A theorem which states that the analytic and topological "indices" are equal for any elliptic differential operator on an -dimensional compactsmooth boundaryless manifold.
For their discovery and proof is this theorem, Atiyah and Singer shared the 2004
Abel prize.
Atiyah, M. F. and Singer, I. M. "The Index of Elliptic Operators on Compact Manifolds." Bull. Amer. Math. Soc.69,
322-433, 1963.Atiyah, M. F. and Singer, I. M. "The Index
of Elliptic Operators I, II, III." Ann. Math.87, 484-604, 1968.Petkovšek,
M.; Wilf, H. S.; and Zeilberger, D. A=B.
Wellesley, MA: A K Peters, p. 4, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.Rognes,
J. "On the Atiyah-Singer Index Theorem." http://www.abelprisen.no/nedlastning/2004/popular_english_2004.pdf.
-dimensional compact smooth 
boundaryless manifold.
