Editing Hilbert's problems (section)

==Follow-ups==
Since 1900, mathematicians and mathematical organizations have announced problem lists but, with few exceptions, these have not had nearly as much influence nor generated as much work as Hilbert's problems.

One exception consists of three conjectures made by [[André Weil]] in the late 1940s (the [[Weil conjectures]]). In the fields of [[algebraic geometry]], number theory and the links between the two, the Weil conjectures were very important.<ref>{{Cite journal| last1=Weil | first1=André | author1-link=André Weil | title=Numbers of solutions of equations in finite fields | url=http://www.ams.org/bull/1949-55-05/S0002-9904-1949-09219-4/home.html | doi=10.1090/S0002-9904-1949-09219-4  | mr=0029393 | year=1949 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=55 | pages=497–508 | issue=5| doi-access=free }}</ref><ref name="Browder American Mathematical Society">{{cite book | last=Browder | first=Felix E. | title=Mathematical developments arising from Hilbert problems. | publisher=American Mathematical Society | publication-place=Providence | date=1976 | isbn=0-8218-1428-1 | oclc=2331329 | page=}}</ref> The first of these was proved by [[Bernard Dwork]]; a completely different proof of the first two, via [[Étale cohomology|ℓ-adic cohomology]], was given by [[Alexander Grothendieck]]. The last and deepest of the Weil conjectures (an analogue of the Riemann hypothesis) was proved by [[Pierre Deligne]]. Both Grothendieck and Deligne were awarded the [[Fields medal]]. However, the Weil conjectures were, in their scope, more like a single Hilbert problem, and Weil never intended them as a programme for all mathematics. This is somewhat ironic, since arguably Weil was the mathematician of the 1940s and 1950s who best played the Hilbert role, being conversant with nearly all areas of (theoretical) mathematics and having figured importantly in the development of many of them.

[[Paul Erdős]] posed hundreds, if not thousands, of mathematical [[Paul Erdős#Erdős's problems|problems]], many of them profound. Erdős often offered monetary rewards; the size of the reward depended on the perceived difficulty of the problem.<ref name="Chung Graham 1999 Erdös ">{{cite book | last1=Chung | first1=Fan R. K. | last2=Graham | first2=Ronald L. | title=Erdös on Graphs: his legacy of unsolved problems | publisher=A K Peters/CRC Press | publication-place=Natick, Mass | date=1999-06-01 | isbn=978-1-56881-111-6| oclc=42809520 | page=}}</ref>

The end of the millennium, which was also the centennial of Hilbert's announcement of his problems, provided a natural occasion to propose "a new set of Hilbert problems". Several mathematicians accepted the challenge, notably Fields Medalist [[Steve Smale]], who responded to a request by [[Vladimir Arnold]] to propose a list of 18 problems ([[Smale's problems]]).

At least in the mainstream media, the ''de facto'' 21st&nbsp;century analogue of Hilbert's problems is the list of seven [[Millennium Prize Problems]] chosen during 2000 by the [[Clay Mathematics Institute]]. Unlike the Hilbert problems, where the primary award was the admiration of Hilbert in particular and mathematicians in general, each prize problem includes a million-dollar bounty. As with the Hilbert problems, one of the prize problems (the [[Poincaré conjecture]]) was solved relatively soon after the problems were announced.

The Riemann hypothesis is noteworthy for its appearance on the list of Hilbert problems, Smale's list, the list of Millennium Prize Problems, and even the Weil conjectures, in its geometric guise. Although it has been attacked by major mathematicians of our day, many experts believe that it will still be part of unsolved problems lists for many centuries. Hilbert himself declared: "If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proved?"<ref name="Clawson">{{cite book |title=Mathematical Mysteries: The beauty and magic of numbers |first=Calvin C. |last=Clawson |date=8 December 1999 |page=258 |isbn=9780738202594 |lccn=99-066854 |publisher=Basic Books}}</ref>

In 2008, [[DARPA]] announced its own list of 23&nbsp;problems that it hoped could lead to major mathematical breakthroughs, "thereby strengthening the scientific and technological capabilities of the [[United States Department of Defense|DoD]]".<ref name="Cooney 2008 DARPA 23 questions">{{cite web | last=Cooney | first=Michael | title=The world's 23 toughest math questions | website=Network World | date=30 September 2008 | url=https://www.networkworld.com/article/905224/security-the-world-s-23-toughest-math-questions.html | access-date=7 April 2024}}</ref><ref name="DARPA_math_2019">{{Cite web |date=2008-09-26 |title=DARPA Mathematical Challenges |url=https://www.fbo.gov/?s=opportunity&mode=form&id=c120bc7171c203aa5f4b3903aa08e558&tab=core&_cview=0 |url-status=dead |archive-url=https://web.archive.org/web/20190112150040/https://www.fbo.gov/?s=opportunity&mode=form&id=c120bc7171c203aa5f4b3903aa08e558&tab=core&_cview=0 |archive-date=2019-01-12 |access-date=2021-03-31}}</ref> The DARPA list also includes a few problems from Hilbert's list, e.g. the Riemann hypothesis.