Editing Atiyah–Singer index theorem (section)

{{Short description|Mathematical result in differential geometry}}
{{Infobox mathematical statement
| name = Atiyah–Singer index theorem
| image = 
| caption = 
| field = [[Differential geometry]]
| conjectured by = 
| conjecture date = 
| first proof by = [[Michael Atiyah]] and [[Isadore Singer]]
| first proof date = 1963
| open problem = 
| known cases = 
| implied by =
| equivalent to = 
| generalizations = 
| consequences = [[Chern–Gauss–Bonnet theorem]]<br>[[Grothendieck–Riemann–Roch theorem]]<br>[[Hirzebruch signature theorem]]<br>[[Rokhlin's theorem]]
}}

In [[differential geometry]], the '''Atiyah–Singer index theorem''', proved by [[Michael Atiyah]] and [[Isadore Singer]] (1963),{{sfn|Atiyah|Singer|1963}} states that for an [[elliptic operator|elliptic differential operator]] on a [[compact manifold]], the '''analytical index''' (related to the dimension of the space of solutions) is equal to the '''topological index''' (defined in terms of some topological data).  It includes many other theorems, such as the [[Chern–Gauss–Bonnet theorem]] and [[Riemann–Roch theorem]], as special cases, and has applications to [[theoretical physics]].{{sfn|Kayani|2020}}{{sfn|Hamilton|2020|p=11}}