Uralʹskij gosudarstvennyj universitet im A. M. Gorʹkogo (Ekaterinbourg |date=1969 |publisher=S. l. |language=ru}}</ref><ref>{{cite book| title = Свердловская тетрадь: Сб. нерешённых задач по теории полугрупп |location= [[Свердловск]] |date = 1979 |publisher= [[Уральский государственный университет]] }}</ref><ref>{{cite book| title = Свердловская тетрадь: Сб. нерешённых задач по теории полугрупп |location= [[Свердловск]] |date = 1989 |publisher= [[Уральский государственный университет]] }}</ref>
* The [[Dniester]] Notebook ({{Langx|ru|Днестровская тетрадь}}) lists several hundred unsolved problems in algebra, particularly [[ring theory]] and [[Modulus (algebraic number theory)|modulus theory]].<ref>{{citation |title=ДНЕСТРОВСКАЯ ТЕТРАДЬ |url=http://math.nsc.ru/LBRT/a1/files/dnestr93.pdf |year=1993 |trans-title=DNIESTER NOTEBOOK |publisher=The Russian Academy of Sciences |language=ru}}</ref><ref>{{citation |title=DNIESTER NOTEBOOK: Unsolved Problems in the Theory of Rings and Modules |url=https://math.usask.ca/~bremner/research/publications/dniester.pdf |website=[[University of Saskatchewan]] |access-date=2019-08-15}}</ref>
* The [[Erlagol]] Notebook ({{Langx|ru|Эрлагольская тетрадь}}) lists unsolved problems in algebra and [[model theory]].<ref>{{citation |title=Эрлагольская тетрадь |url=http://uamt.conf.nstu.ru/erl_note.pdf |year=2018 |trans-title=Erlagol notebook |publisher=The Novosibirsk State University |language=ru}}</ref>

== Unsolved problems ==

=== Algebra ===
{{Main|Algebra}}
[[File:Regular tetrahedron inscribed in a sphere.svg|thumb|In the [[Bloch sphere]] representation of a [[qubit]], a [[SIC-POVM]] forms a [[regular tetrahedron]]. Zauner conjectured that analogous structures exist in complex [[Hilbert space]]s of all finite dimensions.]]

* [[Birch–Tate conjecture]] on the relation between the order of the [[Center (group theory)|center]] of the [[Steinberg group (K-theory)|Steinberg group]] of the [[ring of integers]] of a [[Algebraic number field|number field]] to the field's [[Dedekind zeta function]].
* [[Bombieri–Lang conjecture]]s on densities of rational points of [[algebraic surface]]s and [[Algebraic variety|algebraic varieties]] defined on [[Algebraic number field|number fields]] and their [[field extension]]s.
* [[Connes embedding problem]] in [[Von Neumann algebra]] theory
* [[Crouzeix's conjecture]]: the [[matrix norm]] of a complex function <math>f</math> applied to a complex matrix <math>A</math> is at most twice the [[Infimum and supremum|supremum]] of <math>|f(z)|</math> over the [[numerical range|field of values]] of <math>A</math>.
* [[Determinantal conjecture]] on the [[determinant]] of the sum of two [[Normal matrix|normal matrices]].
* [[Eilenberg–Ganea conjecture]]: a group with [[cohomological dimension]] 2 also has a 2-dimensional [[Eilenberg–MacLane space]] <math>K(G, 1)</math>.
* [[Farrell–Jones conjecture]] on whether certain [[assembly map]]s are [[isomorphisms]].
** [[Farrell–Jones conjecture#Bost conjecture|Bost conjecture]]: a specific case of the Farrell–Jones conjecture
* [[Finite lattice representation problem]]: is every finite [[Lattice (order)|lattice]] isomorphic to the [[Quotient (universal algebra)#Congruence lattice|congruence lattice]] of some finite [[Universal algebra|algebra]]?<ref>{{cite journal |last1=Dowling |first1=T. A. |title=A class of geometric lattices based on finite groups|journal=[[Journal of Combinatorial Theory]] |series=Series B |date=February 1973 |volume=14 |issue=1 |pages=61–86 |doi=10.1016/S0095-8956(73)80007-3 | doi-access=free }}</ref>
* [[Goncharov conjecture]] on the [[cohomology]] of certain [[Motivic cohomology|motivic complexes]].
* [[Green's conjecture]]: the [[Clifford's theorem on special divisors|Clifford index]] of a non-[[hyperelliptic curve]] is determined by the extent to which it, as a [[Canonical bundle|canonical curve]], has [[Linear relation|linear syzygies]].
* [[Grothendieck–Katz p-curvature conjecture]]: a conjectured [[Hasse principle|local–global principle]] for [[Linear differential equation|linear ordinary differential equations]].
* [[Hadamard conjecture]]: for every positive integer <math>k</math>, a [[Hadamard matrix]] of order <math>4k</math> exists.
** [[Williamson conjecture]]: the problem of finding Williamson matrices, which can be used to construct Hadamard matrices.
* [[Hadamard's maximal determinant problem]]: what is the largest [[determinant]] of a matrix with entries all equal to 1 or −1?
* [[Hilbert's fifteenth problem]]: put [[Schubert calculus]] on a rigorous foundation.
* [[Hilbert's sixteenth problem]]: what are the possible configurations of the [[Connected space|connected components]] of [[Harnack's curve theorem|M-curves]]?
* [[Homological conjectures in commutative algebra]]
* [[Jacobson's conjecture]]: the intersection of all powers of the [[Jacobson radical]] of a left-and-right [[Noetherian ring]] is precisely 0.
* [[Kaplansky's conjectures]]
* [[Köthe conjecture]]: if a ring has no [[nil ideal]] other than <math>\{0\}</math>, then it has no nil [[Ideal (ring theory)|one-sided ideal]] other than <math>\{0\}</math>.
* [[Monomial conjecture]] on [[Noetherian ring|Noetherian]] [[local ring]]s
* Existence of [[perfect cuboid]]s and associated [[cuboid conjectures]]
* [[Pierce–Birkhoff conjecture]]: every piecewise-polynomial <math>f:\mathbb{R}^{n}\rightarrow\mathbb{R}</math> is the maximum of a finite set of minimums of finite collections of polynomials.
* [[Rota's basis conjecture]]: for matroids of rank <math>n</math> with <math>n</math> disjoint bases <math>B_{i}</math>, it is possible to create an <math>n \times n</math> matrix whose rows are <math>B_{i}</math> and whose columns are also bases.
* [[Serre's conjecture II (algebra)|Serre's conjecture II]]: if <math>G</math> is a [[Simply connected space|simply connected]] [[semisimple algebraic group]] over a perfect [[Field (mathematics)|field]] of [[cohomological dimension]] at most <math>2</math>, then the [[Galois cohomology]] set <math>H^{1}(F, G)</math> is zero.
* [[Serre's multiplicity conjectures|Serre's positivity conjecture]] that if <math>R</math> is a commutative [[regular local ring]], and <math>P, Q</math> are [[prime ideal]]s of <math>R</math>, then <math>\dim (R/P) + \dim (R/Q) = \dim (R)</math> implies <math>\chi(R/P, R/Q) > 0</math>.
* [[Uniform boundedness conjecture for rational points]]: do [[algebraic curve]]s of [[Geometric genus|genus]] <math>g \geq 2</math> over [[Algebraic number field|number fields]] <math>K</math> have at most some bounded number <math>N(K, g)</math> of <math>K</math>-[[rational point]]s?
* [[Wild problem]]s: problems involving classification of pairs of <math>n\times n</math> matrices under simultaneous conjugation.
* [[Zariski–Lipman conjecture]]: for a [[complex algebraic variety]] <math>V</math> with [[coordinate ring]] <math>R</math>, if the [[Derivation (algebra)|derivations]] of <math>R</math> are a [[free module]] over <math>R</math>, then <math>V</math> is [[Smooth algebraic variety|smooth]].
* Zauner's conjecture: do [[SIC-POVM]]s exist in all dimensions?
*[[Zilber–Pink conjecture]] that if <math>X</math> is a mixed [[Shimura variety]] or [[Abelian variety#Semiabelian variety|semiabelian variety]] defined over <math>\mathbb{C}</math>, and <math>V \subseteq X</math> is a subvariety, then <math>V</math> contains only finitely many atypical subvarieties.

==== Group theory ====
{{Main|Group theory }}
[[File:FreeBurnsideGroupExp3Gens2.png|thumb|The [[free Burnside group]] <math>B(2,3)</math> is finite; in its [[Cayley graph]], shown here, each of its 27 elements is represented by a vertex. The question of which other groups <math>B(m,n)</math> are finite remains open.]]
* [[Andrews–Curtis conjecture]]: every balanced [[Presentation of a group|presentation]] of the [[trivial group]] can be transformed into a trivial presentation by a sequence of [[Nielsen transformation]]s on [[Presentation of a group#Definition|relators]] and conjugations of relators
* [[Burnside problem#Bounded Burnside problem|Bounded Burnside problem]]: for which positive integers ''m'', ''n'' is the free Burnside group {{nowrap|B(''m'',''n'')}} finite? In particular, is {{nowrap|B(2, 5)}} finite?
* Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems<ref>{{citation |last=Aschbacher |first=Michael |author-link=Michael Aschbacher |title=On Conjectures of Guralnick and Thompson |journal=[[Journal of Algebra]] |volume=135 |issue=2 |pages=277–343 |year=1990 |doi=10.1016/0021-8693(90)90292-V}}</ref>
* [[Herzog–Schönheim conjecture]]: if a finite system of left [[coset]]s of subgroups of a group <math>G</math> form a partition of <math>G</math>, then the finite indices of said subgroups cannot be distinct.
* The [[inverse Galois problem]]: is every finite group the Galois group of a Galois extension of the rationals?
* [[Isomorphism problem of Coxeter groups]]
* Are there an infinite number of [[Leinster group]]s?
* Does [[Monstrous moonshine#Generalized moonshine|generalized moonshine]] exist?
* Is every [[finitely presented group|finitely presented]] [[periodic group]] finite?
* Is every group [[surjunctive group|surjunctive]]?
* Is every discrete, countable group [[sofic group|sofic]]?
* [[Problems in loop theory and quasigroup theory]] consider generalizations of groups

==== Representation theory ====
* [[Arthur's conjectures]]
* [[Dade's conjecture]] relating the numbers of [[Character theory|characters]] of [[Modular representation theory#Blocks and the structure of the group algebra|blocks]] of a finite group to the numbers of characters of blocks of local [[subgroup]]s.
* [[Demazure conjecture]] on [[Group representation|representations]] of [[algebraic group]]s over the integers.
* [[Kazhdan–Lusztig polynomial#Kazhdan–Lusztig conjectures|Kazhdan–Lusztig conjectures]] relating the values of the [[Kazhdan–Lusztig polynomial]]s at 1 with [[Group representation|representations]] of complex [[Semisimple Lie algebra#Semisimple and reductive groups|semisimple Lie groups]] and [[Semisimple Lie algebra|Lie algebras]].
* [[McKay conjecture]]: in a group <math>G</math>, the number of [[Character theory#Definitions|irreducible complex characters]] of degree not divisible by a [[prime number]] <math>p</math> is equal to the number of irreducible complex characters of the [[centralizer and normalizer|normalizer]] of any [[Sylow theorems|Sylow <math>p</math>-subgroup]] within <math>G</math>.

=== Analysis ===
{{Main|Mathematical analysis}}
* The [[Brennan conjecture]]: estimating the integral of powers of the moduli of the derivative of [[conformal map]]s into the open unit disk, on certain subsets of <math>\mathbb{C}</math>
* [[Fuglede's conjecture]] on whether nonconvex sets in <math>\mathbb{R}</math> and <math>\mathbb{R}^{2}</math> are spectral if and only if they tile by [[Translation (geometry)|translation]].
* [[Goodman's conjecture]] on the coefficients of [[multivalent function|multivalued function]]s
* [[Invariant subspace problem]] – does every [[bounded operator]] on a complex [[Banach space]] send some non-trivial [[Closed set|closed]] subspace to itself?
* Kung–Traub conjecture on the optimal order of a multipoint iteration without memory<ref>{{citation |last1=Kung |first1=H. T. |last2=Traub |first2=Joseph Frederick |author-link1=H. T. Kung |author-link2=Joseph F. Traub |title=Optimal order of one-point and multipoint iteration |journal=[[Journal of the ACM]] |year=1974 |volume=21 |number=4 |pages=643–651|doi=10.1145/321850.321860 |s2cid=74921 }}</ref>
* [[Lehmer's conjecture]] on the Mahler measure of non-cyclotomic polynomials<ref>{{citation | first=Chris | last=Smyth | chapter=The Mahler measure of algebraic numbers: a survey | pages=322–349 | editor1-first=James | editor1-last=McKee | editor2-last=Smyth | editor2-first=Chris | title=Number Theory and Polynomials | series=London Mathematical Society Lecture Note Series | volume=352 | publisher=[[Cambridge University Press]] | year=2008 | isbn=978-0-521-71467-9 }}</ref>
* The [[mean value problem]]: given a [[complex number|complex]] [[polynomial]] <math>f</math> of [[Degree of a polynomial|degree]] <math>d \ge 2</math> and a complex number <math>z</math>, is there a [[critical point (mathematics)|critical point]] <math>c</math> of <math>f</math> such that <math>|f(z)-f(c)| \le |f'(z)||z-c|</math>?
* The [[Pompeiu problem]] on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy<ref>{{SpringerEOM|title=Pompeiu problem|id=Pompeiu_problem&oldid=14506|author-last1=Berenstein|author-first1=Carlos A.}}</ref>
* [[Sendov's conjecture]]: if a complex polynomial with degree at least <math>2</math> has all roots in the closed [[unit disk]], then each root is within distance <math>1</math> from some [[Critical point (mathematics)|critical point]].
* [[Analytic capacity#Vitushkin's conjecture|Vitushkin's conjecture]] on compact subsets of <math>\mathbb{C}</math> with [[analytic capacity]] <math>0</math>
* What is the exact value of [[Landau's constants]], including [[Bloch's theorem (complex variables)#Bloch's and Landau's constants|Bloch's constant]]?

* Regularity of solutions of [[Euler equations (fluid dynamics)|Euler equations]]
* Convergence of [https://mathworld.wolfram.com/FlintHillsSeries.html Flint Hills series]
* Regularity of solutions of [[Vlasov–Maxwell equations]]

=== Combinatorics ===
{{Main|Combinatorics}}
* The [[1/3–2/3 conjecture]] – does every finite [[partially ordered set]] that is not [[totally ordered]] contain two elements ''x'' and ''y'' such that the probability that ''x'' appears before ''y'' in a random [[linear extension]] is between 1/3 and 2/3?<ref>{{citation
 | last1 = Brightwell | first1 = Graham R.
 | last2 = Felsner | first2 = Stefan
 | last3 = Trotter | first3 = William T.
 | doi = 10.1007/BF01110378
 | mr = 1368815
 | issue = 4
 | journal = [[Order (journal)|Order]]
 | pages = 327–349
 | title = Balancing pairs and the cross product conjecture
 | volume = 12
 | year = 1995| citeseerx = 10.1.1.38.7841
 | s2cid = 14793475
 }}.</ref>
* The [[Dittert conjecture]] concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition
* [[Problems in Latin squares]] – open questions concerning [[Latin squares]]
* The [[lonely runner conjecture]] – if <math>k</math> runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance <math>1/k</math> from each other runner) at some time?<ref>{{cite journal
| last=Tao | first=Terence | author-link=Terence Tao
| title=Some remarks on the lonely runner conjecture
| journal=Contributions to Discrete Mathematics
| volume=13
| issue=2
| pages=1–31
| date=2018
| arxiv=1701.02048
| doi=10.11575/cdm.v13i2.62728 | doi-access=free}}</ref>
* [[Map folding]] – various problems in map folding and stamp folding.
* [[No-three-in-line problem]] – how many points can be placed in the <math>n \times n</math> grid so that no three of them lie on a line?
* [[Rudin's conjecture]] on the number of squares in finite [[arithmetic progression]]s<ref>{{cite journal|journal=LMS Journal of Computation and Mathematics|volume=17|issue=1|year=2014|pages=58–76|title=On a conjecture of Rudin on squares in arithmetic progressions|author=González-Jiménez, Enrique|author2=Xarles, Xavier|doi=10.1112/S1461157013000259|arxiv=1301.5122|s2cid=11615385 }}</ref>
* The [[sunflower (mathematics)|sunflower conjecture]] – can the number of <math>k</math> size sets required for the existence of a sunflower of <math>r</math> sets be bounded by an exponential function in <math>k</math> for every fixed <math>r>2</math>?
* Frankl's [[union-closed sets conjecture]] – for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets<ref>{{citation
 | last1 = Bruhn
 | first1 = Henning
 | last2 = Schaudt
 | first2 = Oliver
 | doi = 10.1007/s00373-014-1515-0
 | issue = 6
 | journal = Graphs and Combinatorics
 | mr = 3417215
 | pages = 2043–2074
 | title = The journey of the union-closed sets conjecture
 | url = http://www.zaik.uni-koeln.de/~schaudt/UCSurvey.pdf
 | volume = 31
 | year = 2015
 | arxiv = 1309.3297
 | s2cid = 17531822
 | access-date = 2017-07-18
 | archive-url = https://web.archive.org/web/20170808104232/http://www.zaik.uni-koeln.de/~schaudt/UCSurvey.pdf
 | archive-date = 2017-08-08
 | url-status = live
 }}</ref>

* Give a combinatorial interpretation of the [[Kronecker coefficient]]s<ref>{{citation
 | last = Murnaghan | first = F. D.
 | doi = 10.2307/2371542
 | issue = 1
 | journal = [[American Journal of Mathematics]]
 | mr = 1507301
 | pages = 44–65
 | title = The Analysis of the Direct Product of Irreducible Representations of the Symmetric Groups
 | volume = 60
 | year = 1938| pmc = 1076971
 | pmid=16577800
 | jstor = 2371542
 }}</ref>
* The values of the [[Dedekind number]]s <math>M(n)</math> for <math>n \ge 10</math><ref>{{Cite web |url=http://www.sfu.ca/~tyusun/ThesisDedekind.pdf |title=Dedekind Numbers and Related Sequences |access-date=2020-04-30 |archive-date=2015-03-15 |archive-url=https://web.archive.org/web/20150315021125/http://www.sfu.ca/~tyusun/ThesisDedekind.pdf }}</ref>
* The values of the [[Ramsey numbers]], particularly <math>R(5, 5)</math>
* The values of the [[Van der Waerden number]]s
* Finding a function to model n-step [[self-avoiding walk]]s<ref>{{Cite journal|last1=Liśkiewicz|first1=Maciej|last2=Ogihara|first2=Mitsunori|last3=Toda|first3=Seinosuke|date=2003-07-28|title=The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes|journal=Theoretical Computer Science|volume=304|issue=1|pages=129–156|doi=10.1016/S0304-3975(03)00080-X|s2cid=33806100 }}</ref>

=== Dynamical systems ===
{{Main|Dynamical system}}
[[File:Mandel zoom 07 satellite.jpg|thumb|A detail of the [[Mandelbrot set]]. It is not known whether the Mandelbrot set is [[locally connected space|locally connected]] or not.]]
* [[Arnold–Givental conjecture]] and [[Symplectomorphism#Arnold conjecture|Arnold conjecture]] – relating symplectic geometry to Morse theory.
* [[Quantum chaos#Berry–Tabor conjecture|Berry–Tabor conjecture]] in [[quantum chaos]]
* [[Stefan Banach|Banach's]] problem – is there an [[Measure-preserving dynamical system|ergodic system]] with simple Lebesgue spectrum?<ref>S. M. Ulam, Problems in Modern Mathematics. Science Editions John Wiley & Sons, Inc., New York, 1964, page 76.</ref>
* [[George David Birkhoff|Birkhoff]] conjecture – if a [[dynamical billiards|billiard table]] is strictly convex and integrable, is its boundary necessarily an ellipse?<ref>{{cite journal |last1=Kaloshin |first1=Vadim |author-link1=Vadim Kaloshin |last2=Sorrentino |first2=Alfonso |title=On the local Birkhoff conjecture for convex billiards |doi=10.4007/annals.2018.188.1.6 |volume=188 |number=1 |year=2018 |pages=315–380 |journal=[[Annals of Mathematics]]|arxiv=1612.09194 |s2cid=119171182 }}</ref>
* [[Collatz conjecture]] (also known as the <math>3n + 1</math> conjecture)
* [[Eden's conjecture]] that the [[Infimum and supremum|supremum]] of the local [[Lyapunov dimension]]s on the global [[attractor]] is achieved on a stationary point or an unstable periodic orbit embedded into the attractor.
* [[Alexandre Eremenko|Eremenko's]] conjecture: every component of the [[escaping set]] of an [[entire function|entire]] [[transcendental function|transcendental]] function is unbounded.
* [[Fatou conjecture]] that a quadratic family of maps from the [[complex plane]] to itself is hyperbolic for an open dense set of parameters.
* [[Hillel Furstenberg|Furstenberg]] conjecture – is every invariant and [[ergodicity|ergodic]] measure for the <math>\times 2,\times 3</math> action on the circle either Lebesgue or atomic?
* [[Kaplan–Yorke conjecture]] on the dimension of an [[attractor]] in terms of its [[Lyapunov exponent]]s
* [[Grigory Margulis|Margulis]] conjecture – measure classification for diagonalizable actions in higher-rank groups.
* [[Hilbert–Arnold problem]] – is there a [[uniformly bounded|uniform bound]] on [[limit cycles]] in generic finite-parameter families of [[vector fields]] on a sphere?
* [[MLC conjecture]] – is the Mandelbrot set locally connected?
* Many problems concerning an [[Outer billiard#open problems|outer billiard]], for example showing that outer billiards relative to almost every convex polygon have unbounded orbits.
* Quantum unique ergodicity conjecture on the distribution of large-frequency [[eigenfunction]]s of the [[Laplace operator|Laplacian]] on a [[curvature|negatively-curved]] [[manifold]]<ref>{{citation |last=Sarnak |first=Peter |author-link=Peter Sarnak |title=Recent progress on the quantum unique ergodicity conjecture |journal=[[Bulletin of the American Mathematical Society]] |volume=48 |issue=2 |year=2011 |pages=211–228 |doi=10.1090/S0273-0979-2011-01323-4 |mr=2774090|doi-access=free }}</ref>
* [[Vladimir Abramovich Rokhlin|Rokhlin's]] multiple mixing problem – are all [[Mixing (mathematics)|strongly mixing]] systems also strongly 3-mixing?<ref>Paul Halmos, Ergodic theory. Chelsea, New York, 1956.</ref>
* [[Weinstein conjecture]] – does a regular compact [[contact type]] [[level set]] of a [[Hamiltonian function|Hamiltonian]] on a [[symplectic manifold]] carry at least one periodic orbit of the Hamiltonian flow?

* Does every positive integer generate a [[juggler sequence]] terminating at 1?
* [[Lyapunov function|Lyapunov function: Lyapunov's second method for stability]] – For what classes of [[Ordinary differential equation|ODEs]], describing dynamical systems, does Lyapunov's second method, formulated in the classical and canonically generalized forms, define the necessary and sufficient conditions for the (asymptotical) stability of motion?
* Is every [[reversible cellular automaton]] in three or more dimensions locally reversible?<ref>{{cite conference |last=Kari |first=Jarkko |author-link=Jarkko Kari |year=2009 |title=Structure of Reversible Cellular Automata |conference=International Conference on Unconventional Computation |series=[[Lecture Notes in Computer Science]] |publisher=Springer |volume=5715 |page=6 |bibcode=2009LNCS.5715....6K |doi=10.1007/978-3-642-03745-0_5 |isbn=978-3-642-03744-3 |doi-access=free |contribution=Structure of reversible cellular automata}}</ref>

=== Games and puzzles ===
{{Main|Game theory}}

====Combinatorial games====
{{Main|Combinatorial game theory}}
* [[Sudoku]]:
** How many puzzles have exactly one solution?<ref name="openq"/>
** How many puzzles with exactly one solution are [[Glossary of Sudoku#Other terminology|minimal]]?<ref name="openq"/>
** What is the [[Mathematics of Sudoku#Maximum number of givens|maximum number of givens]] for a [[Glossary of Sudoku#Other terminology|minimal]] puzzle?<ref name="openq">{{Cite web |title=Open Q – Solving and rating of hard Sudoku |url=http://english.log-it-ex.com/2.html |archive-url=https://web.archive.org/web/20171110030932/http://english.log-it-ex.com/2.html |archive-date=10 November 2017 |website=english.log-it-ex.com}}</ref>
* [[Tic-tac-toe variants]]:
** Given the width of a tic-tac-toe board, what is the smallest dimension such that X is guaranteed to have a winning strategy? (See also [[Hales–Jewett theorem]] and [[Nd game|''n''<sup>''d''</sup> game]])<ref>{{cite web |url=https://www.youtube.com/watch?v=FwJZa-helig |title=Higher-Dimensional Tic-Tac-Toe |website=[[PBS Digital Studios|PBS Infinite Series]] |publisher=[[YouTube]] |date=2017-09-21 |access-date=2018-07-29 |archive-url=https://web.archive.org/web/20171011000653/https://www.youtube.com/watch?v=FwJZa-helig |archive-date=2017-10-11 |url-status=live }}</ref>
* [[Chess]]:
** What is the outcome of a perfectly played game of chess? (See also [[first-move advantage in chess]])
* [[Go (game)|Go]]:
** What is the perfect value of [[Komi (Go)|Komi]]?
* Are the nim-sequences of all finite [[octal game]]s eventually periodic?
* Is the nim-sequence of [[Grundy's game]] eventually periodic?

====Games with imperfect information====
* [[Rendezvous problem]]

=== Geometry ===
{{Main|Geometry}}

==== Algebraic geometry ====
{{Main|Algebraic geometry}}
* [[Abundance conjecture]]: if the [[canonical bundle]] of a [[projective variety]] with [[Canonical singularity#Pairs|Kawamata log terminal singularities]] is [[Nef line bundle|nef]], then it is semiample.
* [[Bass conjecture]] on the [[Finitely generated group|finite generation]] of certain [[Algebraic K-theory|algebraic K-groups]].
* [[Bass–Quillen conjecture]] relating [[vector bundle]]s over a [[Regular local ring#Regular ring|regular]] [[Noetherian ring]] and over the [[polynomial ring]] <math>A[t_{1}, \ldots, t_{n}]</math>.
* [[Deligne conjecture]]: any one of numerous named for [[Pierre Deligne]].
** [[Deligne's conjecture on Hochschild cohomology]] about the [[operad]]ic structure on [[Hochschild homology|Hochschild cochain complex]].
* [[Dixmier conjecture]]: any [[endomorphism]] of a [[Weyl algebra]] is an [[automorphism]].
* [[Fröberg conjecture]] on the [[Hilbert series and Hilbert polynomial|Hilbert functions]] of a set of forms.
* [[Fujita conjecture]] regarding the line bundle <math>K_{M} \otimes L^{\otimes m}</math> constructed from a [[Positive form#Positive line bundles|positive]] [[Holomorphic vector bundle|holomorphic line bundle]] <math>L</math> on a [[Compact space|compact]] [[complex manifold]] <math>M</math> and the [[Canonical bundle|canonical line bundle]] <math>K_{M}</math> of <math>M</math>
* [[General elephant|General elephant problem]]: do [[general elephant]]s have at most [[du Val singularity|Du Val singularities]]?
* Hartshorne's conjectures<ref>{{cite journal|title=On two conjectures of Hartshorne's |last1=Barlet |first1=Daniel |last2=Peternell |first2=Thomas |last3=Schneider |first3=Michael |doi=10.1007/BF01453563 |journal=[[Mathematische Annalen]] |year=1990 |volume=286 |issue=1–3 |pages=13–25|s2cid=122151259 }}</ref>
* In [[Spherical geometry|spherical]] or [[hyperbolic geometry]], must polyhedra with the same volume and [[Dehn invariant]] be [[scissors-congruent]]?<ref>{{citation
 |last        = Dupont
 |first       = Johan L.
 |doi         = 10.1142/9789812810335
 |isbn        = 978-981-02-4507-8
 |mr          = 1832859
 |page        = 6
 |publisher   = World Scientific Publishing Co., Inc., River Edge, NJ
 |series      = Nankai Tracts in Mathematics
 |title       = Scissors congruences, group homology and characteristic classes
 |url         = http://home.math.au.dk/dupont/scissors.ps
 |volume      = 1
 |year        = 2001
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20160429152252/http://home.math.au.dk/dupont/scissors.ps
 |archive-date = 2016-04-29
}}.</ref>
* [[Jacobian conjecture]]: if a [[polynomial mapping]] over a [[Characteristic (algebra)|characteristic]]-0 field has a constant nonzero [[Jacobian matrix and determinant|Jacobian determinant]], then it has a [[Morphism of algebraic varieties|regular]] (i.e. with polynomial components) inverse function.
* [[Manin conjecture]] on the distribution of [[rational point]]s of bounded [[Height function|height]] in certain subsets of [[Fano variety|Fano varieties]]
* [[Maulik–Nekrasov–Okounkov–Pandharipande conjecture]] on an equivalence between [[Gromov–Witten invariant|Gromov–Witten theory]] and [[Donaldson–Thomas theory]]<ref>{{citation
 |last1=Maulik |first1=Davesh
 |last2=Nekrasov |first2=Nikita |author-link2=Nikita Nekrasov
 |last3=Okounov |first3=Andrei |author-link3=Andrei Okounov
 |last4=Pandharipande |first4=Rahul |author-link4=Rahul Pandharipande
 |title=Gromov–Witten theory and Donaldson–Thomas theory, I
 |arxiv=math/0312059
 |date=2004-06-05|bibcode=2003math.....12059M
 }}</ref>
* [[Nagata's conjecture on curves]], specifically the minimal degree required for a [[Algebraic curve|plane algebraic curve]] to pass through a collection of very general points with prescribed [[Multiplicity (mathematics)|multiplicities]].
* [[Nagata–Biran conjecture]] that if <math>X</math> is a smooth [[algebraic surface]] and <math>L</math> is an [[ample line bundle]] on <math>X</math> of degree <math>d</math>, then for sufficiently large <math>r</math>, the [[Seshadri constant]] satisfies <math>\varepsilon(p_1,\ldots,p_r;X,L) = d/\sqrt{r}</math>.
* [[Nakai conjecture]]: if a [[complex algebraic variety]] has a ring of [[differential operator]]s generated by its contained [[Derivation (differential algebra)|derivations]], then it must be [[Singular point of an algebraic variety|smooth]].
* [[Parshin's conjecture]]: the higher [[Algebraic K-theory|algebraic K-groups]] of any [[Smooth morphism|smooth]] [[projective variety]] defined over a [[finite field]] must vanish up to torsion.
* [[Section conjecture]] on splittings of [[group homomorphism]]s from [[fundamental group]]s of complete [[Curve#Differential geometry|smooth curves]] over finitely-generated [[Field (mathematics)|fields]] <math>k</math> to the [[Galois group]] of <math>k</math>.
* [[Standard conjectures]] on algebraic cycles
* [[Tate conjecture]] on the connection between [[algebraic cycle]]s on [[Algebraic variety|algebraic varieties]] and [[Galois module|Galois representations]] on [[Étale cohomology|étale cohomology groups]].
* [[Virasoro conjecture]]: a certain [[generating function]] encoding the [[Gromov–Witten invariant]]s of a [[Singular point of an algebraic variety|smooth]] [[projective variety]] is fixed by an action of half of the [[Virasoro algebra]].
* Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of [[algebraic variety|varieties]] at [[singular point of an algebraic variety|singular points]]<ref>{{cite journal|last=Zariski |first=Oscar |author-link=Oscar Zariski |title=Some open questions in the theory of singularities |journal=[[Bulletin of the American Mathematical Society]] |volume=77 |issue=4 |year=1971 |pages=481–491 |doi=10.1090/S0002-9904-1971-12729-5 |mr=0277533|doi-access=free }}</ref>

* Are infinite sequences of [[Flip (mathematics)|flips]] possible in dimensions greater than 3?
* [[Resolution of singularities]] in characteristic <math>p</math>

====Covering and packing====
* [[Borsuk's conjecture|Borsuk's problem]] on upper and lower bounds for the number of smaller-diameter subsets needed to cover a [[bounded set|bounded]] ''n''-dimensional set.
* The [[covering problem of Rado]]: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?<ref>{{citation|last1=Bereg|first1=Sergey|last2=Dumitrescu|first2=Adrian|last3=Jiang|first3=Minghui|doi=10.1007/s00453-009-9298-z|issue=3|journal=Algorithmica|mr=2609053|pages=538–561|title=On covering problems of Rado|volume=57|year=2010|s2cid=6511998}}</ref>
* The [[Circle packing in an equilateral triangle|Erdős–Oler conjecture]]: when <math>n</math> is a [[triangular number]], packing <math>n-1</math> circles in an equilateral triangle requires a triangle of the same size as packing <math>n</math> circles.<ref>{{citation|last=Melissen|first=Hans|doi=10.2307/2324212|issue=10|journal=American Mathematical Monthly|mr=1252928|pages=916–925|title=Densest packings of congruent circles in an equilateral triangle|volume=100|year=1993|jstor=2324212}}</ref>
* The [[disk covering problem]] abount finding the smallest [[real number]]  <math>r(n)</math> such that <math>n</math> [[disk (mathematics)|disks]] of radius <math>r(n)</math> can be arranged in such a way as to cover the [[unit disk]].
* The [[kissing number problem]] for dimensions other than 1, 2, 3, 4, 8 and 24<ref>{{citation |first=John H. |last=Conway |author-link=John Horton Conway |author2=Neil J.A. Sloane |author-link2=Neil Sloane |year=1999 |title=Sphere Packings, Lattices and Groups |edition=3rd |publisher=Springer-Verlag |location=New York |isbn=978-0-387-98585-5|pages=[https://books.google.com/books?id=upYwZ6cQumoC&pg=PA21 21–22]}}</ref>
* [[Reinhardt's conjecture]]: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets<ref>{{citation
 | last = Hales | first = Thomas | author-link = Thomas Callister Hales
 | arxiv = 1703.01352
 | title = The Reinhardt conjecture as an optimal control problem
 | year = 2017}}</ref>
* [[Sphere packing]] problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions.
* [[Square packing in a square]]: what is the asymptotic growth rate of wasted space?<ref>{{citation|last1=Brass|first1=Peter|last2=Moser|first2=William|last3=Pach|first3=János|author3-link=János Pach|isbn=978-0387-23815-9|mr=2163782|page=45|publisher=Springer|location=New York|title=Research Problems in Discrete Geometry|url=https://books.google.com/books?id=WehCspo0Qa0C&pg=PA45|year=2005}}</ref>
* [[Ulam's packing conjecture]] about the identity of the worst-packing convex solid<ref>{{citation |last=Gardner |first=Martin |date=1995 |title=New Mathematical Diversions (Revised Edition) |location=Washington |publisher=Mathematical Association of America |page=251 }}</ref>
* The [[Tammes problem]] for numbers of nodes greater than 14 (except 24).<ref>{{cite journal |last1=Musin |first1=Oleg R. |last2=Tarasov |first2=Alexey S. |title=The Tammes Problem for N = 14 |journal=Experimental Mathematics |date=2015 |volume=24 |issue=4 |pages=460–468 |doi=10.1080/10586458.2015.1022842|s2cid=39429109 }}</ref>

==== Differential geometry ====
{{Main|Differential geometry}}
* The [[spherical Bernstein's problem]], a generalization of [[Bernstein's problem]]
* [[Carathéodory conjecture]]: any convex, closed, and twice-differentiable surface in three-dimensional [[Euclidean space]] admits at least two [[umbilical point]]s.
* [[Cartan–Hadamard conjecture]]: can the classical [[isoperimetric inequality]] for subsets of Euclidean space be extended to spaces of nonpositive curvature, known as [[Hadamard manifold|Cartan–Hadamard manifolds]]?
* [[Chern's conjecture (affine geometry)]] that the [[Euler characteristic]] of a [[Closed manifold|compact]] [[affine manifold]] vanishes.
* [[Chern's conjecture for hypersurfaces in spheres]], a number of closely related conjectures.
* Closed curve problem: find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed.<ref>{{citation
 | last = Barros | first = Manuel
 | jstor = 2162098
 | journal = [[Proceedings of the American Mathematical Society]]
 | pages = 1503–1509
 | title = General Helices and a Theorem of Lancret
 | volume = 125
 | issue = 5
 | year = 1997| doi = 10.1090/S0002-9939-97-03692-7
 | doi-access = free
 }}</ref>
* The [[filling area conjecture]], that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length<ref>{{citation
 | last = Katz | first = Mikhail G.
 | doi = 10.1090/surv/137
 | isbn = 978-0-8218-4177-8
 | mr = 2292367
 | page = 57
 | publisher = American Mathematical Society, Providence, RI
 | series = Mathematical Surveys and Monographs
 | title = Systolic geometry and topology
 | url = https://books.google.com/books?id=R5_zBwAAQBAJ&pg=PA57
 | volume = 137
 | year = 2007}}</ref>
* The [[Hopf conjecture]]s relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds<ref>{{citation
 | last = Rosenberg | first = Steven
 | doi = 10.1017/CBO9780511623783
 | isbn = 978-0-521-46300-3
 | location = Cambridge
 | mr = 1462892
 | pages = 62–63
 | publisher = Cambridge University Press
 | series = London Mathematical Society Student Texts
 | title = The Laplacian on a Riemannian Manifold: An introduction to analysis on manifolds
 | url = https://books.google.com/books?id=gzJ6Vn0y7XQC&pg=PA62
 | volume = 31
 | year = 1997}}</ref>
* [[Osserman conjecture]]: that every [[Osserman manifold]] is either [[flat manifold|flat]] or locally [[Isometry|isometric]] to a rank-one [[symmetric space]]<ref>{{citation | last = Nikolayevsky | first = Y. | journal = Differential Geometry and Its Applications | title = Two theorems on Osserman manifolds | volume = 18 | pages = 239–253 | year = 2003 | issue = 3 | doi = 10.1016/S0926-2245(02)00160-2}}</ref>
* [[Yau's conjecture on the first eigenvalue]] that the first [[Eigenvalues and eigenvectors|eigenvalue]] for the [[Laplace–Beltrami operator]] on an embedded [[Minimal surface|minimal hypersurface]] of <math>S^{n+1}</math> is <math>n</math>.

==== Discrete geometry ====
{{Main|Discrete geometry }}
[[File:Kissing-3d.png|thumb|In three dimensions, the [[kissing number problem|kissing number]] is 12, because 12 non-overlapping unit spheres can be put into contact with a central unit sphere. (Here, the centers of outer spheres form the vertices of a [[regular icosahedron]].) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24.]]
* The [[big-line-big-clique conjecture]] on the existence of either many collinear points or many mutually visible points in large planar point sets<ref>{{citation
 | last1 = Ghosh | first1 = Subir Kumar
 | last2 = Goswami | first2 = Partha P.
 | arxiv = 1012.5187
 | doi = 10.1145/2543581.2543589
 | issue = 2
 | journal = ACM Computing Surveys
 | pages = 22:1–22:29
 | title = Unsolved problems in visibility graphs of points, segments, and polygons
 | volume = 46
 | year = 2013| s2cid = 8747335
 }}</ref>
* The [[Hadwiger conjecture (combinatorial geometry)|Hadwiger conjecture]] on covering ''n''-dimensional convex bodies with at most 2<sup>''n''</sup> smaller copies<ref>{{citation|title=Results and Problems in Combinatorial Geometry|first1=V.|last1=Boltjansky|first2=I.|last2=Gohberg|publisher=Cambridge University Press|year=1985|contribution=11. Hadwiger's Conjecture|pages=44–46}}.</ref>
* Solving the [[happy ending problem]] for arbitrary <math>n</math><ref>{{citation
 | last1 = Morris | first1 = Walter D.
 | last2 = Soltan | first2 = Valeriu
 | doi = 10.1090/S0273-0979-00-00877-6
 | issue = 4
 | journal = Bull. Amer. Math. Soc.
 | mr = 1779413
 | pages = 437–458
 | title = The Erdős-Szekeres problem on points in convex position—a survey
 | volume = 37
 | year = 2000| doi-access = free
 }}; {{citation
 | last = Suk | first = Andrew
 | arxiv = 1604.08657
 | doi = 10.1090/jams/869
 | journal = J. Amer. Math. Soc.
 | title = On the Erdős–Szekeres convex polygon problem
 | year = 2016
 | volume=30
 | issue = 4
 | pages=1047–1053| s2cid = 15732134
 }}</ref>
*Improving lower and upper bounds for the [[Heilbronn triangle problem]].
* [[Kalai's 3^d conjecture|Kalai's 3<sup>''d''</sup> conjecture]] on the least possible number of faces of [[point symmetry|centrally symmetric]] [[polytopes]].<ref name="kalai">{{citation
 | last = Kalai | first = Gil | author-link = Gil Kalai
 | doi = 10.1007/BF01788696
 | issue = 1
 | journal = [[Graphs and Combinatorics]]
 | mr = 1554357
 | pages = 389–391
 | title = The number of faces of centrally-symmetric polytopes
 | volume = 5
 | year = 1989| s2cid = 8917264 }}.</ref>
* The [[Kobon triangle problem]] on triangles in line arrangements<ref>{{cite journal
 | last1 = Moreno | first1 = José Pedro
 | last2 = Prieto-Martínez | first2 = Luis Felipe
 | hdl = 10486/705416
 | issue = 1
 | journal = La Gaceta de la Real Sociedad Matemática Española
 | language = es
 | mr = 4225268
 | pages = 111–130
 | title = El problema de los triángulos de Kobon
 | trans-title = The Kobon triangles problem
 | volume = 24
 | year = 2021}}</ref>
* The [[Kusner conjecture]]: at most <math>2d</math> points can be equidistant in <math>L^1</math> spaces<ref>{{citation
 | last = Guy | first = Richard K. | author-link = Richard K. Guy
 | issue = 3
 | journal = [[American Mathematical Monthly]]
 | mr = 1540158
 | pages = 196–200
 | title = An olla-podrida of open problems, often oddly posed
 | jstor = 2975549
 | volume = 90
 | year = 1983
 | doi = 10.2307/2975549 }}</ref>
* The [[McMullen problem]] on projectively transforming sets of points into [[convex position]]<ref>{{citation
 | last = Matoušek | first = Jiří | author-link = Jiří Matoušek (mathematician)
 | doi = 10.1007/978-1-4613-0039-7
 | isbn = 978-0-387-95373-1
 | mr = 1899299
 | page = 206
 | publisher = Springer-Verlag, New York
 | series = Graduate Texts in Mathematics
 | title = Lectures on discrete geometry
 | volume = 212
 | year = 2002}}</ref>
*[[Opaque forest problem]] on finding [[opaque set]]s for various planar shapes
* [[Unit distance graph#Counting unit distances|How many unit distances]] can be determined by a set of {{mvar|n}} points in the Euclidean plane?<ref>{{citation
 | last1 = Brass | first1 = Peter
 | last2 = Moser | first2 = William
 | last3 = Pach | first3 = János
 | contribution = 5.1 The Maximum Number of Unit Distances in the Plane
 | isbn = 978-0-387-23815-9
 | mr = 2163782
 | pages = 183–190
 | publisher = Springer, New York
 | title = Research problems in discrete geometry
 | year = 2005}}</ref>
* Finding matching upper and lower bounds for [[K-set (geometry)|''k''-sets]] and halving lines<ref>{{citation
 | last = Dey | first = Tamal K. | author-link = Tamal Dey
 | doi = 10.1007/PL00009354
 | journal = [[Discrete & Computational Geometry]]
 | mr = 1608878
 | pages = 373–382
 | title = Improved bounds for planar ''k''-sets and related problems
 | volume = 19
 | issue = 3
 | year = 1998| doi-access = free
 }}; {{citation
 | last = Tóth | first = Gábor
 | doi = 10.1007/s004540010022
 | issue = 2
 | journal = [[Discrete & Computational Geometry]]
 | mr = 1843435
 | pages = 187–194
 | title = Point sets with many ''k''-sets
 | volume = 26
 | year = 2001| doi-access = free
 }}.</ref>
* [[Tripod packing]]:<ref>{{citation|last1=Aronov|first1=Boris|author1-link=Boris Aronov|last2=Dujmović|first2=Vida|author2-link=Vida Dujmović|last3=Morin|first3=Pat|author3-link= Pat Morin |last4=Ooms|first4=Aurélien|last5=Schultz Xavier da Silveira |first5=Luís Fernando|issue=1|journal=[[Electronic Journal of Combinatorics]]|page=P1.8|title=More Turán-type theorems for triangles in convex point sets |url=https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i1p8 |volume=26 |year=2019 |bibcode=2017arXiv170610193A |arxiv=1706.10193 |access-date=2019-02-18 |archive-url=https://web.archive.org/web/20190218082023/https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i1p8|archive-date=2019-02-18|url-status=live|doi-access=free|doi=10.37236/7224}}</ref> how many tripods can have their apexes packed into a given cube?

====Euclidean geometry====
{{Main|Euclidean geometry}}
* The [[Atiyah conjecture on configurations]] on the invertibility of a certain <math>n</math>-by-<math>n</math> matrix depending on <math>n</math> points in <math>\mathbb{R}^{3}</math><ref>{{Citation | last1=Atiyah | first1=Michael | author1-link=Michael Atiyah | title=Configurations of points | doi=10.1098/rsta.2001.0840 | mr=1853626 | year=2001 | journal= Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences| issn=1364-503X | volume=359 | issue=1784 | pages=1375–1387| bibcode=2001RSPTA.359.1375A | s2cid=55833332 }}</ref>
* [[Bellman's lost-in-a-forest problem]] – find the shortest route that is guaranteed to reach the boundary of a given shape, starting at an unknown point of the shape with unknown orientation<ref>{{citation |last1=Finch |first1=S. R. |last2=Wetzel |first2=J. E. |title=Lost in a forest |volume=11 |issue=8 |year=2004 |journal=[[American Mathematical Monthly]] |pages=645–654 |mr=2091541 |doi=10.2307/4145038 |jstor=4145038}}</ref>
* [[Borromean rings]] — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?<ref>{{citation
 | last = Howards | first = Hugh Nelson
 | arxiv = 1406.3370
 | doi = 10.1142/S0218216513500831
 | issue = 14
 | journal = Journal of Knot Theory and Its Ramifications
 | mr = 3190121
 | pages = 1350083, 15
 | title = Forming the Borromean rings out of arbitrary polygonal unknots
 | volume = 22
 | year = 2013| s2cid = 119674622
 }}</ref>
* [[Connelly’s blooming conjecture]]: Does every net of a convex polyhedron have a [[Blooming (geometry)|blooming]]?<ref>{{citation
 | last1 = Miller | first1 = Ezra
 | last2 = Pak | first2 = Igor | author2-link = Igor Pak
 | doi = 10.1007/s00454-008-9052-3
 | issue = 1–3
 | journal = [[Discrete & Computational Geometry]]
 | mr = 2383765
 | pages = 339–388
 | title = Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings
 | volume = 39
 | year = 2008| doi-access = free
 }}. Announced in 2003.</ref>
* Danzer's problem and Conway's dead fly problem – do [[Danzer set]]s of bounded density or bounded separation exist?<ref>{{citation |last1=Solomon |first1=Yaar |last2=Weiss |first2=Barak |arxiv=1406.3807 |doi=10.24033/asens.2303 |issue=5 |journal=Annales Scientifiques de l'École Normale Supérieure |mr=3581810 |pages=1053–1074 |title=Dense forests and Danzer sets |volume=49 |year=2016 |s2cid=672315}}; {{citation |last=Conway |first=John H. |author-link=John Horton Conway |publisher=[[On-Line Encyclopedia of Integer Sequences]] |title=Five $1,000 Problems (Update 2017) |url=https://oeis.org/A248380/a248380.pdf |archive-url=https://web.archive.org/web/20190213123825/https://oeis.org/A248380/a248380.pdf |archive-date=2019-02-13 |access-date=2019-02-12 |url-status=live}}</ref>
* [[Dissection into orthoschemes]] – is it possible for [[simplex|simplices]] of every dimension?<ref>{{citation |last1=Brandts |first1=Jan |last2=Korotov |first2=Sergey |last3=Křížek |first3=Michal |last4=Šolc |first4=Jakub |doi=10.1137/060669073 |issue=2 |journal=SIAM Review |mr=2505583 |pages=317–335 |title=On nonobtuse simplicial partitions |volume=51 |year=2009 |url=https://pure.uva.nl/ws/files/836396/73198_315330.pdf |bibcode=2009SIAMR..51..317B |s2cid=216078793 |access-date=2018-11-22 |archive-date=2018-11-04 |archive-url=https://web.archive.org/web/20181104211116/https://pure.uva.nl/ws/files/836396/73198_315330.pdf |url-status=live}}. See in particular Conjecture 23, p. 327.</ref>
* [[Ehrhart's volume conjecture]]: a convex body <math>K</math> in <math>n</math> dimensions containing a single lattice point in its interior as its [[center of mass]] cannot have volume greater than <math>(n+1)^{n}/n!</math>
* [[Falconer's conjecture]]: sets of Hausdorff dimension greater than <math>d/2</math> in <math>\mathbb{R}^d</math> must have a distance set of nonzero [[Lebesgue measure]]<ref>{{citation |last1=Arutyunyants |first1=G. |last2=Iosevich |first2=A. |editor-last=Pach |editor-first=János |editor-link=János Pach |contribution=Falconer conjecture, spherical averages and discrete analogs |doi=10.1090/conm/342/06127 |mr=2065249 |pages=15–24 |publisher=Amer. Math. Soc., Providence, RI|series=Contemp. Math. |title=Towards a Theory of Geometric Graphs |volume=342 |year=2004 |isbn=978-0-8218-3484-8 |doi-access=free}}</ref>
* The values of the [[Hermite constant]]s for dimensions other than 1–8 and 24
* What is the lowest number of faces possible for a [[holyhedron]]?
* [[Inscribed square problem]], also known as [[Toeplitz' conjecture]] and the square peg problem – does every [[Jordan curve]] have an inscribed square?<ref name="matschke">{{citation|last=Matschke|first=Benjamin|date=2014|title=A survey on the square peg problem|journal=[[Notices of the American Mathematical Society]]|volume=61|issue=4|pages=346–352|doi=10.1090/noti1100|doi-access=free}}</ref>
* The [[Kakeya conjecture]] –&nbsp;do <math>n</math>-dimensional sets that contain a unit line segment in every direction necessarily have [[Hausdorff dimension]] and [[Minkowski dimension]] equal to <math>n</math>?<ref>{{citation |last1=Katz |first1=Nets |author1-link=Nets Katz|last2=Tao|first2=Terence|author2-link=Terence Tao|title=Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000) |doi=10.5565/PUBLMAT_Esco02_07|series=Publicacions Matemàtiques|mr=1964819 |pages=161–179 |contribution=Recent progress on the Kakeya conjecture |year=2002 |citeseerx=10.1.1.241.5335 |s2cid=77088}}</ref>
* The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the [[Weaire–Phelan structure]] as a solution to the Kelvin problem<ref>{{citation |title=The Kelvin Problem |editor-first=Denis |editor-last=Weaire |editor-link=Denis Weaire |publisher=CRC Press |year=1997 |isbn=978-0-7484-0632-6 |page=1 |url=https://books.google.com/books?id=otokU4KQnXIC&pg=PA1}}</ref>
* [[Lebesgue's universal covering problem]] on the minimum-area convex shape in the plane that can cover any shape of diameter one<ref>{{citation |last1=Brass |first1=Peter |last2=Moser |first2=William |last3=Pach |first3=János |location=New York |mr=2163782 |page=457 |publisher=Springer |title=Research problems in discrete geometry|url=https://books.google.com/books?id=cT7TB20y3A8C&pg=PA457 |year=2005 |isbn=978-0-387-29929-7}}</ref>
* [[Mahler volume|Mahler's conjecture]] on the product of the volumes of a [[central symmetry|centrally symmetric]] [[convex body]] and its [[Polar set|polar]].<ref>{{Cite journal|last1=Mahler|first1=Kurt|title=Ein Minimalproblem für konvexe Polygone |journal=Mathematica (Zutphen) B|pages=118–127|year=1939}}</ref>
* [[Moser's worm problem]] – what is the smallest area of a shape that can cover every unit-length curve in the plane?<ref>{{citation |last1=Norwood |first1=Rick |last2=Poole |first2=George |last3=Laidacker |first3=Michael |doi=10.1007/BF02187832 |issue=2 |journal=[[Discrete & Computational Geometry]] |mr=1139077 |pages=153–162 |title=The worm problem of Leo Moser |volume=7 |year=1992 |doi-access=free}}</ref>
* The [[moving sofa problem]] – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?<ref>{{citation |last=Wagner |first=Neal R. |date=1976 |title=The Sofa Problem |journal=The American Mathematical Monthly |doi=10.2307/2977022 |jstor=2977022 |volume=83 |issue=3 |pages=188–189 |url=http://www.cs.utsa.edu/~wagner/pubs/corner/corner_final.pdf |access-date=2014-05-14 |archive-url=https://web.archive.org/web/20150420160001/http://www.cs.utsa.edu/~wagner/pubs/corner/corner_final.pdf |archive-date=2015-04-20 |url-status=live}}</ref>
* In [[parallelohedron]]:
** Can every spherical non-convex polyhedron that tiles space by translation have its faces grouped into patches with the same combinatorial structure as a parallelohedron?<ref>{{cite journal|last1=Senechal|first1=Marjorie|author1-link=Marjorie Senechal|last2=Galiulin|first2=R. V.|hdl=2099/1195|issue=10|journal=Structural Topology|language=en,fr|mr=768703|pages=5–22|title=An introduction to the theory of figures: the geometry of E. S. Fedorov|year=1984}}</ref>
** Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a [[Voronoi diagram]]?<ref>{{cite journal|last1=Grünbaum|first1=Branko|author1-link=Branko Grünbaum|last2=Shephard|first2=G. C.|author2-link=Geoffrey Colin Shephard|doi=10.1090/S0273-0979-1980-14827-2|issue=3|journal=Bulletin of the American Mathematical Society|mr=585178|pages=951–973|series=New Series|title=Tilings with congruent tiles|volume=3|year=1980|doi-access=free}}</ref>
* Does every convex polyhedron have [[Prince Rupert's cube#Generalizations|Rupert's property]]?<ref name=cyz>{{citation |first1=Ying |last1=Chai |first2=Liping |last2=Yuan |first3=Tudor |last3=Zamfirescu |title=Rupert Property of Archimedean Solids |journal=[[The American Mathematical Monthly]] |volume=125 |issue=6 |pages=497–504 |date=June–July 2018 |doi=10.1080/00029890.2018.1449505| s2cid=125508192}}</ref><ref name=styu>{{citation|title=An algorithmic approach to Rupert's problem |first1=Jakob |last1=Steininger |first2=Sergey |last2=Yurkevich| date=December 27, 2021 |arxiv=2112.13754}}</ref>
* [[Shephard's conjecture|Shephard's problem (a.k.a. Dürer's conjecture)]] – does every [[convex polyhedron]] have a [[net (polyhedron)|net]], or simple edge-unfolding?<ref>{{citation |last1=Demaine |first1=Erik D. |author1-link=Erik Demaine |last2=O'Rourke |first2=Joseph |author2-link=Joseph O'Rourke (professor) |date=2007 |title=Geometric Folding Algorithms: Linkages, Origami, Polyhedra |title-link=Geometric Folding Algorithms |publisher=Cambridge University Press |contribution=Chapter 22. Edge Unfolding of Polyhedra |pages=306–338}}</ref><ref>{{Cite journal |last=Ghomi |first=Mohammad |date=2018-01-01 |title=Dürer's Unfolding Problem for Convex Polyhedra |journal=Notices of the American Mathematical Society |volume=65 |issue=1 |pages=25–27 |doi=10.1090/noti1609 |issn=0002-9920 |doi-access=free}}</ref>
* Is there a non-convex polyhedron without self-intersections with [[Szilassi polyhedron|more than seven faces]], all of which share an edge with each other?
* The [[Thomson problem]] – what is the minimum energy configuration of <math>n</math> mutually-repelling particles on a unit sphere?<ref>{{citation|last=Whyte|first=L. L.|doi=10.2307/2306764|journal=The American Mathematical Monthly|mr=0050303|pages=606–611|title=Unique arrangements of points on a sphere|volume=59|issue=9|year=1952|jstor=2306764}}</ref>
* Convex [[uniform 5-polytope]]s – find and classify the complete set of these shapes<ref>{{citation |author=ACW |date=May 24, 2012 |title=Convex uniform 5-polytopes |url=http://www.openproblemgarden.org/op/convex_uniform_5_polytopes |work=Open Problem Garden |access-date=2016-10-04 |archive-url=https://web.archive.org/web/20161005164840/http://www.openproblemgarden.org/op/convex_uniform_5_polytopes |archive-date=October 5, 2016 |url-status=live}}.</ref>

=== Graph theory ===
{{Main|Graph theory}}

==== Algebraic graph theory ====
* [[Babai's problem]]: which groups are Babai invariant groups?
* [[Brouwer's conjecture]] on upper bounds for sums of [[eigenvalues and eigenvectors|eigenvalues]] of [[Laplacian matrix|Laplacians]] of graphs in terms of their number of edges

==== Games on graphs ====
* Does there exist a graph <math>G</math> such that the [[dominating set|dominating number]] <math>\gamma(G)</math> equals the [[eternal dominating set|eternal dominating number]] <math>\gamma</math><sub>∞</sub><math>(G)</math> of <math>G</math> and <math>\gamma(G)</math> is less than the [[clique cover|clique covering number]] of <math>G</math>? <ref>{{cite journal
 |last1=Klostermeyer |first1=W.
 |last2=Mynhardt |first2=C.
 |year=2015
 |title=Protecting a graph with mobile guards
 |journal=Applicable Analysis and Discrete Mathematics
 |volume=10 |pages=21
 |arxiv=1407.5228
 |doi=10.2298/aadm151109021k
}}.</ref>
* [[Graham's pebbling conjecture]] on the pebbling number of Cartesian products of graphs<ref>{{cite journal
 | last = Pleanmani | first = Nopparat
 | doi = 10.1142/s179383091950068x
 | issue = 6
 | journal = Discrete Mathematics, Algorithms and Applications
 | mr = 4044549
 | pages = 1950068, 7
 | title = Graham's pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph
 | volume = 11
 | year = 2019| s2cid = 204207428
 }}</ref>
* Meyniel's conjecture that [[cop number]] is <math>O(\sqrt n)</math><ref>{{cite journal
 | last1 = Baird | first1 = William
 | last2 = Bonato | first2 = Anthony
 | arxiv = 1308.3385
 | doi = 10.4310/JOC.2012.v3.n2.a6
 | issue = 2
 | journal = Journal of Combinatorics
 | mr = 2980752
 | pages = 225–238
 | title = Meyniel's conjecture on the cop number: a survey
 | volume = 3
 | year = 2012| s2cid = 18942362
 }}</ref>
* Suppose Alice has a winning strategy for the [[graph coloring game|vertex coloring game]] on a graph <math>G</math> with <math>k</math> colors. Does she have one for <math>k+1</math> colors?<ref name=zhu1999>{{cite journal
| last         = Zhu
| first        = Xuding
| date         = 1999
| title        = The Game Coloring Number of Planar Graphs
| journal      = Journal of Combinatorial Theory, Series B
| volume       = 75
| issue        = 2
| pages        =245–258
| doi=10.1006/jctb.1998.1878
| doi-access= free
}}</ref>

==== Graph coloring and labeling ====
[[File:Erdős–Faber–Lovász conjecture.svg|thumb|An instance of the Erdős–Faber–Lovász conjecture: a graph formed from four cliques of four vertices each, any two of which intersect in a single vertex, can be four-colored.]]
* The [[Graph factorization#1-factorization conjecture|1-factorization conjecture]] that if <math>n</math> is odd or even and <math>k \geq n, n - 1</math> respectively, then a <math>k</math>-[[regular graph]] with <math>2n</math> vertices is [[Graph factorization#1-factorization|1-factorable]].
** The [[Graph factorization#Perfect 1-factorization|perfect 1-factorization conjecture]] that every [[complete graph]] on an even number of vertices admits a [[Graph factorization#Perfect 1-factorization|perfect 1-factorization]].
* [[Cereceda's conjecture]] on the diameter of the space of colorings of degenerate graphs<ref>{{citation
 | last1 = Bousquet | first1 = Nicolas
 | last2 = Bartier | first2 = Valentin
 | editor1-last = Bender | editor1-first = Michael A.
 | editor2-last = Svensson | editor2-first = Ola
 | editor3-last = Herman | editor3-first = Grzegorz
 | contribution = Linear Transformations Between Colorings in Chordal Graphs
 | doi = 10.4230/LIPIcs.ESA.2019.24
 | pages = 24:1–24:15
 | publisher = Schloss Dagstuhl – Leibniz-Zentrum für Informatik
 | series = LIPIcs
 | title = 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany
 | volume = 144
 | year = 2019| doi-access = free
 | isbn = 978-3-95977-124-5
 | s2cid = 195791634
 }}</ref>
* The [[Earth–Moon problem]]: what is the maximum chromatic number of biplanar graphs?<ref>{{citation
 | last = Gethner | first = Ellen | author-link = Ellen Gethner
 | editor1-last = Gera | editor1-first = Ralucca | editor1-link = Ralucca Gera
 | editor2-last = Haynes | editor2-first = Teresa W. | editor2-link = Teresa W. Haynes
 | editor3-last = Hedetniemi | editor3-first = Stephen T.
 | contribution = To the Moon and beyond
 | doi = 10.1007/978-3-319-97686-0_11
 | mr = 3930641
 | pages = 115–133
 | publisher = Springer International Publishing
 | series = Problem Books in Mathematics
 | title = Graph Theory: Favorite Conjectures and Open Problems, II
 | year = 2018| isbn = 978-3-319-97684-6 }}</ref>
* The [[Erdős–Faber–Lovász conjecture]] on coloring unions of cliques<ref>{{citation
 | last1 = Chung | first1 = Fan | author-link1 = Fan Chung
 | last2 = Graham | first2 = Ron | author-link2 = Ronald Graham
 | title = Erdős on Graphs: His Legacy of Unsolved Problems
 | year = 1998
 | publisher = A K Peters
 | pages = 97–99}}.</ref>
* The [[graceful labeling|graceful tree conjecture]] that every tree admits a graceful labeling
** [[Graceful labeling|Rosa's conjecture]] that all [[Cactus graph#Triangular cactus|triangular cacti]] are graceful or nearly-graceful
* The [[Gyárfás–Sumner conjecture]] on χ-boundedness of graphs with a forbidden induced tree<ref>{{citation
 | last1 = Chudnovsky | first1 = Maria | author1-link = Maria Chudnovsky
 | last2 = Seymour | first2 = Paul | author2-link = Paul Seymour (mathematician)
 | doi = 10.1016/j.jctb.2013.11.002
 | journal = [[Journal of Combinatorial Theory]]
 | mr = 3171779
 | pages = 11–16
 | series = Series B
 | title = Extending the Gyárfás-Sumner conjecture
 | volume = 105
 | year = 2014| doi-access = free
 }}</ref>
* The [[Hadwiger conjecture (graph theory)|Hadwiger conjecture]] relating coloring to clique minors<ref>{{citation
 | last = Toft | first = Bjarne
 | journal = Congressus Numerantium
 | mr = 1411244
 | pages = 249–283
 | title = A survey of Hadwiger's conjecture
 | volume = 115
 | year = 1996}}.</ref>
* The [[Hadwiger–Nelson problem]] on the chromatic number of unit distance graphs<ref>{{citation
 | last1 = Croft | first1 = Hallard T.
 | last2 = Falconer | first2 = Kenneth J.
 | last3 = Guy | first3 = Richard K. | author-link3 = Richard K. Guy
 | title = Unsolved Problems in Geometry
 | publisher = Springer-Verlag
 | year = 1991}}, Problem G10.</ref>
* [[Petersen graph#Petersen coloring conjecture|Jaeger's Petersen-coloring conjecture]]: every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph<ref>{{citation
 | last1 = Hägglund
 | first1 = Jonas
 | last2 = Steffen
 | first2 = Eckhard
 | issue = 1
 | journal = Ars Mathematica Contemporanea
 | mr = 3047618
 | pages = 161–173
 | title = Petersen-colorings and some families of snarks
 | url = http://amc-journal.eu/index.php/amc/article/viewFile/288/247
 | volume = 7
 | year = 2014
 | doi = 10.26493/1855-3974.288.11a
 | access-date = 2016-09-30
 | archive-url = https://web.archive.org/web/20161003070647/http://amc-journal.eu/index.php/amc/article/viewFile/288/247
 | archive-date = 2016-10-03
 | url-status = live
 | doi-access = free
 }}.</ref>
* The [[list coloring conjecture]]: for every graph, the list chromatic index equals the chromatic index<ref>{{citation |last1=Jensen |first1=Tommy R. |last2=Toft |first2=Bjarne |year=1995 |title=Graph Coloring Problems |location=New York |publisher=Wiley-Interscience |isbn=978-0-471-02865-9 |chapter=12.20 List-Edge-Chromatic Numbers |pages=201–202}}.</ref>
* The [[Overfull graph#Overfull conjecture|overfull conjecture]] that a graph with maximum degree <math>\Delta(G) \geq n/3</math> is [[Vizing's theorem|class 2]] if and only if it has an [[Overfull graph|overfull subgraph]] <math>S</math> satisfying <math>\Delta(S) = \Delta(G)</math>.
* The [[total coloring conjecture]] of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree<ref>{{citation
 | last1 = Molloy | first1 = Michael
 | last2 = Reed | first2 = Bruce | author1-link = Bruce Reed (mathematician)
 | doi = 10.1007/PL00009820
 | issue = 2
 | journal = [[Combinatorica]]
 | mr = 1656544
 | pages = 241–280
 | title = A bound on the total chromatic number
 | volume = 18
 | year = 1998| citeseerx = 10.1.1.24.6514
 | s2cid = 9600550
 }}.</ref>

==== Graph drawing and embedding ====
* The [[Albertson conjecture]]: the crossing number can be lower-bounded by the crossing number of a [[complete graph]] with the same [[chromatic number]]<ref>{{citation|first1=János|last1=Barát|first2=Géza|last2=Tóth|year=2010|title=Towards the Albertson Conjecture|arxiv=0909.0413|journal=Electronic Journal of Combinatorics|volume=17|issue=1|page=R73|bibcode=2009arXiv0909.0413B|doi-access=free|doi=10.37236/345}}.</ref>
* [[Conway's thrackle conjecture]]<ref>{{citation |last1=Fulek |first1=Radoslav |last2=Pach |first2=János |author-link2=János Pach |title=A computational approach to Conway's thrackle conjecture|journal=[[Computational Geometry (journal)|Computational Geometry]] |volume=44 |year=2011|issue=6–7 |pages=345–355 |mr=2785903 |doi=10.1016/j.comgeo.2011.02.001|doi-access=free|arxiv=1002.3904 }}.</ref> that [[thrackle]]s cannot have more edges than vertices
* The [[GNRS conjecture]] on whether minor-closed graph families have <math>\ell_1</math> embeddings with bounded distortion<ref>{{citation
 | last1 = Gupta | first1 = Anupam
 | last2 = Newman | first2 = Ilan
 | last3 = Rabinovich | first3 = Yuri
 | last4 = Sinclair | first4 = Alistair | author4-link = Alistair Sinclair
 | doi = 10.1007/s00493-004-0015-x
 | issue = 2
 | journal = [[Combinatorica]]
 | mr = 2071334
 | pages = 233–269
 | title = Cuts, trees and <math>\ell_1</math>-embeddings of graphs
 | volume = 24
 | year = 2004| citeseerx = 10.1.1.698.8978
 | s2cid = 46133408
 }}</ref>
* [[Harborth's conjecture]]: every planar graph can be drawn with integer edge lengths<ref>{{citation|title=Pearls in Graph Theory: A Comprehensive Introduction|title-link= Pearls in Graph Theory |series=Dover Books on Mathematics|last1=Hartsfield|first1=Nora|last2=Ringel|first2=Gerhard|author2-link=Gerhard Ringel|publisher=Courier Dover Publications|year=2013|isbn=978-0-486-31552-2|at=[https://books.google.com/books?id=VMjDAgAAQBAJ&pg=PA247 p. 247]|mr=2047103}}.</ref>
* [[Negami's conjecture]] on projective-plane embeddings of graphs with planar covers<ref>{{citation | last = Hliněný | first = Petr | doi = 10.1007/s00373-010-0934-9 | issue = 4 | journal = [[Graphs and Combinatorics]] | mr = 2669457 | pages = 525–536 | title = 20 years of Negami's planar cover conjecture | url = http://www.fi.muni.cz/~hlineny/papers/plcover20-gc.pdf | volume = 26 | year = 2010 | citeseerx = 10.1.1.605.4932 | s2cid = 121645 | access-date = 2016-10-04 | archive-url = https://web.archive.org/web/20160304030722/http://www.fi.muni.cz/~hlineny/papers/plcover20-gc.pdf | archive-date = 2016-03-04 | url-status = live }}.</ref>
* The [[Greedy embedding#Planar graphs|strong Papadimitriou–Ratajczak conjecture]]: every polyhedral graph has a convex greedy embedding<ref>{{citation | last1 = Nöllenburg | first1 = Martin | last2 = Prutkin | first2 = Roman | last3 = Rutter | first3 = Ignaz | doi = 10.20382/jocg.v7i1a3 | issue = 1 | journal = [[Journal of Computational Geometry]] | mr = 3463906 | pages = 47–69 | title = On self-approaching and increasing-chord drawings of 3-connected planar graphs | volume = 7 | year = 2016| arxiv = 1409.0315 | s2cid = 1500695 }}</ref>
* [[Turán's brick factory problem]] – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?<ref>{{citation | last1 = Pach | first1 = János | author1-link = János Pach | last2 = Sharir | first2 = Micha | author2-link = Micha Sharir | contribution = 5.1 Crossings—the Brick Factory Problem | pages = 126–127 | publisher = [[American Mathematical Society]] | series = Mathematical Surveys and Monographs | title = Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures | volume = 152 | year = 2009}}.</ref>

* [[Universal point set]]s of subquadratic size for planar graphs<ref>{{citation | last1 = Demaine | first1 = E. | author1-link = Erik Demaine | last2 = O'Rourke | first2 = J. | author2-link = Joseph O'Rourke (professor) | contribution = Problem 45: Smallest Universal Set of Points for Planar Graphs | title = The Open Problems Project | url = http://cs.smith.edu/~orourke/TOPP/P45.html | year = 2002–2012 | access-date = 2013-03-19 | archive-url = https://web.archive.org/web/20120814154255/http://cs.smith.edu/~orourke/TOPP/P45.html | archive-date = 2012-08-14 | url-status = live }}.</ref>

==== Restriction of graph parameters ====
* [[Conway's 99-graph problem]]: does there exist a [[strongly regular graph]] with parameters (99,14,1,2)?<ref>{{citation
 | last = Conway
 | first = John H.
 | author-link = John Horton Conway
 | access-date = 2019-02-12
 | publisher = Online Encyclopedia of Integer Sequences
 | title = Five $1,000 Problems (Update 2017)
 | url = https://oeis.org/A248380/a248380.pdf
 | archive-url = https://web.archive.org/web/20190213123825/https://oeis.org/A248380/a248380.pdf
 | archive-date = 2019-02-13
 | url-status = live
 }}</ref>
* [[Degree diameter problem]]: given two positive integers <math>d, k</math>, what is the largest graph of diameter <math>k</math> such that all vertices have degrees at most <math>d</math>?
* Jørgensen's conjecture that every 6-vertex-connected ''K''<sub>6</sub>-minor-free graph is an [[apex graph]]<ref>{{citation |last1=mdevos |title=Jorgensen's Conjecture |date=December 7, 2019 |url=http://www.openproblemgarden.org/op/jorgensens_conjecture |work=Open Problem Garden |archive-url=https://web.archive.org/web/20161114232136/http://www.openproblemgarden.org/op/jorgensens_conjecture |access-date=2016-11-13 |archive-date=2016-11-14 |last2=Wood |first2=David |url-status=live}}.</ref>
* Does a [[Moore graph]] with girth 5 and degree 57 exist?<ref>{{citation
 | last=Ducey
 | first=Joshua E.
 | doi=10.1016/j.disc.2016.10.001
 | issue=5
 | journal=[[Discrete Mathematics (journal)|Discrete Mathematics]]
 | mr=3612450
 | pages=1104–1109
 | title=On the critical group of the missing Moore graph
 | volume=340
 | year=2017
 | arxiv=1509.00327
 | s2cid=28297244}}
</ref>
* Do there exist infinitely many [[strongly regular graph|strongly regular]] [[geodetic graph]]s, or any strongly regular geodetic graphs that are not Moore graphs?<ref>{{citation
 | last1 = Blokhuis | first1 = A.
 | last2 = Brouwer | first2 = A. E. | author-link = Andries Brouwer
 | doi = 10.1007/BF00191941
 | issue = 1–3
 | journal = [[Geometriae Dedicata]]
 | mr = 925851
 | pages = 527–533
 | title = Geodetic graphs of diameter two
 | volume = 25
 | year = 1988| s2cid = 189890651
 }}</ref>

==== Subgraphs ====
* [[Barnette's conjecture]]: every cubic bipartite three-connected planar graph has a Hamiltonian cycle<ref>{{citation
 | last = Florek | first = Jan
 | doi = 10.1016/j.disc.2010.01.018
 | issue = 10–11
 | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
 | mr = 2601261
 | pages = 1531–1535
 | title = On Barnette's conjecture
 | volume = 310
 | year = 2010}}.</ref>
* [[Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean plane]] that the Steiner ratio is <math>\sqrt{3}/2</math>
* [[Graph toughness|Chvátal's toughness conjecture]], that there is a number {{mvar|t}} such that every {{mvar|t}}-tough graph is Hamiltonian<ref>{{citation
 | last1 = Broersma | first1 = Hajo
 | last2 = Patel | first2 = Viresh
 | last3 = Pyatkin | first3 = Artem
 | doi = 10.1002/jgt.21734
 | issue = 3
 | journal = [[Journal of Graph Theory]]
 | mr = 3153119
 | pages = 244–255
 | title = On toughness and Hamiltonicity of $2K_2$-free graphs
 | volume = 75
 | year = 2014| s2cid = 1377980
 | url = https://ris.utwente.nl/ws/files/6416631/jgt21734.pdf
 }}</ref>
* The [[cycle double cover conjecture]]: every bridgeless graph has a family of cycles that includes each edge twice<ref>{{citation
 | last = Jaeger | first = F.
 | contribution = A survey of the cycle double cover conjecture
 | doi = 10.1016/S0304-0208(08)72993-1
 | pages = 1–12
 | series = North-Holland Mathematics Studies
 | title = Annals of Discrete Mathematics 27 – Cycles in Graphs
 | volume = 27
 | year = 1985| isbn = 978-0-444-87803-8
 }}.</ref>
* The [[Erdős–Gyárfás conjecture]] on cycles with power-of-two lengths in cubic graphs<ref>{{citation|title=Erdös-Gyárfás conjecture for cubic planar graphs |first1=Christopher Carl |last1=Heckman |first2=Roi |last2=Krakovski |volume=20 |issue=2 |year=2013 |at=P7 |journal=Electronic Journal of Combinatorics |doi-access=free |doi=10.37236/3252}}.</ref>
* The [[Erdős–Hajnal conjecture]] on large cliques or independent sets in graphs with a forbidden induced subgraph<ref>{{citation
 | last = Chudnovsky
 | first = Maria
 | author-link = Maria Chudnovsky
 | arxiv = 1606.08827
 | doi = 10.1002/jgt.21730
 | issue = 2
 | journal = [[Journal of Graph Theory]]
 | mr = 3150572
 | zbl = 1280.05086
 | pages = 178–190
 | title = The Erdös–Hajnal conjecture—a survey
 | url = http://www.columbia.edu/~mc2775/EHsurvey.pdf
 | volume = 75
 | year = 2014
 | s2cid = 985458
 | access-date = 2016-09-22
 | archive-url = https://web.archive.org/web/20160304102611/http://www.columbia.edu/~mc2775/EHsurvey.pdf
 | archive-date = 2016-03-04
 | url-status = live
 }}.</ref>
* The [[linear arboricity]] conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree<ref>{{citation
 | last1 = Akiyama | first1 = Jin | author1-link = Jin Akiyama
 | last2 = Exoo | first2 = Geoffrey
 | last3 = Harary | first3 = Frank
 | doi = 10.1002/net.3230110108
 | issue = 1
 | journal = Networks
 | mr = 608921
 | pages = 69–72
 | title = Covering and packing in graphs. IV. Linear arboricity
 | volume = 11
 | year = 1981}}.</ref>
* The [[Lovász conjecture]] on Hamiltonian paths in symmetric graphs<ref>{{Cite book |last=Babai |first=László |url=http://newtraell.cs.uchicago.edu/files/tr_authentic/TR-94-10.ps |title=Handbook of Combinatorics |date=June 9, 1994 |chapter=Automorphism groups, isomorphism, reconstruction |format=PostScript |author-link=László Babai |archive-url=https://web.archive.org/web/20070613201449/http://www.cs.uchicago.edu/research/publications/techreports/TR-94-10 |archive-date=13 June 2007}}</ref>
* The [[Oberwolfach problem]] on which 2-regular graphs have the property that a complete graph on the same number of vertices can be decomposed into edge-disjoint copies of the given graph.<ref>{{citation
 | last1 = Lenz | first1 = Hanfried
 | last2 = Ringel | first2 = Gerhard
 | doi = 10.1016/0012-365X(91)90416-Y
 | issue = 1–3
 | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
 | mr = 1140782
 | pages = 3–16
 | title = A brief review on Egmont Köhler's mathematical work
 | volume = 97
 | year = 1991}}</ref>
* What is the largest possible [[pathwidth]] of an {{mvar|n}}-vertex [[cubic graph]]?<ref>{{citation
 | last1 = Fomin | first1 = Fedor V.
 | last2 = Høie | first2 = Kjartan
 | doi = 10.1016/j.ipl.2005.10.012
 | issue = 5
 | journal = Information Processing Letters
 | mr = 2195217
 | pages = 191–196
 | title = Pathwidth of cubic graphs and exact algorithms
 | volume = 97
 | year = 2006}}
</ref>
* The [[reconstruction conjecture]] and [[new digraph reconstruction conjecture]] on whether a graph is uniquely determined by its vertex-deleted subgraphs.<ref>{{cite conference |last=Schwenk |first=Allen |year=2012 |title=Some History on the Reconstruction Conjecture |url=http://faculty.nps.edu/rgera/conjectures/jmm2012/Schwenk,%20%20Some%20History%20on%20the%20RC.pdf |conference=Joint Mathematics Meetings |archive-url=https://web.archive.org/web/20150409233306/http://faculty.nps.edu/rgera/Conjectures/jmm2012/Schwenk,%20%20Some%20History%20on%20the%20RC.pdf |archive-date=2015-04-09 |access-date=2018-11-26}}</ref><ref>{{citation
 | last = Ramachandran | first = S.
 | doi = 10.1016/S0095-8956(81)80019-6
 | issue = 2
 | journal = [[Journal of Combinatorial Theory]]
 | mr = 630977
 | pages = 143–149
 | series = Series B
 | title = On a new digraph reconstruction conjecture
 | volume = 31
 | year = 1981| doi-access = free
 }}</ref>
* The [[snake-in-the-box]] problem: what is the longest possible [[induced path]] in an <math>n</math>-dimensional [[hypercube]] graph?
* [[Sumner's conjecture]]: does every <math>(2n-2)</math>-vertex tournament contain as a subgraph every <math>n</math>-vertex oriented tree?<ref>{{citation
 | last1 = Kühn | first1 = Daniela | author1-link = Daniela Kühn
 | last2 = Mycroft | first2 = Richard
 | last3 = Osthus | first3 = Deryk
 | arxiv = 1010.4430
 | doi = 10.1112/plms/pdq035
 | issue = 4
 | journal = Proceedings of the London Mathematical Society | series = Third Series
 | mr = 2793448 | zbl=1218.05034
 | pages = 731–766
 | title = A proof of Sumner's universal tournament conjecture for large tournaments
 | volume = 102
 | year = 2011| s2cid = 119169562 }}.</ref>
* [[Szymanski's conjecture]]: every [[permutation]] on the <math>n</math>-dimensional doubly-[[Directed graph|directed]] [[hypercube graph]] can be routed with edge-disjoint [[Path (graph theory)|paths]].
* [[Tuza's conjecture]]: if the maximum number of disjoint triangles is <math>\nu</math>, can all triangles be hit by a set of at most <math>2\nu</math> edges?<ref>{{cite journal
 | last = Tuza | first = Zsolt
 | doi = 10.1007/BF01787705
 | issue = 4
 | journal = Graphs and Combinatorics
 | mr = 1092587
 | pages = 373–380
 | title = A conjecture on triangles of graphs
 | volume = 6
 | year = 1990| s2cid = 38821128
 }}</ref>
* [[Vizing's conjecture]] on the [[domination number]] of [[cartesian product of graphs|cartesian products of graphs]]<ref>{{citation
 | last1 = Brešar | first1 = Boštjan
 | last2 = Dorbec | first2 = Paul
 | last3 = Goddard | first3 = Wayne
 | last4 = Hartnell | first4 = Bert L.
 | last5 = Henning | first5 = Michael A.
 | last6 = Klavžar | first6 = Sandi
 | last7 = Rall | first7 = Douglas F.
 | doi = 10.1002/jgt.20565
 | issue = 1
 | journal = [[Journal of Graph Theory]]
 | mr = 2864622
 | pages = 46–76
 | title = Vizing's conjecture: a survey and recent results
 | volume = 69
 | year = 2012| citeseerx = 10.1.1.159.7029
 | s2cid = 9120720
 }}.</ref>
* [[Zarankiewicz problem]]: how many edges can there be in a [[bipartite graph]] on a given number of vertices with no [[complete bipartite graph|complete bipartite subgraphs]] of a given size?

==== Word-representation of graphs ====
*Are there any graphs on ''n'' vertices whose [[Word-representable graph|representation]] requires more than floor(''n''/2) copies of each letter?<ref name="KL15">{{Cite book |last1=Kitaev |first1=Sergey | author1-link = Sergey Kitaev|url=https://link.springer.com/book/10.1007/978-3-319-25859-1 |title=Words and Graphs |last2=Lozin |first2=Vadim |year=2015 |isbn=978-3-319-25857-7 |series=Monographs in Theoretical Computer Science. An EATCS Series |doi=10.1007/978-3-319-25859-1 |via=link.springer.com |s2cid=7727433}}</ref><ref name="K17">{{Cite conference |last=Kitaev |first=Sergey |date=2017-05-16 |title=A Comprehensive Introduction to the Theory of Word-Representable Graphs |conference=[[International Conference on Developments in Language Theory]] |language=en |doi=10.1007/978-3-319-62809-7_2|arxiv=1705.05924v1 }}</ref><ref name="KP18">{{Cite journal|title=Word-Representable Graphs: a Survey|first1=S. V.|last1=Kitaev|first2=A. V.|last2=Pyatkin|date=April 1, 2018|journal=Journal of Applied and Industrial Mathematics|volume=12|issue=2|pages=278–296|via=Springer Link|doi=10.1134/S1990478918020084|s2cid=125814097 }}</ref><ref name="KP18-2">{{Cite journal |last1=Kitaev |first1=Sergey V. |last2=Pyatkin |first2=Artem V. |date=2018 |title=Графы, представимые в виде слов. Обзор результатов |trans-title=Word-representable graphs: A survey |url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=da&paperid=894&option_lang=rus |journal=Дискретн. анализ и исслед. опер. |language=ru |volume=25 |issue=2 |pages=19–53 |doi=10.17377/daio.2018.25.588}}</ref>
*Characterise (non-)[[Word-representable graph|word-representable]] [[planar graph]]s<ref name="KL15"/><ref name="K17"/><ref name="KP18"/><ref name="KP18-2"/>
*Characterise [[word-representable graph]]s in terms of (induced) forbidden subgraphs.<ref name="KL15"/><ref name="K17"/><ref name="KP18"/><ref name="KP18-2"/>
*Characterise [[Word-representable graph|word-representable]] near-triangulations containing the complete graph ''K''<sub>4</sub> (such a characterisation is known for ''K''<sub>4</sub>-free planar graphs<ref name="Glen2019">{{cite arXiv |eprint=1605.01688|author1=Marc Elliot Glen|title=Colourability and word-representability of near-triangulations|class=math.CO|year=2016}}</ref>)
*Classify graphs with representation number 3, that is, graphs that can be [[Word-representable graph|represented]] using 3 copies of each letter, but cannot be represented using 2 copies of each letter<ref name="Kit2013-3-repr">{{Cite arXiv|last=Kitaev |first=Sergey |date=2014-03-06 |title=On graphs with representation number 3 |class=math.CO |eprint=1403.1616v1 }}</ref>
*Is it true that out of all [[bipartite graph]]s, [[crown graph]]s require longest word-representants?<ref name="GKP18">{{cite journal|url = https://www.sciencedirect.com/science/article/pii/S0166218X18301045 | doi=10.1016/j.dam.2018.03.013 | volume=244 | title=On the representation number of a crown graph | year=2018 | journal=Discrete Applied Mathematics | pages=89–93 | last1 = Glen | first1 = Marc | last2 = Kitaev | first2 = Sergey | last3 = Pyatkin | first3 = Artem| arxiv=1609.00674 | s2cid=46925617 }}</ref>
*Is the [[line graph]] of a non-[[Word-representable graph|word-representable]] graph always non-[[Word-representable graph|word-representable]]?<ref name="KL15"/><ref name="K17"/><ref name="KP18"/><ref name="KP18-2"/>
*Which (hard) problems on graphs can be translated to words [[Word-representable graph|representing]] them and solved on words (efficiently)?<ref name="KL15"/><ref name="K17"/><ref name="KP18"/><ref name="KP18-2"/>

==== Miscellaneous graph theory ====
* The [[implicit graph conjecture]] on the existence of implicit representations for slowly-growing [[Hereditary property#In graph theory|hereditary families of graphs]]<ref>{{citation|first=Jeremy P.|last=Spinrad|title=Efficient Graph Representations|year=2003|isbn=978-0-8218-2815-1|chapter=2. Implicit graph representation|pages=17–30|publisher=American Mathematical Soc. |chapter-url=https://books.google.com/books?id=RrtXSKMAmWgC&pg=PA17}}.</ref>
* [[Ryser's conjecture]] relating the maximum [[Matching in hypergraphs|matching]] size and minimum [[Vertex cover in hypergraphs|transversal]] size in [[hypergraph]]s
* The [[second neighborhood problem]]: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one?<ref>{{Cite web |title=Seymour's 2nd Neighborhood Conjecture |url=https://faculty.math.illinois.edu/~west/openp/2ndnbhd.html |url-status=live |archive-url=https://web.archive.org/web/20190111175310/https://faculty.math.illinois.edu/~west/openp/2ndnbhd.html |archive-date=11 January 2019 |access-date=17 August 2022 |website=faculty.math.illinois.edu}}</ref>
* [[Sidorenko's conjecture]] on [[Homomorphism density|homomorphism densities]] of graphs in [[graphon]]s
* Tutte's conjectures:
** every bridgeless graph has a [[nowhere-zero flows|nowhere-zero 5-flow]]<ref>{{cite web |last=mdevos |date=May 4, 2007 |title=5-flow conjecture |url=http://www.openproblemgarden.org/op/5_flow_conjecture |url-status=live |archive-url=https://web.archive.org/web/20181126134833/http://www.openproblemgarden.org/op/5_flow_conjecture |archive-date=November 26, 2018 |website=Open Problem Garden}}</ref>
** every [[Petersen graph|Petersen]]-[[Graph minor|minor]]-free bridgeless graph has a nowhere-zero 4-flow<ref>{{cite web |last=mdevos |date=March 31, 2010 |title=4-flow conjecture |url=http://www.openproblemgarden.org/op/4_flow_conjecture |url-status=live |archive-url=https://web.archive.org/web/20181126134908/http://www.openproblemgarden.org/op/4_flow_conjecture |archive-date=November 26, 2018 |website=Open Problem Garden}}</ref>
* [[Woodall's conjecture]] that the minimum number of edges in a [[dicut]] of a [[directed graph]] is equal to the maximum number of disjoint [[dijoin]]s

=== Model theory and formal languages ===
{{Main|Model theory|formal languages}}
* The [[Stable group|Cherlin–Zilber conjecture]]: A simple group whose first-order theory is [[Stable theory|stable]] in <math>\aleph_0</math> is a simple algebraic group over an algebraically closed field.
* [[Generalized star height problem]]: can all [[regular language]]s be expressed using [[Regular expression#Expressive power and compactness|generalized regular expressions]] with limited nesting depths of [[Kleene star]]s?
* For which number fields does [[Hilbert's tenth problem]] hold?
* Kueker's conjecture<ref>{{cite journal |last1=Hrushovski |first1=Ehud |year=1989 |title=Kueker's conjecture for stable theories |journal=Journal of Symbolic Logic |volume=54 |issue=1| pages=207–220 |doi=10.2307/2275025| jstor=2275025 |s2cid=41940041}}</ref>
* The main gap conjecture, e.g. for uncountable [[First order theory|first order theories]], for [[Abstract elementary class|AECs]], and for <math>\aleph_1</math>-saturated models of a countable theory.<ref name=":0">{{cite book |vauthors=Shelah S |title=Classification Theory |publisher=North-Holland |year=1990}}</ref>
* Shelah's categoricity conjecture for <math>L_{\omega_1,\omega}</math>: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.<ref name=":0" />
* Shelah's eventual categoricity conjecture: For every cardinal <math>\lambda</math> there exists a cardinal <math>\mu(\lambda)</math> such that if an [[Abstract elementary class|AEC]] ''K'' with LS(''K'')<math>{} \le \lambda</math> is categorical in a cardinal above <math>\mu(\lambda)</math> then it is categorical in all cardinals above <math>\mu(\lambda)</math>.<ref name=":0" /><ref>{{Cite book
| title = Classification theory for abstract elementary classes
| last = Shelah
| first = Saharon
| publisher = College Publications
| year = 2009
| isbn = 978-1-904987-71-0
}}</ref>
* The stable field conjecture: every infinite field with a [[Stable theory|stable]] first-order theory is separably closed.
* The stable forking conjecture for simple theories<ref>{{cite journal | last1 = Peretz | first1 = Assaf | year = 2006 | title = Geometry of forking in simple theories | journal = Journal of Symbolic Logic| volume = 71 | issue = 1| pages = 347–359 | doi=10.2178/jsl/1140641179| arxiv = math/0412356| s2cid = 9380215 }}</ref>
* [[Tarski's exponential function problem]]: is the [[Theory (mathematical logic)|theory]] of the [[real number]]s with the [[exponential function]] [[Decidability (logic)#Decidability of a theory|decidable]]?
* The universality problem for ''C''-free graphs: For which finite sets ''C'' of graphs does the class of ''C''-free countable graphs have a universal member under strong embeddings?<ref>{{cite journal |last1=Cherlin |first1=Gregory |last2=Shelah |first2=Saharon | author-link2=Saharon Shelah|date=May 2007 |title=Universal graphs with a forbidden subtree |journal=[[Journal of Combinatorial Theory]] | series=Series B |arxiv=math/0512218 |doi=10.1016/j.jctb.2006.05.008 | doi-access=free |volume=97 |issue=3 |pages=293–333|s2cid=10425739 }}</ref>
* The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?<ref>Džamonja, Mirna, "Club guessing and the universal models." ''On PCF'', ed. M. Foreman, (Banff, Alberta, 2004).</ref>
* [[Vaught conjecture]]: the number of [[Countable set|countable]] models of a [[First-order logic|first-order]] [[complete theory]] in a countable [[Formal language|language]] is either finite, <math>\aleph_0</math>, or <math>2^{\aleph_0}</math>.

* Assume ''K'' is the class of models of a countable first order theory omitting countably many [[Type (model theory)|types]]. If ''K'' has a model of cardinality <math>\aleph_{\omega_1}</math> does it have a model of cardinality continuum?<ref>{{cite journal |last=Shelah |first=Saharon |author-link=Saharon Shelah |date=1999 |title=Borel sets with large squares |journal=[[Fundamenta Mathematicae]] |arxiv=math/9802134 |volume=159 |issue=1 |pages=1–50|bibcode=1998math......2134S |doi=10.4064/fm-159-1-1-50 |s2cid=8846429 }}</ref>
* Do the [[Henson graph]]s have the [[finite model property]]?
* Does a finitely presented homogeneous structure for a finite relational language have finitely many [[reduct]]s?
* Does there exist an [[o-minimal]] first order theory with a trans-exponential (rapid growth) function?
* If the class of atomic models of a complete first order theory is [[Categorical (model theory)|categorical]] in the <math>\aleph_n</math>, is it categorical in every cardinal?<ref>{{cite book |last=Baldwin |first=John T. |date=July 24, 2009 |title=Categoricity |publisher=[[American Mathematical Society]] |isbn=978-0-8218-4893-7 |url=http://www.math.uic.edu/~jbaldwin/pub/AEClec.pdf |access-date=February 20, 2014 |archive-url=https://web.archive.org/web/20100729073738/http://www.math.uic.edu/%7Ejbaldwin/pub/AEClec.pdf |archive-date=July 29, 2010 |url-status=live }}</ref><ref>{{cite arXiv |last=Shelah |first=Saharon |title=Introduction to classification theory for abstract elementary classes |year=2009 |class=math.LO |eprint=0903.3428 }}</ref>
* Is every infinite, minimal field of characteristic zero [[algebraically closed field|algebraically closed]]? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)
* Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?<ref>Gurevich, Yuri, "Monadic Second-Order Theories," in [[Jon Barwise|J. Barwise]], [[Solomon Feferman|S. Feferman]], eds., ''Model-Theoretic Logics'' (New York: Springer-Verlag, 1985), 479–506.</ref>
* Is the theory of the field of Laurent series over <math>\mathbb{Z}_p</math> [[Decidability (logic)|decidable]]? of the field of polynomials over <math>\mathbb{C}</math>?
* Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?<ref>Makowsky J, "Compactness, embeddings and definability," in ''Model-Theoretic Logics'', eds Barwise and Feferman, Springer 1985 pps. 645–715.</ref>

* Determine the structure of Keisler's order.<ref>{{cite journal | last1 = Keisler | first1 = HJ | year = 1967 | title = Ultraproducts which are not saturated | journal = J. Symb. Log. | volume = 32 | issue = 1| pages = 23–46 | doi=10.2307/2271240| jstor = 2271240 | s2cid = 250345806 }}</ref><ref>{{Cite arXiv |eprint=1208.2140 |class=math.LO |first1=Maryanthe |last1=Malliaris |first2=Saharon |last2=Shelah |author-link=Maryanthe Malliaris |author-link2=Saharon Shelah |title=A Dividing Line Within Simple Unstable Theories |date=10 August 2012}} {{Cite arXiv |title=A Dividing Line within Simple Unstable Theories |eprint=1208.2140 |last1=Malliaris |first1=M. |last2=Shelah |first2=S. |date=2012 |class=math.LO }}</ref>

=== Probability theory ===
{{Main|Probability theory}}
* [[Ibragimov–Iosifescu conjecture for φ-mixing sequences]]

=== Number theory ===
{{Main articles|Category:Unsolved problems in number theory}}
{{See also|Number theory }}

==== General ====
[[File:Perfect number Cuisenaire rods 6 exact.svg|thumb|6 is a [[perfect number]] because it is the sum of its proper positive divisors, 1, 2 and 3. It is not known how many perfect numbers there are, nor if any of them is odd.]]
*[[Special values of L-functions|Beilinson's conjectures]]
* [[Brocard's problem]]: are there any integer solutions to <math>n! + 1 = m^{2}</math> other than <math>n = 4, 5, 7</math>?
* [[Büchi's problem]] on sufficiently large sequences of square numbers with constant second difference.
* [[Carmichael's totient function conjecture]]: do all values of [[Euler's totient function]] have [[Multiplicity (mathematics)|multiplicity]] greater than <math>1</math>?
* [[Casas-Alvero conjecture]]: if a polynomial of degree <math>d</math> defined over a [[Field (mathematics)|field]] <math>K</math> of [[Characteristic (algebra)|characteristic]] <math>0</math> has a factor in common with its first through <math>d - 1</math>-th derivative, then must <math>f</math> be the <math>d</math>-th power of a linear polynomial?
* [[Aliquot sequence#Catalan-Dickson conjecture|Catalan–Dickson conjecture on aliquot sequences]]: no [[aliquot sequence]]s are infinite but non-repeating.
* [[Erdős–Ulam problem]]: is there a [[dense set]] of points in the plane all at rational distances from one-another?
* [[Van der Corput's method#Exponent pairs|Exponent pair conjecture]]: for all <math>\varepsilon > 0</math>, is the pair <math>(\varepsilon, 1/2 + \varepsilon)</math> an [[Van der Corput's method#Exponent pairs|exponent pair]]?
* The [[Gauss circle problem]]: how far can the number of integer points in a circle centered at the origin be from the area of the circle?
*[[Grand Riemann hypothesis]]: do the nontrivial zeros of all [[automorphic L-function]]s lie on the critical line <math>1/2 + it</math> with real <math>t</math>?
**[[Generalized Riemann hypothesis]]: do the nontrivial zeros of all [[Dirichlet L-function]]s lie on the critical line <math>1/2 + it</math> with real <math>t</math>?
***[[Riemann hypothesis]]: do the nontrivial zeros of the [[Riemann zeta function]] lie on the critical line <math>1/2 + it</math> with real <math>t</math>?
* [[Grimm's conjecture]]: each element of a set of consecutive [[composite number]]s can be assigned a distinct [[prime number]] that divides it.
* [[Hall's conjecture]]: for any <math>\varepsilon > 0</math>, there is some constant <math>c(\varepsilon)</math> such that either <math>y^2 = x^3</math> or <math>|y^2 - x^3| > c(\varepsilon)x^{1/2 - \varepsilon}</math>.
* [[Hardy–Littlewood zeta function conjectures]]
* [[Hilbert–Pólya conjecture]]: the nontrivial zeros of the [[Riemann zeta function]] correspond to [[Eigenvalues and eigenvectors|eigenvalues]] of a [[self-adjoint operator]].
* [[Hilbert's eleventh problem]]: classify [[quadratic form]]s over [[algebraic number field]]s.
* [[Hilbert's ninth problem]]: find the most general [[reciprocity law]] for the [[Hilbert symbol|norm residues]] of <math>k</math>-th order in a general [[algebraic number field]], where <math>k</math> is a power of a prime. 
* [[Hilbert's twelfth problem]]: extend the [[Kronecker–Weber theorem]] on [[Abelian extension]]s of <math>\mathbb{Q}</math> to any base number field.
* Keating–Snaith conjecture concerning the asymptotics of an integral involving the [[Riemann zeta function]]<ref>{{citation
 |last=Conrey |first=Brian |author-link=Brian Conrey
 |doi=10.1090/bull/1525
 |title=Lectures on the Riemann zeta function (book review)
 |journal=[[Bulletin of the American Mathematical Society]]
 |volume=53 |number=3 |pages=507–512 |year=2016|doi-access=free}}</ref>
*[[Lehmer's totient problem]]: if <math>\phi(n)</math> divides <math>n - 1</math>, must <math>n</math> be prime?
* [[Leopoldt's conjecture]]: a [[p-adic number|p-adic]] analogue of the [[Dirichlet's unit theorem#The regulator|regulator]] of an [[algebraic number field]] does not vanish.
* [[Lindelöf hypothesis]] that for all <math>\varepsilon > 0</math>, <math>\zeta(1/2 + it) = o(t^\varepsilon)</math>
** The [[Bombieri–Vinogradov theorem|density hypothesis]] for zeroes of the Riemann zeta function
* [[Littlewood conjecture]]: for any two real numbers <math>\alpha, \beta</math>, <math>\liminf_{n \rightarrow \infty} n\,\Vert n\alpha\Vert\,\Vert n\beta\Vert = 0</math>, where <math>\Vert x\Vert</math> is the distance from <math>x</math> to the nearest integer.
* [[Mahler's 3/2 problem]] that no real number <math>x</math> has the property that the fractional parts of <math>x(3/2)^n</math> are less than <math>1/2</math> for all positive integers <math>n</math>.
* [[Montgomery's pair correlation conjecture]]: the normalized pair [[correlation function]] between pairs of zeros of the [[Riemann zeta function]] is the same as the pair correlation function of [[Random matrix#Gaussian ensembles|random Hermitian matrices]].
* [[n conjecture|''n'' conjecture]]: a generalization of the ''abc'' conjecture to more than three integers.
** [[abc conjecture|''abc'' conjecture]]: for any <math>\varepsilon > 0</math>, <math>\operatorname{rad}(abc)^{1+\varepsilon} < c</math> is true for only finitely many positive <math>a, b, c</math> such that <math>a + b = c</math>.
** [[Szpiro's conjecture]]: for any <math>\varepsilon > 0</math>, there is some constant <math>C(\varepsilon)</math> such that, for any elliptic curve <math>E</math> defined over <math>\mathbb{Q}</math> with minimal discriminant <math>\Delta</math> and conductor <math>f</math>, we have <math>|\Delta| \leq C(\varepsilon) \cdot f^{6+\varepsilon}</math>.
* [[Newman's conjecture]]: the [[Partition function (number theory)|partition function]] satisfies any arbitrary congruence infinitely often.
* [[Divisor summatory function#Piltz divisor problem|Piltz divisor problem]] on bounding <math>\Delta_k(x) = D_k(x) - xP_k(\log(x))</math>
** [[Divisor summatory function#Dirichlet's divisor problem|Dirichlet's divisor problem]]: the specific case of the Piltz divisor problem for <math>k = 1</math>
* [[Ramanujan–Petersson conjecture]]: a number of related conjectures that are generalizations of the original conjecture.
* [[Sato–Tate conjecture]]: also a number of related conjectures that are generalizations of the original conjecture.
* [[Scholz conjecture]]: the length of the shortest [[addition chain]] producing <math>2^n - 1</math> is at most <math>n - 1</math> plus the length of the shortest addition chain producing <math>n</math>.
* Do [[Siegel zero]]s exist?
* [[Singmaster's conjecture]]: is there a finite upper bound on the multiplicities of the entries greater than 1 in [[Pascal's triangle]]?<ref>{{citation |last=Singmaster |first=David |title=Research Problems: How often does an integer occur as a binomial coefficient? |journal=[[American Mathematical Monthly]] |volume=78 |issue=4 |pages=385–386 |year=1971 |doi=10.2307/2316907 |jstor=2316907 |mr=1536288 |author-link=David Singmaster}}.</ref>
* [[Vojta's conjecture]] on [[Height function|heights]] of points on [[Algebraic variety|algebraic varieties]] over [[algebraic number field]]s.

* Are there infinitely many [[perfect number]]s?
*Do any [[odd perfect number]]s exist?
*Do [[quasiperfect number]]s exist?
*Do any non-power of 2 [[almost perfect number]]s exist?
*Are there 65, 66, or 67 [[idoneal number]]s?
* Are there any pairs of [[amicable numbers]] which have opposite parity?
* Are there any pairs of [[betrothed numbers]] which have same parity?
* Are there any pairs of [[relatively prime]] [[amicable numbers]]?
* Are there infinitely many [[amicable numbers]]?
* Are there infinitely many [[betrothed numbers]]?
* Are there infinitely many [[Giuga number]]s?
* Does every [[rational number]] with an odd denominator have an [[odd greedy expansion]]?
* Do any [[Lychrel number]]s exist?
* Do any odd [[noncototient]]s exist?
* Do any odd [[weird number]]s exist?
* Do any [[Superperfect number#Generalizations|(2, 5)-perfect numbers]] exist?
* Do any [[Generalized taxicab number|Taxicab(5, 2, n)]] exist for ''n''&nbsp;>&nbsp;1?
* Is there a [[covering system]] with odd distinct moduli?<ref>{{citation
 | last1 = Guo | first1 = Song
 | last2 = Sun | first2 = Zhi-Wei
 | doi = 10.1016/j.aam.2005.01.004
 | issue = 2
 | journal = Advances in Applied Mathematics
 | mr = 2152886
 | pages = 182–187
 | title = On odd covering systems with distinct moduli
 | volume = 35
 | year = 2005| arxiv = math/0412217
 | s2cid = 835158
 }}</ref>
* Is <math>\pi</math> a [[normal number]] (i.e., is each digit 0–9 equally frequent)?<ref>{{cite web|url=http://www2.lbl.gov/Science-Articles/Archive/pi-random.html|title=Are the Digits of Pi Random? Berkeley Lab Researcher May Hold Key|access-date=2016-03-18|archive-url=https://web.archive.org/web/20160327035021/http://www2.lbl.gov/Science-Articles/Archive/pi-random.html|archive-date=2016-03-27|url-status=live}}</ref>
* Are all [[Irrational number|irrational]] [[Algebraic number|algebraic]] numbers normal?
* Is 10 a [[solitary number]]?
* Can a 3×3 [[magic square]] be constructed from 9 distinct perfect square numbers?<ref>{{Cite journal |last=Robertson |first=John P. |date=1996-10-01 |title=Magic Squares of Squares |journal=Mathematics Magazine |volume=69 |issue=4 |pages=289–293 |doi=10.1080/0025570X.1996.11996457 |issn=0025-570X}}</ref>
* Find the value of the [[De Bruijn–Newman constant]].

==== Additive number theory ====
{{Main|Additive number theory }}
{{See also|Problems involving arithmetic progressions}}
* [[Erdős conjecture on arithmetic progressions]] that if the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long [[arithmetic progression]]s.
* [[Erdős–Turán conjecture on additive bases]]: if <math>B</math> is an [[additive basis]] of order <math>2</math>, then the number of ways that positive integers <math>n</math> can be expressed as the sum of two numbers in <math>B</math> must tend to infinity as <math>n</math> tends to infinity.
* [[Gilbreath's conjecture]] on consecutive applications of the unsigned [[Finite difference|forward difference]] operator to the sequence of [[prime number]]s.
* [[Goldbach's conjecture]]: every even natural number greater than <math>2</math> is the sum of two [[prime number]]s.
* [[Lander, Parkin, and Selfridge conjecture]]: if the sum of <math>m</math> <math>k</math>-th powers of positive integers is equal to a different sum of <math>n</math> <math>k</math>-th powers of positive integers, then <math>m + n \geq k</math>.
* [[Lemoine's conjecture]]: all odd integers greater than <math>5</math> can be represented as the sum of an odd [[prime number]] and an even [[semiprime]].
* [[Minimum overlap problem]] of estimating the minimum possible maximum number of times a number appears in the termwise difference of two equally large sets partitioning the set <math>\{1, \ldots, 2n\}</math>
* [[Pollock's conjectures]]
* Does every nonnegative integer appear in [[Recamán's sequence]]?
* [[Skolem problem]]: can an algorithm determine if a [[constant-recursive sequence]] contains a zero?
* The values of ''g''(''k'') and ''G''(''k'') in [[Waring's problem]]

* Do the [[Ulam number]]s have a positive density?

* Determine growth rate of ''r''<sub>''k''</sub>(''N'') (see [[Szemerédi's theorem]])

==== Algebraic number theory ====
{{Main|Algebraic number theory }}
* [[Class number problem]]: are there infinitely many [[Class number problem#Real quadratic fields|real quadratic number fields]] with [[unique factorization]]?
* [[Fontaine–Mazur conjecture]]: actually numerous conjectures, all proposed by [[Jean-Marc Fontaine]] and [[Barry Mazur]].
* [[Gan–Gross–Prasad conjecture]]: a [[Restricted representation|restriction]] problem in [[Representation of a Lie group|representation theory of real or p-adic Lie groups]].
* [[Greenberg's conjectures]]
* [[Hermite's problem]]: is it possible, for any natural number <math>n</math>, to assign a sequence of [[natural number]]s to each [[real number]] such that the sequence for <math>x</math> is eventually [[Periodic sequence|periodic]] if and only if <math>x</math> is [[Algebraic number|algebraic]] of degree <math>n</math>?
* [[Kummer–Vandiver conjecture]]: primes <math>p</math> do not divide the [[Ideal class group#Properties|class number]] of the maximal real [[Field extension|subfield]] of the <math>p</math>-th [[cyclotomic field]].
* Lang and Trotter's conjecture on [[Supersingular prime (algebraic number theory)|supersingular primes]] that the number of [[Supersingular prime (algebraic number theory)|supersingular primes]] less than a constant <math>X</math> is within a constant multiple of <math>\sqrt{X}/\ln{X}</math>
* [[Selberg's 1/4 conjecture]]: the [[Eigenvalues and eigenvectors|eigenvalues]] of the [[Laplace operator]] on [[Maass wave form]]s of [[congruence subgroup]]s are at least <math>1/4</math>.
* [[Stark conjectures]] (including [[Brumer–Stark conjecture]])

* Characterize all algebraic number fields that have some [[Algebraic number field#Bases for number fields|power basis]].

====Computational number theory====
{{Main|Computational number theory}}
* Can [[integer factorization]] be done in [[polynomial time]]?

==== Diophantine approximation and transcendental number theory ====
{{Further|Diophantine approximation|Transcendental number theory}}
[[File:gamma-area.svg|right|thumb|The area of the blue region converges to the [[Euler–Mascheroni constant]], which may or may not be a rational number.]]
* [[Schanuel's conjecture]] on the [[transcendence degree]] of certain [[Field extension|field extensions]] of the rational numbers.<ref name="waldschmidt">{{citation |last=Waldschmidt |first=Michel |title=Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables |pages=14, 16 |year=2013 |url=https://books.google.com/books?id=Wrj0CAAAQBAJ&pg=PA14 |publisher=Springer |isbn=978-3-662-11569-5}}</ref> In particular: Are <math>\pi</math> and <math>e</math> [[Algebraic independence|algebraically independent]]? Which nontrivial combinations of [[Transcendental number|transcendental numbers]] (such as <math>e + \pi, e\pi, \pi^e, \pi^{\pi}, e^e</math>) are themselves transcendental?<ref>{{Cite conference |last=Waldschmidt |first=Michel |date=2008 |title=An introduction to irrationality and transcendence methods. |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/AWSLecture5.pdf |conference=2008 Arizona Winter School |archive-url=https://web.archive.org/web/20141216004531/http://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/AWSLecture5.pdf |archive-date=16 December 2014 |access-date=15 December 2014}}</ref><ref>{{Citation |last=Albert |first=John |title=Some unsolved problems in number theory |url=http://www2.math.ou.edu/~jalbert/courses/openprob2.pdf |access-date=15 December 2014 |archive-url=https://web.archive.org/web/20140117150133/http://www2.math.ou.edu/~jalbert/courses/openprob2.pdf |archive-date=17 January 2014}}</ref>
* The [[four exponentials conjecture]]: the transcendence of at least one of four exponentials of combinations of irrationals<ref name="waldschmidt" />
* Are [[Euler–Mascheroni constant|Euler's constant]] <math>\gamma</math> and [[Catalan's constant]] <math>G</math> irrational? Are they transcendental? Is [[Apéry's constant]] <math>\zeta(3)</math> transcendental?<ref>For some background on the numbers in this problem, see articles by [[Eric W. Weisstein]] at [[Wolfram MathWorld|''Wolfram'' ''MathWorld'']] (all articles accessed 22 August 2024): 

*[https://mathworld.wolfram.com/Euler-MascheroniConstant.html Euler's Constant]
*[https://mathworld.wolfram.com/CatalansConstant.html Catalan's Constant]
*[https://mathworld.wolfram.com/AperysConstant.html Apéry's Constant]
*[http://mathworld.wolfram.com/IrrationalNumber.html irrational numbers] ({{Webarchive|url=https://web.archive.org/web/20150327024040/http://mathworld.wolfram.com/IrrationalNumber.html|date=2015-03-27}})
*[http://mathworld.wolfram.com/TranscendentalNumber.html transcendental numbers] ({{Webarchive|url=https://web.archive.org/web/20141113174913/http://mathworld.wolfram.com/TranscendentalNumber.html|date=2014-11-13}})
*[http://mathworld.wolfram.com/IrrationalityMeasure.html irrationality measures] ({{Webarchive|url=https://web.archive.org/web/20150421203736/http://mathworld.wolfram.com/IrrationalityMeasure.html|date=2015-04-21}})</ref><ref name=":1">{{Cite arXiv |last=Waldschmidt |first=Michel |date=2003-12-24 |title=Open Diophantine Problems |eprint=math/0312440 |language=en}}</ref>
* Which transcendental numbers are [[Period (algebraic geometry)|(exponential) periods]]?<ref>{{Citation |last1=Kontsevich |first1=Maxim |title=Periods |date=2001 |work=Mathematics Unlimited — 2001 and Beyond |pages=771–808 |editor-last=Engquist |editor-first=Björn |url=https://link.springer.com/chapter/10.1007/978-3-642-56478-9_39 |access-date=2024-08-22 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-642-56478-9_39 |isbn=978-3-642-56478-9 |last2=Zagier |first2=Don |editor2-last=Schmid |editor2-first=Wilfried}}</ref>
* How well can [[Quadratic equation|non-quadratic]] irrational numbers be approximated? What is the [[irrationality measure]] of specific (suspected) transcendental numbers such as <math>\pi</math> and <math>\gamma</math>?<ref name=":1" />
* Which irrational numbers have [[simple continued fraction]] terms whose [[geometric mean]] converges to [[Khinchin's constant]]?<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Khinchin's Constant |url=https://mathworld.wolfram.com/KhinchinsConstant.html |access-date=2024-09-22 |website=mathworld.wolfram.com |language=en}}</ref>

==== Diophantine equations ====
{{Further|Diophantine equation}}
* [[Beal's conjecture]]: for all integral solutions to <math>A^x + B^y = C^z</math> where <math>x, y, z > 2</math>, all three numbers <math>A, B, C</math> must share some prime factor.
* [[Congruent number problem]] (a corollary to [[Birch and Swinnerton-Dyer conjecture]], per [[Tunnell's theorem]]): determine precisely what rational numbers are [[congruent number]]s.
* Erdős–Moser problem: is <math>1^1 + 2^1 = 3^1</math> the only solution to the [[Erdős–Moser equation]]?
* [[Erdős–Straus conjecture]]: for every <math>n \geq 2</math>, there are positive integers <math>x, y, z</math> such that <math>4/n = 1/x + 1/y + 1/z</math>.
* [[Fermat–Catalan conjecture]]: there are finitely many distinct solutions <math>(a^m, b^n, c^k)</math> to the equation <math>a^m + b^n = c^k</math> with <math>a, b, c</math> being positive [[coprime integers]] and <math>m, n, k</math> being positive integers satisfying <math>1/m + 1/n + 1/k < 1</math>.
* [[Goormaghtigh conjecture]] on solutions to <math>(x^m - 1)/(x - 1) = (y^n - 1)/(y - 1)</math> where <math>x > y > 1</math> and <math>m, n > 2</math>.
* The [[Markov number#Other properties|uniqueness conjecture for Markov numbers]]<ref>{{citation
 | last = Aigner | first = Martin
 | doi = 10.1007/978-3-319-00888-2
 | isbn = 978-3-319-00887-5
 | location = Cham
 | mr = 3098784
 | publisher = Springer
 | title = Markov's theorem and 100 years of the uniqueness conjecture
 | year = 2013}}</ref> that every [[Markov number]] is the largest number in exactly one normalized solution to the Markov [[Diophantine equation]].
* [[Pillai's conjecture]]: for any <math>A, B, C</math>, the equation <math>Ax^m - By^n = C</math> has finitely many solutions when <math>m, n</math> are not both <math>2</math>.
* Which integers can be written as the [[Sums of three cubes|sum of three perfect cubes]]?<ref>{{Cite arXiv |eprint = 1604.07746|last1 = Huisman |first1 = Sander G.|title = Newer sums of three cubes|class = math.NT|year = 2016}}</ref>
* [[Sum of four cubes problem|Can every integer be written as a sum of four perfect cubes?]]

==== Prime numbers ====
{{Main|Prime numbers}}
{{Prime number conjectures}}
[[File:Goldbach partitions of the even integers from 4 to 50 rev4b.svg|thumb=Goldbach_partitions_of_the_even_integers_from_4_to_28_300px.png|[[Goldbach's conjecture]] states that all even integers greater than 2 can be written as the sum of two primes. Here this is illustrated for the even integers from 4 to 28.]]
* [[Agoh–Giuga conjecture]] on the [[Bernoulli number]]s that <math>p</math> is prime if and only if <math>pB_{p-1} \equiv -1 \pmod p</math>
* [[Agrawal's conjecture]] that given [[Coprime integers|coprime positive integers]] <math>n</math> and <math>r</math>, if <math>(X - 1)^n \equiv X^n - 1 \pmod{n, X^r - 1}</math>, then either <math>n</math> is prime or <math>n^{2} \equiv 1 \pmod{r}</math>
* [[Artin's conjecture on primitive roots]] that if an integer is neither a perfect square nor <math>-1</math>, then it is a [[Primitive root modulo n|primitive root]] modulo infinitely many [[prime number]]s <math>p</math>
* [[Brocard's conjecture]]: there are always at least <math>4</math> [[prime number]]s between consecutive squares of prime numbers, aside from <math>2^{2}</math> and <math>3^{2}</math>.
* [[Bunyakovsky conjecture]]: if an integer-coefficient polynomial <math>f</math> has a positive leading coefficient, is irreducible over the integers, and has no common factors over all <math>f(x)</math> where <math>x</math> is a positive integer, then <math>f(x)</math> is prime infinitely often.
* [[Catalan's Mersenne conjecture]]: some [[Double Mersenne number#Catalan–Mersenne number conjecture|Catalan–Mersenne number]] is composite and thus all Catalan–Mersenne numbers are composite after some point.
* [[Dickson's conjecture]]: for a finite set of linear forms <math>a_1 + b_1 n, \ldots, a_k + b_k n</math> with each <math>b_i \geq 1</math>, there are infinitely many <math>n</math> for which all forms are [[prime number|prime]], unless there is some [[Modular arithmetic|congruence]] condition preventing it.
* Dubner's conjecture: every even number greater than <math>4208</math> is the sum of two [[prime number|primes]] which both have a [[Twin prime|twin]].
* [[Elliott–Halberstam conjecture]] on the distribution of [[prime number]]s in [[arithmetic progression]]s.
* [[Powerful number#Mathematical properties|Erdős–Mollin–Walsh conjecture]]: no three consecutive numbers are all [[Powerful number|powerful]].
* [[Feit–Thompson conjecture]]: for all distinct [[prime number]]s <math>p</math> and <math>q</math>, <math>(p^q - 1)/(p - 1)</math> does not divide <math>(q^p - 1)/(q - 1)</math>
* Fortune's conjecture that no [[Fortunate number]] is composite.
* The [[Gaussian moat]] problem: is it possible to find an infinite sequence of distinct [[Gaussian prime number]]s such that the difference between consecutive numbers in the sequence is bounded?
* [[Gillies' conjecture]] on the distribution of [[Prime number|prime]] divisors of [[Mersenne prime|Mersenne numbers]].
* [[Landau's problems]]
** [[Goldbach conjecture]]: all even [[natural number]]s greater than <math>2</math> are the sum of two [[prime number]]s.
** [[Legendre's conjecture]]: for every positive integer <math>n</math>, there is a prime between <math>n^{2}</math> and <math>(n+1)^{2}</math>.
** [[Twin prime#Twin prime conjecture|Twin prime conjecture]]: there are infinitely many [[twin prime]]s.
** Are there infinitely many primes of the form <math>n^{2} + 1</math>?
* Problems associated to [[Linnik's theorem]]
* [[Mersenne conjectures#New Mersenne conjecture|New Mersenne conjecture]]: for any odd [[natural number]] <math>p</math>, if any two of the three conditions <math>p = 2^k \pm 1</math> or <math>p = 4^k \pm 3</math>, <math>2^p - 1</math> is prime, and <math>(2^{p} + 1)/3</math> is prime are true, then the third condition is also true.
* [[Polignac's conjecture]]: for all positive even numbers <math>n</math>, there are infinitely many [[prime gap]]s of size <math>n</math>.
* [[Schinzel's hypothesis H]] that for every finite collection <math>\{f_1, \ldots, f_k\}</math> of nonconstant [[irreducible polynomial]]s over the integers with positive leading coefficients, either there are infinitely many positive integers <math>n</math> for which <math>f_1(n), \ldots, f_k(n)</math> are all [[prime number|primes]], or there is some fixed divisor <math>m > 1</math> which, for all <math>n</math>, divides some <math>f_i(n)</math>.
* [[Sierpiński number|Selfridge's conjecture]]: is 78,557 the lowest [[Sierpiński number]]?
* Does the [[Wolstenholme's theorem#The converse as a conjecture|converse of Wolstenholme's theorem]] hold for all natural numbers?
* Are all [[Euclid number]]s [[Square-free integer|square-free]]?
* Are all [[Fermat number]]s [[Square-free integer|square-free]]?
* Are all [[Mersenne number]]s of prime index [[Square-free integer|square-free]]?
* Are there any composite ''c'' satisfying 2<sup>''c'' − 1</sup> ≡ 1 (mod ''c''<sup>2</sup>)?
* Are there any [[Wall–Sun–Sun prime]]s?
* Are there any [[Wieferich prime]]s in base 47?
* Are there infinitely many [[balanced prime]]s?
* Are there infinitely many Carol primes?
* Are there infinitely many [[cluster prime]]s?
* Are there infinitely many [[cousin prime]]s?
* Are there infinitely many [[Cullen number|Cullen prime]]s?
* Are there infinitely many [[Euclid number|Euclid prime]]s?
* Are there infinitely many [[Fibonacci prime]]s?
* Are there infinitely many [[Euclid number#Generalization|Kummer prime]]s?
* Are there infinitely many Kynea primes?
* Are there infinitely many [[Lucas number#Lucas primes|Lucas prime]]s?
* Are there infinitely many [[Mersenne prime]]s ([[Lenstra–Pomerance–Wagstaff conjecture]]); equivalently, infinitely many even [[perfect number]]s?
* Are there infinitely many [[Newman–Shanks–Williams prime]]s?
* Are there infinitely many [[palindromic prime]]s to every base?
* Are there infinitely many [[Pell number|Pell primes]]?
* Are there infinitely many [[Pierpont prime]]s?
* Are there infinitely many [[prime quadruplet]]s?
* Are there infinitely many [[prime triplet]]s?
* [[Regular prime|Siegel's conjecture]]: are there infinitely many regular primes, and if so is their [[natural density]] as a subset of all primes <math>e^{-1/2}</math>?
* Are there infinitely many [[sexy prime]]s?
* Are there infinitely many [[safe and Sophie Germain primes]]?
* Are there infinitely many [[Wagstaff prime]]s?
* Are there infinitely many [[Wieferich prime]]s?
* Are there infinitely many [[Wilson prime]]s?
* Are there infinitely many [[Wolstenholme prime]]s?
* Are there infinitely many [[Woodall number#Woodall primes|Woodall prime]]s?
* Can a prime ''p'' satisfy <math>2^{p-1}\equiv 1\pmod{p^2}</math> and <math>3^{p-1}\equiv 1\pmod{p^2}</math> simultaneously?<ref>{{cite arXiv |last=Dobson |first= J. B. |date=1 April 2017 |title=On Lerch's formula for the Fermat quotient |eprint=1103.3907v6|page=23|mode=cs2|class= math.NT }}</ref>
* Does every prime number appear in the [[Euclid–Mullin sequence]]?
* What is the smallest [[Skewes's number]]?
* For any given integer ''a'' > 0, are there infinitely many [[Lucas–Wieferich prime]]s associated with the pair (''a'', −1)? (Specially, when ''a'' = 1, this is the Fibonacci-Wieferich primes, and when ''a'' = 2, this is the Pell-Wieferich primes)
* For any given integer ''a'' > 0, are there infinitely many primes ''p'' such that ''a''<sup>''p'' − 1</sup> ≡ 1 (mod ''p''<sup>2</sup>)?<ref>{{cite book |last=Ribenboim |first=P. |author-link=Paulo Ribenboim |date=2006 |title=Die Welt der Primzahlen |edition=2nd |language=de |publisher=Springer |doi=10.1007/978-3-642-18079-8 |isbn=978-3-642-18078-1 |pages=242–243 |url=https://books.google.com/books?id=XMyzh-2SClUC&q=die+folgenden+probleme+sind+ungel%C3%B6st&pg=PA242|series=Springer-Lehrbuch }}</ref>
* For any given integer ''b'' which is not a perfect power and not of the form −4''k''<sup>4</sup> for integer ''k'', are there infinitely many [[repunit]] primes to base ''b''?
* For any given integers <math>k\geq 1, b\geq 2, c\neq 0</math>, with {{nowrap|1=gcd(''k'', ''c'') = 1}} and {{nowrap|1=gcd(''b'', ''c'') = 1,}} are there infinitely many primes of the form <math>(k\times b^n+c)/\gcd(k+c,b-1)</math> with integer ''n'' ≥ 1?
* Is every [[Fermat number]] <math>2^{2^n} + 1</math> composite for <math>n > 4</math>?
* Is 509,203 the lowest [[Riesel number]]?

=== Set theory ===
{{Main|Set theory}}

Note: These conjectures are about [[model theory|models]] of [[Zermelo-Frankel set theory]] with [[axiom of choice|choice]], and may not be able to be expressed in models of other set theories such as the various [[constructive set theory|constructive set theories]] or [[non-wellfounded set theory]].
* ([[W. Hugh Woodin|Woodin]]) Does the [[generalized continuum hypothesis]] below a [[strongly compact cardinal]] imply the [[generalized continuum hypothesis]] everywhere?
* Does the [[generalized continuum hypothesis]] entail [[Diamondsuit|<math>{\diamondsuit(E^{\lambda^+}_{\operatorname{cf}(\lambda)}})</math>]] for every [[singular cardinal]] <math>\lambda</math>?
* Does the [[generalized continuum hypothesis]] imply the existence of an [[Suslin tree|ℵ<sub>2</sub>-Suslin tree]]?
* If ℵ<sub>ω</sub> is a strong limit cardinal, is <math>2^{\aleph_\omega} < \aleph_{\omega_1}</math> (see [[Singular cardinals hypothesis]])? The best bound, ℵ<sub>ω<sub>4</sub></sub>, was obtained by [[Saharon Shelah|Shelah]] using his [[PCF theory]].
* The problem of finding the ultimate [[core model]], one that contains all [[Large cardinal property|large cardinals]].
* [[W. Hugh Woodin|Woodin's]] [[Ω-logic|Ω-conjecture]]: if there is a [[Class (set theory)|proper class]] of [[Woodin cardinal]]s, then [[Ω-logic]] satisfies an analogue of [[Gödel's completeness theorem]].
* Does the [[consistency]] of the existence of a [[strongly compact cardinal]] imply the consistent existence of a [[supercompact cardinal]]?
* Does there exist a [[Jónsson cardinal|Jónsson algebra]] on ℵ<sub>ω</sub>?
* Is OCA (the [[open coloring axiom]]) consistent with <math>2^{\aleph_{0}}>\aleph_{2}</math>?
* [[Reinhardt cardinal]]s: Without assuming the [[axiom of choice]], can a [[Reinhardt cardinal|nontrivial elementary embedding]] ''V''→''V'' exist?

===Topology===
{{Main|Topology}}
[[File:Ochiai_unknot.svg|thumb|The [[unknotting problem]] asks whether there is an efficient algorithm to identify when the shape presented in a [[knot diagram]] is actually the [[unknot]].]]
* [[Baum–Connes conjecture]]: the [[Baum–Connes conjecture#Formulation|assembly map]] is an [[isomorphism]].
* [[Berge knot|Berge conjecture]] that the only [[Knot (mathematics)|knots]] in the [[3-sphere]] which admit [[lens space]] [[Dehn surgery|surgeries]] are [[Berge knot]]s.
* [[Bing–Borsuk conjecture]]: every <math>n</math>-dimensional [[Homogeneous space|homogeneous]] [[Retraction (topology)|absolute neighborhood retract]] is a [[topological manifold]].
* [[Borel conjecture]]: [[Aspherical space|aspherical]] [[closed manifold]]s are determined up to [[homeomorphism]] by their [[fundamental group]]s.
* [[Halperin conjecture]] on rational [[Serre spectral sequence]]s of certain [[fibration]]s.
* [[Hilbert–Smith conjecture]]: if a [[Locally compact space|locally compact]] [[topological group]] has a [[Continuous function|continuous]], [[Group action#Remarkable properties of actions|faithful group action]] on a [[topological manifold]], then the group must be a [[Lie group]].
* Mazur's conjectures<ref>{{citation |last=Mazur |first=Barry |author-link=Barry Mazur |title=The topology of rational points |journal=[[Experimental Mathematics (journal)|Experimental Mathematics]] |volume=1 |number=1 |year=1992 |pages=35–45 |doi=10.1080/10586458.1992.10504244 |s2cid=17372107 |url=https://projecteuclid.org/euclid.em/1048709114 |access-date=2019-04-07 |archive-url=https://web.archive.org/web/20190407161124/https://projecteuclid.org/euclid.em/1048709114 |archive-date=2019-04-07 |url-status=live }}</ref>
* [[Novikov conjecture]] on the [[Homotopy#Invariance|homotopy invariance]] of certain [[polynomial]]s in the [[Pontryagin class]]es of a [[manifold]], arising from the [[fundamental group]].
* [[Quadrisecant]]s of [[wild knot]]s: it has been conjectured that wild knots always have infinitely many quadrisecants.<ref>{{citation
 | last = Kuperberg | first = Greg | author-link = Greg Kuperberg
 | arxiv = math/9712205
 | doi = 10.1142/S021821659400006X
 | journal = [[Journal of Knot Theory and Its Ramifications]]
 | mr = 1265452
 | pages = 41–50
 | title = Quadrisecants of knots and links
 | volume = 3
 | year = 1994| s2cid = 6103528 }}</ref>
* [[Ravenel conjectures|Telescope conjecture]]: the last of [[Ravenel's conjectures]] in [[stable homotopy theory]] to be resolved.{{efn|A disproof has been announced, with a preprint made available on [[arXiv]].<ref>{{cite arXiv |last1=Burklund |first1=Robert |last2=Hahn |first2=Jeremy |last3=Levy |first3=Ishan |last4=Schlank |first4=Tomer |title=K-theoretic counterexamples to Ravenel's telescope conjecture |date=2023 |class=math.AT |eprint=2310.17459 }}</ref>}}
* [[Unknotting problem]]: can [[unknot]]s be recognized in [[Time complexity#Polynomial time|polynomial time]]?
* [[Volume conjecture]] relating [[quantum invariant]]s of [[Knot (mathematics)|knots]] to the [[hyperbolic geometry]] of their [[knot complement]]s.
* [[Whitehead conjecture]]: every [[Connectedness|connected]] [[CW complex#Inductive construction of CW complexes|subcomplex]] of a two-dimensional [[Aspherical space|aspherical]] [[CW complex]] is aspherical.
* [[Zeeman conjecture]]: given a finite [[Contractible space|contractible]] two-dimensional [[CW complex]] <math>K</math>, is the space <math>K \times [0, 1]</math> [[Collapse (topology)|collapsible]]?

== Problems solved since 1995 ==
[[File:Ricci flow.png|thumb|[[Ricci flow]], here illustrated with a 2D manifold, was the key tool in [[Grigori Perelman]]'s [[Poincaré conjecture#Solution|solution of the Poincaré conjecture]].]]

===Algebra===
* [[Uniform boundedness conjecture for rational points#Mazur's conjecture B|Mazur's conjecture B]] (Vessilin Dimitrov, Ziyang Gao, and Philipp Habegger, 2020)<ref>{{cite journal
 |first1=Vessilin
 |last1=Dimitrov
 |first2=Ziyang
 |last2=Gao
 |first3=Philipp
 |last3=Habegger
 |title=Uniformity in Mordell–Lang for curves
 |journal = [[Annals of Mathematics]]
 |volume = 194
 |year=2021
 |pages=237–298
 |doi=10.4007/annals.2021.194.1.4
 |arxiv=2001.10276
 |s2cid=210932420
 |url=https://hal.sorbonne-universite.fr/hal-03374335/file/Dimitrov%20et%20al.%20-%202021%20-%20Uniformity%20in%20Mordell%E2%80%93Lang%20for%20curves.pdf}}
</ref>
* [[Suita conjecture]] (Qi'an Guan and [[Xiangyu Zhou]], 2015) <ref>{{cite journal
 | jstor=24523356
 | last1=Guan
 | first1=Qi'an
 | last2=Zhou
 | first2=Xiangyu
 | author2-link=Xiangyu Zhou
 | title=A solution of an <math>L^2</math> extension problem with optimal estimate and applications
 | journal=Annals of Mathematics
 | year=2015
 | volume=181
 | issue=3
 | pages=1139–1208
 | doi=10.4007/annals.2015.181.3.6
 | s2cid=56205818
 | arxiv=1310.7169}}
</ref>
* [[Torsion conjecture]] ([[Loïc Merel]], 1996)<ref>{{cite journal
 | last1 = Merel
 | first1 = Loïc
 | year = 1996
 | title = "Bornes pour la torsion des courbes elliptiques sur les corps de nombres" [Bounds for the torsion of elliptic curves over number fields]
 | journal = Inventiones Mathematicae
 | volume = 124
 | issue = 1
 | pages = 437–449
 | doi = 10.1007/s002220050059
 | mr = 1369424
 | bibcode = 1996InMat.124..437M
 | s2cid = 3590991 }}
</ref>
* [[Carlitz–Wan conjecture]] ([[Hendrik Lenstra]], 1995)<ref>{{citation
 | last1=Cohen
 | first1=Stephen D.
 | last2=Fried
 | first2=Michael D.
 | author2-link=Michael D. Fried
 | doi=10.1006/ffta.1995.1027
 | issue=3
 | journal=Finite Fields and Their Applications
 | mr=1341953
 | pages=372–375
 | title=Lenstra's proof of the Carlitz–Wan conjecture on exceptional polynomials: an elementary version
 | volume=1
 | year=1995
 | doi-access=free}}
</ref>
* [[Serre's multiplicity conjectures#Nonnegativity|Serre's nonnegativity conjecture]] ([[Ofer Gabber]], 1995)

===Analysis===
* [[Kadison–Singer problem]] ([[Adam Marcus (mathematician)|Adam Marcus]], [[Daniel Spielman]] and [[Nikhil Srivastava]], 2013)<ref name=Casazza2006>{{cite book|last1=Casazza|first1=Peter G.|last2=Fickus|first2=Matthew|last3=Tremain|first3=Janet C.|last4=Weber|first4=Eric|editor1-last=Han|editor1-first=Deguang|editor2-last=Jorgensen|editor2-first=Palle E. T.|editor3-last=Larson|editor3-first=David Royal|contribution=The Kadison-Singer problem in mathematics and engineering: A detailed account|series=Contemporary Mathematics|date=2006|volume=414|pages=299–355|contribution-url=https://books.google.com/books?id=9b-4uqEGJdoC&pg=PA299|access-date=24 April 2015|title=Large Deviations for Additive Functionals of Markov Chains: The 25th Great Plains Operator Theory Symposium, June 7–12, 2005, University of Central Florida, Florida|publisher=American Mathematical Society.|isbn=978-0-8218-3923-2|doi=10.1090/conm/414/07820}}</ref><ref name=SIAM02.2014>{{cite news|last1=Mackenzie|first1=Dana|title=Kadison–Singer Problem Solved|url=https://www.siam.org/pdf/news/2123.pdf|access-date=24 April 2015|work=SIAM News|issue=January/February 2014|publisher=[[Society for Industrial and Applied Mathematics]]|archive-url=https://web.archive.org/web/20141023120958/http://www.siam.org/pdf/news/2123.pdf|archive-date=23 October 2014|url-status=live}}</ref> (and the [[Hans Georg Feichtinger#Feichtinger's conjecture|Feichtinger's conjecture]], Anderson's paving conjectures, Weaver's discrepancy theoretic <math>KS_r</math> and <math>KS'_r</math> conjectures, Bourgain-Tzafriri conjecture and <math>R_\varepsilon</math>-conjecture)
* [[Ahlfors measure conjecture]] ([[Ian Agol]], 2004)<ref name="Agol">{{cite arXiv | eprint = math/0405568|last1 = Agol |first1 = Ian|title = Tameness of hyperbolic 3-manifolds|year = 2004}}</ref>
* [[Gradient conjecture]] (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)<ref>{{Cite journal
| arxiv=math/9906212
| last1=Kurdyka | first1=Krzysztof
| last2=Mostowski | first2=Tadeusz
| last3=Parusiński | first3=Adam
| title = Proof of the gradient conjecture of R. Thom
| journal=Annals of Mathematics
| pages=763–792
| volume=152
| date=2000
| issue=3
| doi=10.2307/2661354| jstor=2661354 | s2cid=119137528 }}</ref>

===Combinatorics===
* [[Erdős sumset conjecture]] (Joel Moreira, Florian Richter, Donald Robertson, 2018)<ref>{{Cite journal |last1=Moreira |first1=Joel |last2=Richter |first2=Florian K. |last3=Robertson |first3=Donald |title=A proof of a sumset conjecture of Erdős |journal=[[Annals of Mathematics]] |doi=10.4007/annals.2019.189.2.4 |volume=189 |number=2 |pages=605–652 |language=en-US|year=2019 |arxiv=1803.00498 |s2cid=119158401 }}</ref>
* [[Simplicial sphere|McMullen's g-conjecture]] on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim Adiprasito, 2018)<ref>{{citation|last=Stanley|first=Richard P. |editor1-last=Bisztriczky|editor1-first=T.|editor2-last=McMullen|editor2-first=P.|editor3-last=Schneider|editor3-first=R.|editor4-last=Weiss|editor4-first=A. IviÄ‡|contribution=A survey of Eulerian posets|location=Dordrecht|mr=1322068|pages=301–333 |publisher=Kluwer Academic Publishers|series=NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences |title=Polytopes: abstract, convex and computational (Scarborough, ON, 1993)|volume=440|year=1994}}. See in particular [https://books.google.com/books?id=gHjrCAAAQBAJ&pg=PA316 p.&nbsp;316].</ref><ref>{{cite web |last1=Kalai |first1=Gil |title=Amazing: Karim Adiprasito proved the g-conjecture for spheres! |url=https://gilkalai.wordpress.com/2018/12/25/amazing-karim-adiprasito-proved-the-g-conjecture-for-spheres/ |access-date=2019-02-15 |archive-url=https://web.archive.org/web/20190216031650/https://gilkalai.wordpress.com/2018/12/25/amazing-karim-adiprasito-proved-the-g-conjecture-for-spheres/ |archive-date=2019-02-16 |url-status=live |date=2018-12-25 }}</ref>
* [[Hirsch conjecture]] ([[Francisco Santos Leal]], 2010)<ref>{{cite journal |last=Santos |first=Franciscos |date=2012 |title=A counterexample to the Hirsch conjecture |journal=Annals of Mathematics |volume=176 |issue=1 |pages=383–412 |doi=10.4007/annals.2012.176.1.7 |arxiv=1006.2814 |s2cid=15325169 }}</ref><ref>{{cite journal |last=Ziegler |first=Günter M. |date=2012 |title=Who solved the Hirsch conjecture? |journal=Documenta Mathematica |series=Documenta Mathematica Series |volume=6 |issue=Extra Volume "Optimization Stories" |pages=75–85 |doi=10.4171/dms/6/13 |doi-access=free |isbn=978-3-936609-58-5 | url=https://www.math.uni-bielefeld.de/documenta/vol-ismp/22_ziegler-guenter.html}}</ref>
* [[Ira Gessel#Gessel's lattice path conjecture|Gessel's lattice path conjecture]] ([[Manuel Kauers]], [[Christoph Koutschan]], and [[Doron Zeilberger]], 2009)<ref>{{cite journal | last1=Kauers | first1=Manuel | author1-link=Manuel Kauers | last2=Koutschan | first2=Christoph | author2-link=Christoph Koutschan | last3=Zeilberger | first3=Doron | author3-link=Doron Zeilberger | title=Proof of Ira Gessel's lattice path conjecture | journal=Proceedings of the National Academy of Sciences | volume=106 | issue=28 | date=2009-07-14 | issn=0027-8424 | doi=10.1073/pnas.0901678106 | pages=11502–11505 | pmc=2710637 | arxiv=0806.4300 | bibcode=2009PNAS..10611502K | doi-access=free }}</ref>
* [[Stanley–Wilf conjecture]] ([[Gábor Tardos]] and [[Adam Marcus (mathematician)|Adam Marcus]], 2004)<ref>{{cite journal |last1=Chung |first1=Fan |last2=Greene |first2=Curtis |last3=Hutchinson |first3=Joan |date=April 2015 |title=Herbert S. Wilf (1931–2012) |journal=[[Notices of the AMS]] |volume=62 |issue=4 |page=358 |issn=1088-9477 |oclc=34550461 |quote=The conjecture was finally given an exceptionally elegant proof by A. Marcus and G. Tardos in 2004. |doi=10.1090/noti1247 |doi-access=free }}</ref> (and also the Alon–Friedgut conjecture)
* [[Kemnitz's conjecture]] ([[Christian Reiher]], 2003, Carlos di Fiore, 2003)<ref>{{cite journal|title=Kemnitz' conjecture revisited | doi=10.1016/j.disc.2005.02.018 |doi-access=free| volume=297|issue=1–3 |journal=Discrete Mathematics|pages=196–201|year=2005 | last1 = Savchev | first1 = Svetoslav}}</ref>
* [[Cameron–Erdős conjecture]] ([[Ben J. Green]], 2003, Alexander Sapozhenko, 2003)<ref>{{cite journal | last = Green | first = Ben | author-link = Ben J. Green | arxiv = math.NT/0304058 | doi = 10.1112/S0024609304003650 | issue = 6 | journal = The Bulletin of the London Mathematical Society | mr = 2083752 | pages = 769–778 | title = The Cameron–Erdős conjecture | volume = 36 | year = 2004| s2cid = 119615076 }}</ref><ref>{{cite web |url=https://www.ams.org/news?news_id=155 |title=News from 2007 |author=<!--Staff writer(s); no by-line.--> |date=31 December 2007 |website=American Mathematical Society |publisher=AMS |access-date=2015-11-13 |quote=The 2007 prize also recognizes Green for "his many outstanding results including his resolution of the Cameron-Erdős conjecture..." |archive-url=https://web.archive.org/web/20151117030726/http://www.ams.org/news?news_id=155 |archive-date=17 November 2015 |url-status=live }}</ref>

===Dynamical systems===
* [[Zimmer's conjecture]] (Aaron Brown, David Fisher, and Sebastián Hurtado-Salazar, 2017)<ref>{{cite arXiv
 | last1=Brown
 | first1=Aaron
 | last2=Fisher
 | first2=David
 | last3=Hurtado
 | first3=Sebastian
 | date=2017-10-07
 | title=Zimmer's conjecture for actions of {{not a typo|SL(𝑚,ℤ)}}
 | eprint=1710.02735
 | class=math.DS}}
</ref>
* [[Painlevé conjecture]] (Jinxin Xue, 2014)<ref name="Xue1">{{Cite arXiv|title=Noncollision Singularities in a Planar Four-body Problem|last=Xue|first=Jinxin|date=2014|class=math.DS |eprint = 1409.0048}}</ref><ref name="Xue2">{{Cite journal|title=Non-collision singularities in a planar 4-body problem|last=Xue|first=Jinxin|date=2020|journal=[[Acta Mathematica]]|volume=224|issue=2|pages=253–388|doi=10.4310/ACTA.2020.v224.n2.a2|s2cid=226420221}}</ref>

===Game theory===
* Existence of a non-terminating game of [[beggar-my-neighbour]] (Brayden Casella, 2024)<ref>
{{cite web | url= https://richardpmann.com/beggar-my-neighbour-records.html | title= Known Historical Beggar-My-Neighbour Records |author=  Richard P Mann |access-date= 2024-02-10 }}</ref> 
* The [[angel problem]] (Various independent proofs, 2006)<ref>{{Cite web |url=http://homepages.warwick.ac.uk/~masgak/papers/bhb-angel.pdf |title=The angel game in the plane |first=Brian H. |last=Bowditch|date=2006|location=School of Mathematics, [[University of Southampton]] |publisher=warwick.ac.uk [[Warwick University]]|access-date=2016-03-18 |archive-url=https://web.archive.org/web/20160304185616/http://homepages.warwick.ac.uk/~masgak/papers/bhb-angel.pdf |archive-date=2016-03-04 |url-status=live }}</ref><ref>{{Cite web |url=http://home.broadpark.no/~oddvark/angel/Angel.pdf |title=A Solution to the Angel Problem |first=Oddvar |last=Kloster |publisher=SINTEF ICT |location=Oslo, Norway|access-date=2016-03-18 |archive-url=https://web.archive.org/web/20160107125925/http://home.broadpark.no/~oddvark/angel/Angel.pdf |archive-date=2016-01-07 }}</ref><ref>{{Cite journal |url=http://homepages.warwick.ac.uk/~masibe/angel-mathe.pdf |title=The Angel of power 2 wins |first=Andras |last=Mathe |date=2007|journal=[[Combinatorics, Probability and Computing]] |volume=16 |number=3|pages= 363–374|doi=10.1017/S0963548306008303 |doi-broken-date=1 November 2024 |s2cid=16892955 |access-date=2016-03-18 |archive-url=https://web.archive.org/web/20161013034302/http://homepages.warwick.ac.uk/~masibe/angel-mathe.pdf |archive-date=2016-10-13 |url-status=live }}</ref><ref>{{Cite web |last=Gacs |first=Peter |date=June 19, 2007 |title=THE ANGEL WINS |url=http://www.cs.bu.edu/~gacs/papers/angel.pdf |archive-url=https://web.archive.org/web/20160304030433/http://www.cs.bu.edu/~gacs/papers/angel.pdf |archive-date=2016-03-04 |access-date=2016-03-18}}</ref>

===Geometry===
====21st century====
* [[Einstein problem]] (David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, 2024)<ref>{{Cite journal |last1=Smith |first1=David |last2=Myers |first2=Joseph Samuel |last3=Kaplan |first3=Craig S. |last4=Goodman-Strauss |first4=Chaim |date=2024 |title=An aperiodic monotile |url=https://escholarship.org/uc/item/3317z9z9 |journal=Combinatorial Theory |language=en |volume=4 |issue=1 |doi=10.5070/C64163843 |issn=2766-1334}}</ref>
* [[Maximal rank conjecture]] (Eric Larson, 2018)<ref>{{Cite arXiv| eprint=1711.04906 | last1=Larson | first1=Eric | title=The Maximal Rank Conjecture | year=2017 | class=math.AG }}</ref>
* [[Weibel's conjecture]] (Moritz Kerz, Florian Strunk, and Georg Tamme, 2018)<ref>{{citation
 | first1=Moritz
 | last1=Kerz
 | first2=Florian
 | last2=Strunk
 | first3=Georg
 | last3=Tamme
 | title=Algebraic ''K''-theory and descent for blow-ups
 | journal=[[Inventiones Mathematicae]]
 | volume=211
 | year=2018
 | issue=2
 | pages=523–577
 | mr=3748313
 | doi=10.1007/s00222-017-0752-2
 | arxiv=1611.08466| bibcode=2018InMat.211..523K
 | s2cid=253741858
 }}
</ref>
* [[Yau's conjecture]] ([[Antoine Song]], 2018)<ref>{{cite web
 | url = https://www.ams.org/amsmtgs/2251_abstracts/1147-53-499.pdf 
 | title =  Existence of infinitely many minimal hypersurfaces in closed manifolds.
 | author =  Song, Antoine
 | work = www.ams.org
 | quote = "..I will present a solution of the conjecture, which builds on min-max methods developed by F. C. Marques and A. Neves.."
 |  access-date = 19 June 2021}}
</ref><ref>{{Cite web
 | url=https://www.claymath.org/people/antoine-song
 | title = Antoine Song &#124; Clay Mathematics Institute
 | quote="...Building on work of Codá Marques and Neves, in 2018 Song proved Yau's conjecture in complete generality"}}
</ref>
* [[Pentagonal tiling]] (Michaël Rao, 2017)<ref>{{citation|url=https://www.quantamagazine.org/pentagon-tiling-proof-solves-century-old-math-problem-20170711/|magazine=[[Quanta Magazine]]|title=Pentagon Tiling Proof Solves Century-Old Math Problem|first=Natalie|last=Wolchover|date=July 11, 2017|access-date=July 18, 2017|archive-url=https://web.archive.org/web/20170806093353/https://www.quantamagazine.org/pentagon-tiling-proof-solves-century-old-math-problem-20170711/|archive-date=August 6, 2017}}</ref>
* [[Willmore conjecture]] ([[Fernando Codá Marques]] and [[André Neves]], 2012)<ref>{{cite journal|last1=Marques |first1=Fernando C.|first2=André|last2=Neves|title=Min-max theory and the Willmore conjecture|journal=Annals of Mathematics |year=2013|arxiv=1202.6036|doi=10.4007/annals.2014.179.2.6|volume=179|issue=2|pages=683–782|s2cid=50742102}}</ref>
* [[Erdős distinct distances problem]] ([[Larry Guth]], [[Nets Katz|Nets Hawk Katz]], 2011)<ref>{{cite journal
| arxiv=1011.4105
| last1=Guth | first1=Larry
| last2=Katz | first2=Nets Hawk
| title=On the Erdos distinct distance problem in the plane
| journal=Annals of Mathematics
| pages=155–190
| volume=181
| date=2015
| issue=1
| doi=10.4007/annals.2015.181.1.2 | doi-access=free}}</ref>
* [[Squaring the plane|Heterogeneous tiling conjecture (squaring the plane)]] (Frederick V. Henle and James M. Henle, 2008)<ref>{{Cite web |url=http://www.ww.amc12.org/sites/default/files/pdf/pubs/SquaringThePlane.pdf |title=Squaring the Plane |first1=Frederick V. |last1=Henle |first2=James M. |last2=Henle |access-date=2016-03-18 |publisher=www.maa.org [[Mathematics Association of America]]|archive-url=https://web.archive.org/web/20160324074609/http://www.ww.amc12.org/sites/default/files/pdf/pubs/SquaringThePlane.pdf |archive-date=2016-03-24 |url-status=live }}</ref>
* [[Tameness conjecture]] ([[Ian Agol]], 2004)<ref name="Agol"/>
* [[Ending lamination theorem]] ([[Jeffrey Brock|Jeffrey F. Brock]], [[Richard Canary|Richard D. Canary]], [[Yair Minsky|Yair N. Minsky]], 2004)<ref>{{Cite journal
| arxiv=math/0412006
| last1=Brock | first1=Jeffrey F.
| last2=Canary | first2=Richard D.
| last3=Minsky | first3=Yair N. | author-link3=Yair Minsky
| title=The classification of Kleinian surface groups, II: The Ending Lamination Conjecture
| date=2012
| journal=Annals of Mathematics
| volume=176
| issue=1
| pages=1–149
| doi=10.4007/annals.2012.176.1.1 | doi-access=free}}</ref>
* [[Carpenter's rule problem]] ([[Robert Connelly]], [[Erik Demaine]], Günter Rote, 2003)<ref>{{citation
 | last1 = Connelly | first1 = Robert | author1-link = Robert Connelly
 | last2 = Demaine | first2 = Erik D. | author2-link = Erik Demaine
 | last3 = Rote | first3 = Günter
 | doi = 10.1007/s00454-003-0006-7 | doi-access = free
 | issue = 2
 | journal = [[Discrete & Computational Geometry]]
 | mr = 1931840
 | pages = 205–239
 | title = Straightening polygonal arcs and convexifying polygonal cycles
 | url = http://page.mi.fu-berlin.de/~rote/Papers/pdf/Straightening+polygonal+arcs+and+convexifying+polygonal+cycles-DCG.pdf
 | volume = 30
 | year = 2003| s2cid = 40382145 }}</ref>
* [[Lambda g conjecture]] (Carel Faber and [[Rahul Pandharipande]], 2003)<ref>{{Citation
 | first1=C.
 | last1=Faber
 | first2=R.
 | last2=Pandharipande
 | author2-link=Rahul Pandharipande
 | title=Hodge integrals, partition matrices, and the <math>\lambda_g</math> conjecture
 | journal=Ann. of Math.
 | series= 2
 | volume=157
 | issue=1
 | pages=97–124
 | year=2003
 | arxiv=math.AG/9908052
 | doi=10.4007/annals.2003.157.97}}
</ref>
* [[Nagata's conjecture]] (Ivan Shestakov, Ualbai Umirbaev, 2003)<ref>{{cite journal
 | last1 = Shestakov | first1 = Ivan P.
 | last2 = Umirbaev | first2 = Ualbai U.
 | doi = 10.1090/S0894-0347-03-00440-5
 | issue = 1
 | journal = Journal of the American Mathematical Society
 | mr = 2015334
 | pages = 197–227
 | title = The tame and the wild automorphisms of polynomial rings in three variables
 | volume = 17
 | year = 2004}}</ref>
* [[Double bubble conjecture]] ([[Michael Hutchings (mathematician)|Michael Hutchings]], [[Frank Morgan (mathematician)|Frank Morgan]], Manuel Ritoré, Antonio Ros, 2002)<ref>{{cite journal
 | last1 = Hutchings | first1 = Michael
 | last2 = Morgan | first2 = Frank
 | last3 = Ritoré | first3 = Manuel
 | last4 = Ros | first4 = Antonio
 | doi = 10.2307/3062123
 | issue = 2
 | journal = Annals of Mathematics
 | mr = 1906593
 | pages = 459–489
 | series = Second Series
 | title = Proof of the double bubble conjecture
 | volume = 155
 | year = 2002| jstor = 3062123
 | arxiv = math/0406017
 | hdl = 10481/32449
 }}</ref>

====20th century====
* [[Honeycomb theorem]] ([[Thomas Callister Hales]], 1999)<ref>{{Cite journal
| arxiv=math/9906042
| last1=Hales | first1=Thomas C. | author-link1=Thomas Callister Hales
| title=The Honeycomb Conjecture
| journal=[[Discrete & Computational Geometry]]
| volume=25
| pages=1–22
| date=2001
| doi=10.1007/s004540010071 | doi-access=free}}</ref>
* [[Lange's conjecture]] ([[Montserrat Teixidor i Bigas]] and Barbara Russo, 1999)<ref>{{cite journal
 | last1=Teixidor i Bigas
 | first1=Montserrat
 | author1-link=Montserrat Teixidor i Bigas
 | first2=Barbara
 | last2=Russo
 | title=On a conjecture of Lange
 | arxiv=alg-geom/9710019
 | mr=1689352
 | year=1999
 | journal=Journal of Algebraic Geometry
 | issn=1056-3911
 | volume=8
 | issue=3
 | pages=483–496
 | bibcode=1997alg.geom.10019R }}
</ref>
* [[Bogomolov conjecture]] ([[Emmanuel Ullmo]], 1998, [[Shou-Wu Zhang]], 1998)<ref>{{cite journal | last1 = Ullmo | first1 = E | year = 1998 | title = Positivité et Discrétion des Points Algébriques des Courbes | journal = Annals of Mathematics | volume = 147 | issue = 1| pages = 167–179 | doi = 10.2307/120987 | zbl= 0934.14013| jstor = 120987 | arxiv = alg-geom/9606017 | s2cid = 119717506 }}</ref><ref>{{cite journal | last1 = Zhang | first1 = S.-W. | year = 1998 | title = Equidistribution of small points on abelian varieties | journal = Annals of Mathematics | volume = 147 | issue = 1| pages = 159–165 | doi = 10.2307/120986 | jstor = 120986 }}</ref>
* [[Kepler conjecture]] (Samuel Ferguson, [[Thomas Callister Hales]], 1998)<ref>{{cite journal
| arxiv=1501.02155
| last1=Hales | first1=Thomas
| last2=Adams | first2=Mark
| last3=Bauer | first3=Gertrud
| last4=Dang | first4=Dat Tat
| last5=Harrison | first5=John
| last6=Hoang | first6=Le Truong
| last7=Kaliszyk | first7=Cezary
| last8=Magron | first8=Victor
| last9=McLaughlin | first9=Sean
| last10=Nguyen | first10=Tat Thang
| last11=Nguyen | first11=Quang Truong
| last12=Nipkow | first12=Tobias
| last13=Obua | first13=Steven
| last14=Pleso | first14=Joseph
| last15=Rute | first15=Jason
| last16=Solovyev | first16=Alexey
| last17=Ta | first17=Thi Hoai An
| last18=Tran | first18=Nam Trung
| last19=Trieu | first19=Thi Diep
| last20=Urban | first20=Josef
| last21=Ky | first21=Vu
| last22=Zumkeller | first22=Roland
| title=A formal proof of the Kepler conjecture
| journal=Forum of Mathematics, Pi
| volume=5
| date=2017
| pages=e2
| doi=10.1017/fmp.2017.1 | doi-access=free}}</ref>
* [[Dodecahedral conjecture]] ([[Thomas Callister Hales]], Sean McLaughlin, 1998)<ref>{{Cite journal
| arxiv=math/9811079
| last1=Hales | first1=Thomas C.
| last2=McLaughlin | first2=Sean
| title=The dodecahedral conjecture
| journal=Journal of the American Mathematical Society
| volume=23
| date=2010
| issue=2 | pages=299–344
| doi=10.1090/S0894-0347-09-00647-X  | bibcode=2010JAMS...23..299H | doi-access=free}}</ref>

===Graph theory===
* [[Kahn–Kalai conjecture]] ([[Jinyoung Park (mathematician)|Jinyoung Park]] and Huy Tuan Pham, 2022)<ref>{{cite arXiv |last1=Park |first1=Jinyoung |last2=Pham |first2=Huy Tuan |date=2022-03-31 |title=A Proof of the Kahn-Kalai Conjecture |class=math.CO |eprint=2203.17207 }}</ref>
* [[Blankenship–Oporowski conjecture]] on the book thickness of subdivisions ([[Vida Dujmović]], [[David Eppstein]], Robert Hickingbotham, [[Pat Morin]], and [[David Wood (mathematician)|David Wood]], 2021)<ref>{{cite journal
 | last1 = Dujmović | first1 = Vida | author1-link = Vida Dujmović
 | last2 = Eppstein | first2 = David | author2-link = David Eppstein
 | last3 = Hickingbotham | first3 = Robert
 | last4 = Morin | first4 = Pat | author4-link = Pat Morin
 | last5 = Wood | first5 = David R. | author5-link = David Wood (mathematician)
 | arxiv = 2011.04195
 | date = August 2021
 | doi = 10.1007/s00493-021-4585-7
 | journal = [[Combinatorica]]
 | title = Stack-number is not bounded by queue-number| volume = 42 | issue = 2 | pages = 151–164 | s2cid = 226281691 }}</ref>
*[[Graceful labeling|Ringel's conjecture]] that the complete graph <math>K_{2n+1}</math> can be decomposed into <math>2n+1</math> copies of any tree with <math>n</math> edges (Richard Montgomery, [[Benny Sudakov]], Alexey Pokrovskiy, 2020)<ref>{{cite journal|last1=Huang |first1=C.|title=Further results on tree labellings |journal=Utilitas Mathematica |volume=21 |pages=31–48 |year=1982|mr=668845|last2=Kotzig|first2=A.|last3=Rosa|first3=A.|author2-link=Anton Kotzig}}.</ref><ref>{{Cite web |url=https://www.quantamagazine.org/mathematicians-prove-ringels-graph-theory-conjecture-20200219/|title=Rainbow Proof Shows Graphs Have Uniform Parts|last=Hartnett |first=Kevin|website=Quanta Magazine|date=19 February 2020|language=en|access-date=2020-02-29}}</ref>
*Disproof of [[Hedetniemi's conjecture]] on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)<ref>{{cite journal |last1=Shitov |first1=Yaroslav |date=2019-09-01 |df=dmy-all |title=Counterexamples to Hedetniemi's conjecture |journal=Annals of Mathematics |volume=190 |issue=2 |pages=663–667 |arxiv=1905.02167 |doi=10.4007/annals.2019.190.2.6 |jstor=10.4007/annals.2019.190.2.6 |mr= 3997132 |zbl=1451.05087 |s2cid=146120733 |url=https://annals.math.princeton.edu/2019/190-2/p06 |access-date=2021-07-19}}</ref>
* [[Kelmans–Seymour conjecture]] (Dawei He, Yan Wang, and Xingxing Yu, 2020)<ref>{{Cite journal
 | last1=He
 | first1=Dawei
 | last2=Wang
 | first2=Yan
 | last3=Yu
 | first3=Xingxing
 | date=2019-12-11
 | title=The Kelmans-Seymour conjecture I: Special separations
 | url=http://www.sciencedirect.com/science/article/pii/S0095895619301224
 | journal=Journal of Combinatorial Theory, Series B
 | volume=144
 | pages=197–224
 | doi=10.1016/j.jctb.2019.11.008
 | issn=0095-8956
 | arxiv=1511.05020
 | s2cid=29791394}}
</ref><ref>{{Cite journal
 | last1=He
 | first1=Dawei
 | last2=Wang
 | first2=Yan
 | last3=Yu
 | first3=Xingxing
 | date=2019-12-11
 | title=The Kelmans-Seymour conjecture II: 2-Vertices in K4−
 | url=http://www.sciencedirect.com/science/article/pii/S0095895619301212
 | journal=Journal of Combinatorial Theory, Series B
 | volume=144
 | pages=225–264
 | doi=10.1016/j.jctb.2019.11.007
 | issn=0095-8956
 | arxiv=1602.07557| s2cid=220369443
 }}
</ref><ref>{{Cite journal
 | last1=He
 | first1=Dawei
 | last2=Wang
 | first2=Yan
 | last3=Yu
 | first3=Xingxing
 | date=2019-12-09
 | title=The Kelmans-Seymour conjecture III: 3-vertices in K4−
 | url=http://www.sciencedirect.com/science/article/pii/S0095895619301200
 | journal=Journal of Combinatorial Theory, Series B
 | volume=144
 | pages=265–308
 | doi=10.1016/j.jctb.2019.11.006
 | issn=0095-8956
 | arxiv=1609.05747
 | s2cid=119625722}}
</ref><ref>{{Cite journal
 | last1=He
 | first1=Dawei
 | last2=Wang
 | first2=Yan
 | last3=Yu
 | first3=Xingxing
 | date=2019-12-19
 | title=The Kelmans-Seymour conjecture IV: A proof
 | url=http://www.sciencedirect.com/science/article/pii/S0095895619301248
 | journal=Journal of Combinatorial Theory, Series B
 | volume=144
 | pages=309–358
 | doi=10.1016/j.jctb.2019.12.002
 | issn=0095-8956
 | arxiv=1612.07189
 | s2cid=119175309}}
</ref>
* [[Goldberg–Seymour conjecture]] (Guantao Chen, Guangming Jing, and Wenan Zang, 2019)<ref>{{Cite arXiv
 | last1=Zang
 | first1=Wenan
 | last2=Jing
 | first2=Guangming
 | last3=Chen
 | first3=Guantao
 | date=2019-01-29
 | title=Proof of the Goldberg–Seymour Conjecture on Edge-Colorings of Multigraphs
 | class=math.CO
 | language=en
 | eprint=1901.10316v1}}
</ref>
* [[Babai's problem]] (Alireza Abdollahi, Maysam Zallaghi, 2015)<ref>{{cite journal | first= Zallaghi M.|last= Abdollahi A. | year = 2015 | journal = Communications in Algebra | title = Character sums for Cayley graphs | volume = 43| issue = 12| pages = 5159–5167 | doi = 10.1080/00927872.2014.967398 |s2cid= 117651702 }}</ref>
* [[Alspach's conjecture]] (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)
* [[Alon–Saks–Seymour conjecture]] (Hao Huang, [[Benny Sudakov]], 2012)
* [[Read's conjecture|Read–Hoggar conjecture]] ([[June Huh]], 2009)<ref>{{cite journal
 | last=Huh
 | first=June
 | author-link=June Huh
 | title=Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
 | arxiv=1008.4749
 | journal=Journal of the American Mathematical Society
 | volume=25
 | date=2012
 | issue=3
 | pages=907–927
 | doi=10.1090/S0894-0347-2012-00731-0 
 | doi-access=free}}</ref>
* [[Scheinerman's conjecture]] (Jeremie Chalopin and Daniel Gonçalves, 2009)<ref>{{cite conference
 | last1 = Chalopin | first1 = Jérémie
 | last2 = Gonçalves | first2 = Daniel
 | editor-last = Mitzenmacher | editor-first = Michael
 | contribution = Every planar graph is the intersection graph of segments in the plane: extended abstract
 | doi = 10.1145/1536414.1536500
 | pages = 631–638
 | publisher = ACM
 | title = Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 – June 2, 2009
 | year = 2009}}</ref>
* [[Erdős–Menger conjecture]] ([[Ron Aharoni]], Eli Berger 2007)<ref>{{Cite journal
| arxiv=math/0509397
| last1=Aharoni | first1=Ron | author1-link=Ron Aharoni
| last2=Berger | first2=Eli
| title = Menger's theorem for infinite graphs
| journal=Inventiones Mathematicae
| volume=176
| pages=1–62
| date=2009
| issue=1 | doi=10.1007/s00222-008-0157-3 | bibcode=2009InMat.176....1A | doi-access=free}}</ref>
* [[Road coloring conjecture]] ([[Avraham Trahtman]], 2007)<ref>{{cite news |last=Seigel-Itzkovich |first=Judy |title=Russian immigrant solves math puzzle |newspaper=The Jerusalem Post |date=2008-02-08 
|url=http://www.jpost.com/Home/Article.aspx?id=91431 |access-date=2015-11-12}}</ref>
* [[Robertson–Seymour theorem]] ([[Neil Robertson (mathematician)|Neil Robertson]], [[Paul Seymour (mathematician)|Paul Seymour]], 2004)<ref>{{cite book |last=Diestel |first=Reinhard |year=2005 |chapter=Minors, Trees, and WQO |edition=Electronic Edition 2005 |pages=326–367 |publisher=Springer |title=Graph Theory |chapter-url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/preview/Ch12.pdf}}</ref>
* [[Strong perfect graph conjecture]] ([[Maria Chudnovsky]], [[Neil Robertson (mathematician)|Neil Robertson]], [[Paul Seymour (mathematician)|Paul Seymour]] and [[Robin Thomas (mathematician)|Robin Thomas]], 2002)<ref>{{cite journal |url=https://annals.math.princeton.edu/2006/164-1/p02 |title=The strong perfect graph theorem |last1=Chudnovsky |first1=Maria |last2=Robertson |first2=Neil |last3=Seymour |first3=Paul |last4=Thomas |first4=Robin |journal=Annals of Mathematics |year=2002 |volume=164 |pages=51–229 |arxiv=math/0212070 |doi=10.4007/annals.2006.164.51 |bibcode=2002math.....12070C |s2cid=119151552}}</ref>
* [[Toida's conjecture]] (Mikhail Muzychuk, Mikhail Klin, and Reinhard Pöschel, 2001)<ref>Klin, M. H., M. Muzychuk and R. Poschel: The isomorphism problem for circulant graphs via Schur ring theory, Codes and Association Schemes, American Math. Society, 2001.</ref>
* Harary's conjecture on the integral sum number of complete graphs (Zhibo Chen, 1996)<ref>{{Cite journal 
 | url=https://www.researchgate.net/publication/220188021
 | doi=10.1016/0012-365X(95)00163-Q
 | doi-access=free
 | title=Harary's conjectures on integral sum graphs
 | journal=[[Discrete Mathematics (journal)|Discrete Mathematics]]
 | volume=160
 | issue=1–3
 | pages=241–244
 | year=1996
 | last1=Chen
 | first1=Zhibo}}
</ref>

===Group theory===
* [[Hanna Neumann conjecture]] (Joel Friedman, 2011, Igor Mineyev, 2011)<ref>{{Cite journal |last=Friedman |first=Joel |date=January 2015 |title=Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture: with an Appendix by Warren Dicks |url=https://www.cs.ubc.ca/~jf/pubs/web_stuff/shnc_memoirs.pdf |journal=Memoirs of the American Mathematical Society |language=en |volume=233 |issue=1100 |page=0 |doi=10.1090/memo/1100 |s2cid=117941803 |issn=0065-9266}}</ref><ref>{{cite journal
 | last = Mineyev | first = Igor
 | doi = 10.4007/annals.2012.175.1.11
 | issue = 1
 | journal = Annals of Mathematics
 | mr = 2874647
 | pages = 393–414
 | series = Second Series
 | title = Submultiplicativity and the Hanna Neumann conjecture
 | volume = 175
 | year = 2012}}</ref>
* [[Density theorem for Kleinian groups|Density theorem]] (Hossein Namazi, Juan Souto, 2010)<ref>{{Cite journal |url=https://www.researchgate.net/publication/228365532 |doi=10.1007/s11511-012-0088-0|title=Non-realizability and ending laminations: Proof of the density conjecture|journal=Acta Mathematica|volume=209|issue=2|pages=323–395|year=2012|last1=Namazi|first1=Hossein|last2=Souto|first2=Juan|doi-access=free}}</ref>
* Full [[classification of finite simple groups]] ([[Koichiro Harada]], [[Ronald Solomon]], 2008)

===Number theory===
====21st century====
*[[André–Oort conjecture]] ([[Jonathan Pila]], Ananth Shankar, [[Jacob Tsimerman]], 2021)<ref>{{cite arXiv |last1=Pila |first1=Jonathan |last2=Shankar |first2=Ananth |last3=Tsimerman |first3=Jacob |last4=Esnault |first4=Hélène |last5=Groechenig |first5=Michael |date=2021-09-17 |title=Canonical Heights on Shimura Varieties and the André-Oort Conjecture |class=math.NT |eprint=2109.08788}}</ref>
*[[Duffin–Schaeffer theorem]] ([[Dimitris Koukoulopoulos]], [[James Maynard (mathematician)|James Maynard]], 2019)
* [[Vinogradov's mean-value theorem#The conjectured form|Main conjecture in Vinogradov's mean-value theorem]] ([[Jean Bourgain]], Ciprian Demeter, [[Larry Guth]], 2015)<ref>{{cite journal|last1=Bourgain |first1=Jean|first2=Demeter|last2=Ciprian|first3=Guth|last3=Larry|title=Proof of the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three|journal=Annals of Mathematics |year=2015|doi=10.4007/annals.2016.184.2.7|volume=184|issue=2|pages=633–682|hdl=1721.1/115568|bibcode=2015arXiv151201565B|arxiv=1512.01565|s2cid=43929329}}</ref>
* [[Goldbach's weak conjecture]] ([[Harald Helfgott]], 2013)<ref>{{cite arXiv |eprint=1305.2897 |title = Major arcs for Goldbach's theorem|last = Helfgott|first = Harald A. |class=math.NT |year=2013}}</ref><ref>{{cite arXiv |eprint=1205.5252 |title = Minor arcs for Goldbach's problem |last = Helfgott|first = Harald A.|class=math.NT |year=2012}}</ref><ref>{{cite arXiv |eprint=1312.7748 |title = The ternary Goldbach conjecture is true|last = Helfgott|first = Harald A. |class=math.NT |year=2013}}</ref>
*[[Prime gap#Further results|Existence of bounded gaps between arbitrarily large primes]] ([[Yitang Zhang]], [[Polymath Project|Polymath8]], [[James Maynard (mathematician)|James Maynard]], 2013)<ref>{{Cite journal|last=Zhang|first=Yitang|date=2014-05-01|title=Bounded gaps between primes|journal=Annals of Mathematics|volume=179|issue=3|pages=1121–1174|doi=10.4007/annals.2014.179.3.7|issn=0003-486X}}</ref><ref>{{Cite web|title=Bounded gaps between primes – Polymath Wiki|url=https://asone.ai/polymath/index.php?title=Bounded_gaps_between_primes|access-date=2021-08-27|website=asone.ai|archive-date=2020-12-08|archive-url=https://web.archive.org/web/20201208045925/https://asone.ai/polymath/index.php?title=Bounded_gaps_between_primes}}</ref><ref>{{Cite journal|last=Maynard|first=James|date=2015-01-01|title=Small gaps between primes|journal=Annals of Mathematics|pages=383–413|doi=10.4007/annals.2015.181.1.7|arxiv=1311.4600|s2cid=55175056|issn=0003-486X}}</ref>
* [[Sidon sequence|Sidon set problem]] (Javier Cilleruelo, [[Imre Z. Ruzsa]], and Carlos Vinuesa, 2010)<ref>{{cite journal|title=Generalized Sidon sets|doi=10.1016/j.aim.2010.05.010 | volume=225|issue=5|journal=[[Advances in Mathematics]]|pages=2786–2807|year=2010 | last1 = Cilleruelo | first1 = Javier|hdl=10261/31032|s2cid=7385280|doi-access=free|hdl-access=free}}</ref>
* [[Serre's modularity conjecture]] ([[Chandrashekhar Khare]] and [[Jean-Pierre Wintenberger]], 2008)<ref>{{Citation |last1=Khare |first1=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=Serre's modularity conjecture (I) |journal=Inventiones Mathematicae |volume=178 |issue=3 |pages=485–504 |doi=10.1007/s00222-009-0205-7 |bibcode=2009InMat.178..485K |citeseerx=10.1.1.518.4611 |s2cid=14846347 }}</ref><ref>{{Citation |last1=Khare |first1=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=Serre's modularity conjecture (II) |journal=Inventiones Mathematicae |volume=178 |issue=3 |pages=505–586 |doi=10.1007/s00222-009-0206-6 |bibcode=2009InMat.178..505K |citeseerx=10.1.1.228.8022 |s2cid=189820189 }}</ref><ref>{{cite journal |author=<!--Staff writer(s); no by-line.--> |title=2011 Cole Prize in Number Theory |url=https://www.ams.org/notices/201104/rtx110400610p.pdf |journal=[[Notices of the AMS]] |volume=58 |issue=4 |pages=610–611 |issn=1088-9477 |oclc=34550461 |access-date=2015-11-12 |archive-url=https://web.archive.org/web/20151106051835/http://www.ams.org/notices/201104/rtx110400610p.pdf |archive-date=2015-11-06 |url-status=live }}</ref>
* [[Green–Tao theorem]] ([[Ben J. Green]] and [[Terence Tao]], 2004)<ref>{{cite journal |author=<!--Staff writer(s); no by-line.--> |date=May 2010 |title=Bombieri and Tao Receive King Faisal Prize |url=https://www.ams.org/notices/201005/rtx100500642p.pdf |journal=[[Notices of the AMS]] |volume=57 |issue=5 |pages=642–643 |issn=1088-9477 |oclc=34550461 |quote=Working with Ben Green, he proved there are arbitrarily long arithmetic progressions of prime numbers—a result now known as the Green–Tao theorem. |access-date=2016-03-18 |archive-url=https://web.archive.org/web/20160304063504/http://www.ams.org/notices/201005/rtx100500642p.pdf |archive-date=2016-03-04 |url-status=live }}</ref>
* [[Mihăilescu's theorem|Catalan's conjecture]] ([[Preda Mihăilescu]], 2002)<ref>{{cite journal |last=Metsänkylä |first=Tauno |date=5 September 2003 |title=Catalan's conjecture: another old diophantine problem solved |url=https://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf |journal=[[Bulletin of the American Mathematical Society]] |volume=41 |issue=1 |pages=43–57 |issn=0273-0979 |quote=The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihăilescu. |doi=10.1090/s0273-0979-03-00993-5 |access-date=13 November 2015 |archive-url=https://web.archive.org/web/20160304082755/http://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf |archive-date=4 March 2016 |url-status=live }}</ref>
* [[Erdős–Graham problem]] ([[Ernest S. Croot III]], 2000)<ref>{{cite book | last = Croot | first = Ernest S. III | author-link = Ernest S. Croot III | publisher = [[University of Georgia]], Athens | series = Ph.D. thesis | title = Unit Fractions | year = 2000}} {{cite journal | last = Croot | first = Ernest S. III | author-link = Ernest S. Croot III | arxiv = math.NT/0311421 | doi = 10.4007/annals.2003.157.545 | issue = 2 | journal = [[Annals of Mathematics]] | pages = 545–556 | title = On a coloring conjecture about unit fractions | volume = 157 | year = 2003| bibcode = 2003math.....11421C | s2cid = 13514070 }}</ref>

====20th century====
* [[Lafforgue's theorem]] ([[Laurent Lafforgue]], 1998)<ref>{{Citation | last1=Lafforgue | first1=Laurent | title=Chtoucas de Drinfeld et applications | language=fr | trans-title=Drinfelʹd shtukas and applications | url=http://www.math.uni-bielefeld.de/documenta/xvol-icm/07/Lafforgue.MAN.html | mr=1648105 | year=1998 | journal=Documenta Mathematica | issn=1431-0635 | volume=II | pages=563–570 | access-date=2016-03-18 | archive-url=https://web.archive.org/web/20180427200241/https://www.math.uni-bielefeld.de/documenta/xvol-icm/07/Lafforgue.MAN.html | archive-date=2018-04-27 | url-status=live }}</ref>
* [[Fermat's Last Theorem]] ([[Andrew Wiles]] and [[Richard Taylor (mathematician)|Richard Taylor]], 1995)<ref>{{cite journal|last=Wiles|first=Andrew|author-link=Andrew Wiles|year=1995|title=Modular elliptic curves and Fermat's Last Theorem|url=http://math.stanford.edu/~lekheng/flt/wiles.pdf|journal=Annals of Mathematics|volume=141|issue=3|pages=443–551|oclc=37032255|doi=10.2307/2118559|jstor=2118559|citeseerx=10.1.1.169.9076|access-date=2016-03-06|archive-url=https://web.archive.org/web/20110510062158/http://math.stanford.edu/%7Elekheng/flt/wiles.pdf|archive-date=2011-05-10|url-status=live}}</ref><ref>{{cite journal |author=[[Richard Taylor (mathematician)|Taylor R]], [[Andrew Wiles|Wiles A]] |year=1995 |title=Ring theoretic properties of certain Hecke algebras |url=http://www.math.harvard.edu/~rtaylor/hecke.ps |journal=Annals of Mathematics |volume=141 |issue=3 |pages=553–572 |citeseerx=10.1.1.128.531 |doi=10.2307/2118560 |jstor=2118560 |oclc=37032255 |archive-url=https://web.archive.org/web/20000916161311/http://www.math.harvard.edu/~rtaylor/hecke.ps |archive-date=16 September 2000}}</ref>

===Ramsey theory===
* [[Burr–Erdős conjecture]] (Choongbum Lee, 2017)<ref>{{cite journal | last1 = Lee | first1 = Choongbum | year = 2017 | title = Ramsey numbers of degenerate graphs | journal = Annals of Mathematics | volume = 185 | issue = 3| pages = 791–829 | doi = 10.4007/annals.2017.185.3.2 | arxiv = 1505.04773 | s2cid = 7974973 }}</ref>
* [[Boolean Pythagorean triples problem]] ([[Marijn Heule]], Oliver Kullmann, [[Victor W. Marek]], 2016)<ref>{{cite journal |last=Lamb |first=Evelyn |date=26 May 2016 |title=Two-hundred-terabyte maths proof is largest ever |journal=Nature |doi=10.1038/nature.2016.19990 |volume=534 |issue=7605 |pages=17–18 |pmid=27251254 |bibcode=2016Natur.534...17L|doi-access=free }}</ref><ref>{{cite book
 | last1 = Heule | first1 = Marijn J. H. | author1-link=Marijn Heule
 | last2 = Kullmann | first2 = Oliver
 | last3 = Marek | first3 = Victor W. | author3-link=Victor W. Marek
 | editor-last1 = Creignou | editor-first1 = N.
 | editor-last2 = Le Berre | editor-first2 = D. 
 | arxiv = 1605.00723
 | chapter = Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer
 | doi = 10.1007/978-3-319-40970-2_15
 | mr = 3534782
 | pages = 228–245
 | publisher = Springer, [Cham]
 | series = Lecture Notes in Computer Science
 | title = Theory and Applications of Satisfiability Testing – SAT 2016
 | volume = 9710
 | year = 2016| isbn = 978-3-319-40969-6
 | s2cid = 7912943
 }}</ref>

===Theoretical computer science===
*[[Decision tree model#Sensitivity conjecture|Sensitivity conjecture]] for Boolean functions ([[Hao Huang (mathematician)|Hao Huang]], 2019)<ref>{{cite web |author=Linkletter, David |date=27 December 2019 |title=The 10 Biggest Math Breakthroughs of 2019 |url=https://www.popularmechanics.com/science/math/g30346822/biggest-math-breakthroughs-2019/ |access-date=20 June 2021 |work=[[Popular Mechanics]]}}</ref>

===Topology===
*Deciding whether the [[Conway knot]] is a [[slice knot]] ([[Lisa Piccirillo]], 2020)<ref>{{Cite journal |last=Piccirillo |first=Lisa |date=2020 |title=The Conway knot is not slice |url=https://annals.math.princeton.edu/2020/191-2/p05 |journal=[[Annals of Mathematics]] |volume=191 |issue=2 |pages=581–591 |doi=10.4007/annals.2020.191.2.5|s2cid=52398890 }}</ref><ref>{{Cite web |last=Klarreich |first=Erica |author-link=Erica Klarreich |date=2020-05-19 |title=Graduate Student Solves Decades-Old Conway Knot Problem |url=https://www.quantamagazine.org/graduate-student-solves-decades-old-conway-knot-problem-20200519/ |access-date=2022-08-17 |website=[[Quanta Magazine]] |language=en}}</ref>
* [[Virtual Haken conjecture]] ([[Ian Agol]], Daniel Groves, Jason Manning, 2012)<ref>{{Cite journal
| arxiv = 1204.2810v1
| last1 = Agol | first1 = Ian
| title = The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves, and Jason Manning)
| journal=Documenta Mathematica
| volume=18
| date=2013
| pages=1045–1087
| doi = 10.4171/dm/421 | doi-access = free | s2cid = 255586740 | url=https://www.math.uni-bielefeld.de/documenta/vol-18/33.pdf}}</ref> (and by work of [[Daniel Wise (mathematician)|Daniel Wise]] also [[virtually fibered conjecture]])
* [[Hsiang–Lawson's conjecture]] ([[Simon Brendle]], 2012)<ref>{{Cite journal
| arxiv=1203.6597
| last1 = Brendle | first1 = Simon | author1-link=Simon Brendle
| title = Embedded minimal tori in <math>S^3</math> and the Lawson conjecture
| journal=Acta Mathematica
| volume=211
| issue=2
| pages=177–190
| date=2013
| doi=10.1007/s11511-013-0101-2 | doi-access=free}}</ref>
* [[Ehrenpreis conjecture]] ([[Jeremy Kahn]], [[Vladimir Markovic]], 2011)<ref>{{Cite journal
| arxiv=1101.1330
| last1=Kahn | first1=Jeremy | author1-link=Jeremy Kahn
| last2=Markovic | first2=Vladimir | author2-link=Vladimir Markovic
| title=The good pants homology and the Ehrenpreis conjecture
| journal=Annals of Mathematics
| pages=1–72
| volume=182
| date=2015
| issue=1
| doi=10.4007/annals.2015.182.1.1 | doi-access=free}}</ref>
* [[Atiyah conjecture]] for groups with finite subgroups of unbounded order (Austin, 2009)<ref>{{cite journal
| arxiv = 0909.2360
| last1 = Austin |first1 = Tim
| title = Rational group ring elements with kernels having irrational dimension
| journal = Proceedings of the London Mathematical Society
| volume = 107
| issue = 6
| pages = 1424–1448
| date = December 2013
| doi = 10.1112/plms/pdt029 | bibcode = 2009arXiv0909.2360A|s2cid = 115160094}}</ref>
* [[Cobordism hypothesis]] ([[Jacob Lurie]], 2008)<ref>{{cite journal | last1 = Lurie | first1 = Jacob | year = 2009 | title = On the classification of topological field theories | journal = Current Developments in Mathematics | volume = 2008 | pages = 129–280 | doi=10.4310/cdm.2008.v2008.n1.a3| bibcode = 2009arXiv0905.0465L | arxiv = 0905.0465 | s2cid = 115162503 }}</ref>
* [[Spherical space form conjecture]] ([[Grigori Perelman]], 2006)
* [[Poincaré conjecture]] ([[Grigori Perelman]], 2002)<ref name="auto">{{cite press release | publisher=[[Clay Mathematics Institute]] | date=March 18, 2010 | title=Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman | url=http://www.claymath.org/sites/default/files/millenniumprizefull.pdf | access-date=November 13, 2015 | quote=The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman. | archive-url=https://web.archive.org/web/20100322192115/http://www.claymath.org/poincare/ | archive-date=March 22, 2010 | url-status=live }}</ref>
* [[Geometrization conjecture]], ([[Grigori Perelman]],<ref name="auto" /> series of preprints in 2002–2003)<ref>{{Cite arXiv |eprint = 0809.4040|last1 = Morgan |first1 = John |title = Completion of the Proof of the Geometrization Conjecture|last2 = Tian|first2 = Gang|class = math.DG|year = 2008}}</ref>
* [[Nikiel's conjecture]] ([[Mary Ellen Rudin]], 1999)<ref>{{cite journal
| first1=M.E. | last1=Rudin | author-link1=Mary Ellen Rudin
| title=Nikiel's Conjecture
| journal=Topology and Its Applications
| volume=116
| year=2001
| issue=3 | pages=305–331
| doi=10.1016/S0166-8641(01)00218-8 | doi-access=free}}</ref>
* Disproof of the [[Ganea conjecture]] (Iwase, 1997)<ref>{{cite web|url=https://www.researchgate.net/publication/220032558|title=Ganea's Conjecture on Lusternik-Schnirelmann Category|author=Norio Iwase|date=1 November 1998|work=ResearchGate}}</ref>

===Uncategorised===
====2010s====
* [[Erdős discrepancy problem]] ([[Terence Tao]], 2015)<ref>{{Cite arXiv |eprint = 1509.05363v5|last1 = Tao|first1 = Terence | author-link1=Terence Tao|title = The Erdős discrepancy problem|class = math.CO|year = 2015}}</ref>
* [[Umbral moonshine]] conjecture (John F. R. Duncan, Michael J. Griffin, [[Ken Ono]], 2015)<ref>{{cite journal|title=Proof of the umbral moonshine conjecture|first1=John F. R.|last1=Duncan|first2=Michael J.|last2=Griffin|first3=Ken|last3=Ono|date=1 December 2015|journal=Research in the Mathematical Sciences|volume=2|issue=1|page=26|doi=10.1186/s40687-015-0044-7|bibcode=2015arXiv150301472D|arxiv=1503.01472|s2cid=43589605 |doi-access=free }}</ref>
* Anderson conjecture on the finite number of diffeomorphism classes of the collection of 4-manifolds satisfying certain properties ([[Jeff Cheeger]], Aaron Naber, 2014)<ref>{{cite journal
| arxiv=1406.6534
| last1=Cheeger | first1=Jeff
| last2=Naber | first2=Aaron
| title=Regularity of Einstein Manifolds and the Codimension 4 Conjecture
| journal=Annals of Mathematics
| pages=1093–1165
| volume=182
| issue=3
| date=2015
| doi=10.4007/annals.2015.182.3.5 | doi-access=free}}</ref>
* [[Gaussian correlation inequality]] ([[Thomas Royen]], 2014)<ref>{{Cite magazine |last=Wolchover |first=Natalie |date=March 28, 2017 |title=A Long-Sought Proof, Found and Almost Lost |url=https://www.quantamagazine.org/20170328-statistician-proves-gaussian-correlation-inequality/ |url-status=live |magazine=[[Quanta Magazine]] |archive-url=https://web.archive.org/web/20170424133433/https://www.quantamagazine.org/20170328-statistician-proves-gaussian-correlation-inequality/ |archive-date=April 24, 2017 |access-date=May 2, 2017}}</ref>
* Beck's conjecture on discrepancies of set systems constructed from three permutations (Alantha Newman, [[Aleksandar Nikolov (computer scientist)|Aleksandar Nikolov]], 2011)<ref>{{Cite arXiv |eprint = 1104.2922|last1=Newman |first1=Alantha | last2=Nikolov | first2=Aleksandar |title = A counterexample to Beck's conjecture on the discrepancy of three permutations|class = cs.DM|year = 2011}}</ref>
* [[Bloch–Kato conjecture]] ([[Vladimir Voevodsky]], 2011)<ref>{{Cite web |url=https://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p11-p.pdf |title=On motivic cohomology with Z/''l''-coefficients |last=Voevodsky |first=Vladimir |access-date=2016-03-18 |archive-url=https://web.archive.org/web/20160327035457/http://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p11-p.pdf |location=Princeton, NJ |website=annals.math.princeton.edu |publisher=[[Princeton University]] |date=1 July 2011|volume=174|issue=1|pages=401–438|archive-date=2016-03-27 |url-status=live }}</ref> (and [[Quillen–Lichtenbaum conjecture]] and by work of [[Thomas Geisser (mathematician)|Thomas Geisser]] and [[Marc Levine (mathematician)|Marc Levine]] (2001) also [[Norm residue isomorphism theorem#Beilinson–Lichtenbaum conjecture|Beilinson–Lichtenbaum conjecture]]<ref>{{cite journal
 | last1 = Geisser | first1 = Thomas
 | last2 = Levine | first2 = Marc
 | doi = 10.1515/crll.2001.006
 | journal = Journal für die Reine und Angewandte Mathematik
 | mr = 1807268
 | pages = 55–103
 | title = The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky
 | volume = 2001
 | year = 2001| issue = 530
 }}</ref><ref>{{cite web |last=Kahn |first=Bruno |title=Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry |url=https://webusers.imj-prg.fr/~bruno.kahn/preprints/kcag.pdf |url-status=live |archive-url=https://web.archive.org/web/20160327035553/https://webusers.imj-prg.fr/~bruno.kahn/preprints/kcag.pdf |archive-date=2016-03-27 |access-date=2016-03-18 |website=webusers.imj-prg.fr}}</ref>{{Rp|page=359}}<ref>{{cite web|url=https://mathoverflow.net/q/87162 |title=motivic cohomology – Milnor–Bloch–Kato conjecture implies the Beilinson-Lichtenbaum conjecture – MathOverflow|access-date=2016-03-18 }}</ref>)

====2000s====
* [[Kauffman–Harary conjecture]] (Thomas Mattman, Pablo Solis, 2009)<ref>{{Cite journal
| arxiv = 0906.1612
| last1 = Mattman |first1 =  Thomas W.
| last2 = Solis | first2 = Pablo
| title = A proof of the Kauffman-Harary Conjecture
| journal = Algebraic & Geometric Topology
| volume = 9
| issue = 4
| pages = 2027–2039
| year = 2009
| doi = 10.2140/agt.2009.9.2027 | bibcode = 2009arXiv0906.1612M | s2cid = 8447495}}</ref>
* [[Surface subgroup conjecture]] ([[Jeremy Kahn]], [[Vladimir Markovic]], 2009)<ref>{{cite journal
| arxiv=0910.5501
| last1 = Kahn | first1 = Jeremy
| last2 = Markovic | first2 = Vladimir
| title = Immersing almost geodesic surfaces in a closed hyperbolic three manifold
| journal = Annals of Mathematics
| pages=1127–1190
| volume=175
| issue=3
| year=2012
| doi=10.4007/annals.2012.175.3.4 | doi-access=free}}</ref>
* [[Normal scalar curvature conjecture]] and the [[Böttcher–Wenzel conjecture]] (Zhiqin Lu, 2007)<ref>{{cite journal
| first=Zhiqin | last=Lu
| orig-date=2007
| title=Normal Scalar Curvature Conjecture and its applications
| arxiv=0711.3510
| journal=Journal of Functional Analysis
| volume=261
| issue=5
| date=September 2011
| pages=1284–1308
| doi=10.1016/j.jfa.2011.05.002 | doi-access=free}}</ref>
* [[Nirenberg–Treves conjecture]] ([[Nils Dencker]], 2005)<ref>{{citation |last=Dencker |first=Nils |author-link=Nils Dencker |title=The resolution of the Nirenberg–Treves conjecture |journal=[[Annals of Mathematics]] |volume=163 |issue=2 |year=2006 |pages=405–444 |url=https://annals.math.princeton.edu/wp-content/uploads/annals-v163-n2-p02.pdf |doi=10.4007/annals.2006.163.405 |s2cid=16630732 |access-date=2019-04-07 |archive-url=https://web.archive.org/web/20180720145723/http://annals.math.princeton.edu/wp-content/uploads/annals-v163-n2-p02.pdf |archive-date=2018-07-20 |url-status=live }}</ref><ref>{{cite web |url=https://www.claymath.org/research |title=Research Awards |website=[[Clay Mathematics Institute]] |access-date=2019-04-07 |archive-url=https://web.archive.org/web/20190407160116/https://www.claymath.org/research |archive-date=2019-04-07 |url-status=live }}</ref>
* [[Peter Lax|Lax conjecture]] ([[Adrian Lewis (mathematician)|Adrian Lewis]], [[Pablo Parrilo]], Motakuri Ramana, 2005)<ref>{{cite journal
 | last1 = Lewis | first1 = A. S.
 | last2 = Parrilo | first2 = P. A.
 | last3 = Ramana | first3 = M. V.
 | doi = 10.1090/S0002-9939-05-07752-X
 | issue = 9
 | journal = Proceedings of the American Mathematical Society
 | mr = 2146191
 | pages = 2495–2499
 | title = The Lax conjecture is true
 | volume = 133
 | year = 2005| s2cid = 17436983
 }}</ref>
* The [[Langlands–Shelstad fundamental lemma]] ([[Ngô Bảo Châu]] and [[Gérard Laumon]], 2004)<ref>{{cite web |url=http://www.icm2010.in/prize-winners-2010/fields-medal-ngo-bao-chau |title=Fields Medal – Ngô Bảo Châu |author=<!--Staff writer(s); no by-line.--> |date=19 August 2010 |website=International Congress of Mathematicians 2010 |publisher=ICM |access-date=2015-11-12 |quote=Ngô Bảo Châu is being awarded the 2010 Fields Medal for his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods. |archive-url=https://web.archive.org/web/20150924032610/http://www.icm2010.in/prize-winners-2010/fields-medal-ngo-bao-chau |archive-date=24 September 2015 |url-status=live }}</ref>
* [[Milnor conjecture (K-theory)|Milnor conjecture]] ([[Vladimir Voevodsky]], 2003)<ref>{{cite journal |title=Reduced power operations in motivic cohomology |pages=1–57|journal=Publications Mathématiques de l'IHÉS |volume=98 |year=2003 |last1=Voevodsky |first1=Vladimir |doi=10.1007/s10240-003-0009-z |citeseerx=10.1.1.170.4427 |url=http://archive.numdam.org/item/PMIHES_2003__98__1_0/ |access-date=2016-03-18 |url-status=live |archive-url=https://web.archive.org/web/20170728114725/http://archive.numdam.org/item/PMIHES_2003__98__1_0 |archive-date=2017-07-28 |arxiv=math/0107109 |s2cid=8172797}}</ref>
* [[Kirillov's conjecture]] (Ehud Baruch, 2003)<ref>{{cite journal
 | last = Baruch | first = Ehud Moshe
 | doi = 10.4007/annals.2003.158.207
 | issue = 1
 | journal = Annals of Mathematics
 | mr = 1999922
 | pages = 207–252
 | series = Second Series
 | title = A proof of Kirillov's conjecture
 | volume = 158
 | year = 2003}}</ref>
* [[Kouchnirenko's conjecture]] (Bertrand Haas, 2002)<ref>{{Cite journal |last=Haas |first=Bertrand |date=2002 |title=A Simple Counterexample to Kouchnirenko's Conjecture |url=https://www.emis.de/journals/BAG/vol.43/no.1/b43h1haa.pdf |url-status=live |journal=Beiträge zur Algebra und Geometrie |volume=43 |issue=1 |pages=1–8 |archive-url=https://web.archive.org/web/20161007091417/http://www.emis.de/journals/BAG/vol.43/no.1/b43h1haa.pdf |archive-date=2016-10-07 |access-date=2016-03-18}}</ref>
* [[n! conjecture|''n''! conjecture]] ([[Mark Haiman]], 2001)<ref>{{cite journal
 | last = Haiman | first = Mark
 | doi = 10.1090/S0894-0347-01-00373-3
 | issue = 4
 | journal = Journal of the American Mathematical Society
 | mr = 1839919
 | pages = 941–1006
 | title = Hilbert schemes, polygraphs and the Macdonald positivity conjecture
 | volume = 14
 | year = 2001| s2cid = 9253880
 }}</ref> (and also [[Macdonald polynomials#The Macdonald positivity conjecture|Macdonald positivity conjecture]])
* [[Kato's conjecture]] ([[Pascal Auscher]], [[Steve Hofmann]], [[Michael Lacey (mathematician)|Michael Lacey]], [[Alan Gaius Ramsay McIntosh|Alan McIntosh]], and Philipp Tchamitchian, 2001)<ref>{{cite journal
 | last1 = Auscher | first1 = Pascal
 | last2 = Hofmann | first2 = Steve
 | last3 = Lacey | first3 = Michael
 | last4 = McIntosh | first4 = Alan
 | last5 = Tchamitchian | first5 = Ph.
 | doi = 10.2307/3597201
 | issue = 2
 | journal = Annals of Mathematics
 | mr = 1933726
 | pages = 633–654
 | series = Second Series
 | title = The solution of the Kato square root problem for second order elliptic operators on <math>\mathbb{R}^n</math>
 | volume = 156
 | year = 2002| jstor = 3597201
 }}</ref>
* [[Deligne's conjecture on 1-motives]] (Luca Barbieri-Viale, Andreas Rosenschon, [[Morihiko Saito]], 2001)<ref>{{cite journal
| arxiv=math/0102150
| last1=Barbieri-Viale |first1=Luca
| last2=Rosenschon | first2=Andreas
| last3=Saito | first3=Morihiko
| title = Deligne's Conjecture on 1-Motives
| journal=Annals of Mathematics
| pages=593–633
| volume=158
| date=2003
| issue=2
| doi=10.4007/annals.2003.158.593 | doi-access=free}}</ref>
* [[Modularity theorem]] ([[Christophe Breuil]], [[Brian Conrad]], [[Fred Diamond]], and [[Richard Taylor (mathematician)|Richard Taylor]], 2001)<ref>{{Citation | last1=Breuil | first1=Christophe | last2=Conrad | first2=Brian | last3=Diamond | first3=Fred | last4=Taylor | first4=Richard | title=On the modularity of elliptic curves over '''Q''': wild 3-adic exercises | doi=10.1090/S0894-0347-01-00370-8 | mr=1839918 | year=2001 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | volume=14 | issue=4 | pages=843–939| doi-access=free }}</ref>
* [[Erdős–Stewart conjecture]] ([[Florian Luca]], 2001)<ref>{{Cite journal|url=https://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01178-9/S0025-5718-00-01178-9.pdf|doi=10.1090/s0025-5718-00-01178-9|title=On a conjecture of Erdős and Stewart|journal=Mathematics of Computation|volume=70|issue=234|pages=893–897|year=2000|last1=Luca|first1=Florian|access-date=2016-03-18|archive-url=https://web.archive.org/web/20160402030443/http://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01178-9/S0025-5718-00-01178-9.pdf|archive-date=2016-04-02|url-status=live|bibcode=2001MaCom..70..893L}}</ref>
* [[Berry–Robbins problem]] ([[Michael Atiyah]], 2000)<ref>{{cite book
 | last = Atiyah | first = Michael | author-link = Michael Atiyah
 | editor-last = Yau | editor-first = Shing-Tung | editor-link = Shing-Tung Yau
 | contribution = The geometry of classical particles
 | doi = 10.4310/SDG.2002.v7.n1.a1
 | mr = 1919420
 | pages = 1–15
 | publisher = International Press | location = Somerville, Massachusetts
 | series = Surveys in Differential Geometry
 | title = Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer
 | volume = 7
 | year = 2000}}
</ref>

== See also ==
* [[List of conjectures]]
* [[List of unsolved problems in statistics]]
* [[List of unsolved problems in computer science]]
* [[List of unsolved problems in physics]]
* [[Lists of unsolved problems]]
* ''[[Open Problems in Mathematics]]''
* ''[[The Great Mathematical Problems]]''
*[[Scottish Book]]

==Notes==
{{notelist}}

== References ==
{{reflist|colwidth=30em}}

== Further reading ==

=== Books discussing problems solved since 1995 ===

* {{cite book |last=Singh |first=Simon |author-link=Simon Singh |date=2002 |title=Fermat's Last Theorem |publisher=Fourth Estate |isbn=978-1-84115-791-7|title-link=Fermat's Last Theorem (book) }}
* {{cite book |last=O'Shea |first=Donal |author-link=Donal O'Shea| date=2007 |title=The Poincaré Conjecture |publisher=Penguin |isbn=978-1-84614-012-9}}
* {{cite book |last=Szpiro |first=George G. |author-link=George Szpiro| date=2003 |title=Kepler's Conjecture |publisher=Wiley |isbn=978-0-471-08601-7}}
* {{cite book |last=Ronan |first=Mark |author-link=Mark Ronan| date=2006 |title=Symmetry and the Monster |publisher=Oxford |isbn=978-0-19-280722-9}}

=== Books discussing unsolved problems ===
* {{cite book |first1=Fan|last1= Chung|author-link1=Fan Chung |last2=Graham |first2=Ron |author-link2=Ronald Graham| title=Erdös on Graphs: His Legacy of Unsolved Problems|title-link= Erdős on Graphs |publisher=AK Peters |year=1999 |isbn=978-1-56881-111-6}}
* {{cite book |last1=Croft |first1=Hallard T. |last2=Falconer |first2=Kenneth J. |last3=Guy |first3=Richard K. |author-link2=Kenneth Falconer (mathematician) |author-link3=Richard K. Guy |date=1994 |title=Unsolved Problems in Geometry |publisher=Springer |isbn=978-0-387-97506-1 |url-access=registration |url=https://archive.org/details/unsolvedproblems0000crof }}
* {{cite book |last=Guy |first=Richard K. |author-link=Richard K. Guy |date=2004 |title=Unsolved Problems in Number Theory |publisher=Springer |isbn=978-0-387-20860-2}}
* {{cite book |last1=Klee |first1=Victor |author-link1=Victor Klee |last2=Wagon |first2=Stan |author-link2=Stan Wagon |date=1996 |title=Old and New Unsolved Problems in Plane Geometry and Number Theory |url=https://archive.org/details/oldnewunsolvedpr0000klee |url-access=registration |publisher=The Mathematical Association of America |isbn=978-0-88385-315-3}}
* {{cite book |last=du Sautoy |first=Marcus |author-link=Marcus du Sautoy |date=2003 |title=The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics |publisher=Harper Collins |isbn=978-0-06-093558-0 |url-access=registration |url=https://archive.org/details/musicofprimes00marc }}
* {{cite book |last=Derbyshire |first=John |author-link=John Derbyshire |date=2003 |title=Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics |publisher=Joseph Henry Press |isbn=978-0-309-08549-6 |url-access=registration |url=https://archive.org/details/primeobsessionbe00derb_0 }}
* {{cite book |last=Devlin |first=Keith |author-link=Keith Devlin |date=2006 |title=The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time |publisher=Barnes & Noble |isbn=978-0-7607-8659-8}}
* {{cite book |last1=Blondel |first1=Vincent D. |last2=Megrestski |first2=Alexandre |author-link1=Vincent Blondel |date=2004 |title=Unsolved problems in mathematical systems and control theory |publisher=Princeton University Press |isbn=978-0-691-11748-5}}
* {{cite book |first1=Lizhen|last1= Ji|author-link1=Lizhen Ji |first2=Yat-Sun|last2= Poon |first3=Shing-Tung|last3= Yau|author-link3=Shing-Tung Yau |date=2013 |title=Open Problems and Surveys of Contemporary Mathematics (volume 6 in the Surveys in Modern Mathematics series) (Surveys of Modern Mathematics) |publisher=International Press of Boston |isbn=978-1-57146-278-7}}
* {{cite journal |last=Waldschmidt |first=Michel |author-link=Michel Waldschmidt |date=2004 |title=Open Diophantine Problems |journal=Moscow Mathematical Journal |issn=1609-3321 |zbl=1066.11030 |volume=4 |number=1 |pages=245–305 |url=http://www.math.jussieu.fr/~miw/articles/pdf/odp.pdf |doi=10.17323/1609-4514-2004-4-1-245-305 |arxiv=math/0312440 |s2cid=11845578 }}
* {{cite arXiv |last1=Mazurov |first1=V. D. |author-link1=Victor Mazurov |last2=Khukhro |first2=E. I. |eprint=1401.0300v6 |title= Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version) |date= 1 Jun 2015|class=math.GR }}

== External links ==
* [http://faculty.evansville.edu/ck6/integer/unsolved.html 24 Unsolved Problems and Rewards for them]
* [http://www.openproblems.net/ List of links to unsolved problems in mathematics, prizes and research]
* [http://garden.irmacs.sfu.ca/ Open Problem Garden]
* [http://aimpl.org/ AIM Problem Lists]
* [http://cage.ugent.be/~hvernaev/problems/archive.html Unsolved Problem of the Week Archive]. MathPro Press.
* {{cite web|last1=Ball|first1=John M.|author-link=John M. Ball|title=Some Open Problems in Elasticity|url=https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf}}
* {{cite web|last1=Constantin|first1=Peter|author-link=Peter Constantin|title=Some open problems and research directions in the mathematical study of fluid dynamics|url=https://web.math.princeton.edu/~const/2k.pdf}}
* {{cite web|last1=Serre|first1=Denis|author-link=Denis Serre|title=Five Open Problems in Compressible Mathematical Fluid Dynamics|url=http://www.umpa.ens-lyon.fr/~serre/DPF/Ouverts.pdf}}
* [http://unsolvedproblems.org/ Unsolved Problems in Number Theory, Logic and Cryptography]
* [http://www.sci.ccny.cuny.edu/~shpil/gworld/problems/oproblems.html 200 open problems in graph theory] {{Webarchive|url=https://web.archive.org/web/20170515145908/http://www.sci.ccny.cuny.edu/~shpil/gworld/problems/oproblems.html |date=2017-05-15 }}
* [http://cs.smith.edu/~orourke/TOPP/ The Open Problems Project (TOPP)], discrete and computational geometry problems
* [http://math.berkeley.edu/~kirby/problems.ps.gz Kirby's list of unsolved problems in low-dimensional topology]
* [http://www.math.ucsd.edu/~erdosproblems/ Erdös' Problems on Graphs]
* [https://arxiv.org/abs/1409.2823 Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory]
* [http://www.sciencedirect.com/science/article/pii/S0165011414003194 Open problems from the 12th International Conference on Fuzzy Set Theory and Its Applications]
* [http://wwwmath.uni-muenster.de/logik/Personen/rds/list.html List of open problems in inner model theory]
* {{cite web|last1=Aizenman|first1=Michael|author-link=Michael Aizenman|title=Open Problems in Mathematical Physics|url=https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/OPlist.html}}
* [[Barry Simon]]'s [http://math.caltech.edu/SimonPapers/R27.pdf 15 Problems in Mathematical Physics]
* [[Alexandre Eremenko]]. [https://www.math.purdue.edu/~eremenko/uns1.html Unsolved problems in Function Theory]
* [https://www.erdosproblems.com/start Erdos Problems collection]
{{unsolved problems}}

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[[Category:Unsolved problems in mathematics| ]]
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[[Category:Conjectures| ]]
[[Category:Lists of unsolved problems|Mathematics]]
[[Category:Lists of problems]]