Atiyah–Singer index theorem

Atiyah-Singer index theoremindex theoryindex theoremindex theoremsAtiyah-Singer theoremAtiyah–Patodi–Singer theoremAtiyah–Singer index theorem (section on symbol of operator)indexindex theorem for elliptic complexesindex theories
In differential geometry, the Atiyah–Singer index theorem, proved by, states that for an 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).wikipedia
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Friedrich Hirzebruch

HirzebruchFriedrich Ernst Peter HirzebruchHirzebruch, Friedrich
Friedrich Hirzebruch and Armand Borel had proved the integrality of the  genus of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of the Dirac operator (which was rediscovered by Atiyah and Singer in 1961). * - Personal accounts on Atiyah, Bott, Hirzebruch and Singer.
it was also a precursor of the Atiyah–Singer index theorem.

K-theory

K-theoriesK theoryalgebraic K_i
Their first published proof replaced the cobordism theory of the first proof with K-theory, and they used this to give proofs of various generalizations in the papers.
Examples of results gleaned from the K-theory approach include the Grothendieck–Riemann–Roch theorem, Bott periodicity, the Atiyah–Singer index theorem, and the Adams operations.

Elliptic operator

ellipticelliptic differential operatorelliptic partial differential equations
In differential geometry, the Atiyah–Singer index theorem, proved by, states that for an 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).
Weak ellipticity is nevertheless strong enough for the Fredholm alternative, Schauder estimates, and the Atiyah–Singer index theorem.

Michael Atiyah

Sir Michael AtiyahMichael Francis AtiyahAtiyah
* - Personal accounts on Atiyah, Bott, Hirzebruch and Singer.
His best known result, the Atiyah–Singer index theorem, was proved with Singer in 1963 and is used in counting the number of independent solutions to differential equations.

Chern class

Chern characterChern numberfirst Chern class
The Chern classes offer some information about this through, for instance, the Riemann–Roch theorem and the Atiyah–Singer index theorem.

Isadore Singer

Isadore M. SingerSingerI. M. Singer
* - Personal accounts on Atiyah, Bott, Hirzebruch and Singer.
Singer is noted for his work with Michael Atiyah proving the Atiyah–Singer index theorem in 1962, which paved the way for new interactions between pure mathematics and theoretical physics.

Vijay Kumar Patodi

V. K. PatodiPatodiPatodi, Vijay Kumar
He was the first mathematician to apply heat equation methods to the proof of the Index Theorem for elliptic operators.

Riemann–Roch theorem

Riemann-Roch theoremRiemann–Roch formulaRiemann–Roch theorem for algebraic curves
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. Some of the motivating examples included the Riemann–Roch theorem and its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem.
After that the Atiyah–Singer index theorem opened another route to generalization.

Hirzebruch signature theorem

Thom–Hirzebruch signature theorem
Some of the motivating examples included the Riemann–Roch theorem and its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem.
The signature theorem is a special case of the Atiyah–Singer index theorem for

Hirzebruch–Riemann–Roch theorem

Hirzebruch-Riemann-Roch theorem
Some of the motivating examples included the Riemann–Roch theorem and its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem.

Noncommutative geometry

non-commutative geometrynoncommutativenon-commutative
Several generalizations of now-classical index theorems allow for effective extraction of numerical invariants from spectral triples.

Equivariant index theorem

equivariant index theory
If the element is neutral, then the theorem reduces to the usual index theorem.

Elliptic complex

They also arise in connection with the Atiyah-Singer index theorem and Atiyah-Bott fixed point theorem.

Luis Álvarez-Gaumé

Luis Alvarez-Gaume
Álvarez-Gaumé is also known for a physical proof of the Atiyah-Singer theorem using supersymmetry.

Manifold

manifoldsboundarymanifold with boundary
This leads to such functions as the spherical harmonics, and to heat kernel methods of studying manifolds, such as hearing the shape of a drum and some proofs of the Atiyah–Singer index theorem.

Heat equation

heat diffusionheatanalytic theory of heat
gave a new proof of the index theorem using the heat equation, see e.g.
An abstract form of heat equation on manifolds provides a major approach to the Atiyah–Singer index theorem, and has led to much further work on heat equations in Riemannian geometry.

Signature operator

This follows from the Atiyah–Singer index theorem applied to the following signature operator.
The Heat Equation approach to the Atiyah-Singer index theorem can then be used to show that:

Fredholm operator

Fredholm indexFredholmsemi-Fredholm
As the elliptic differential operator D has a pseudoinverse, it is a Fredholm operator.
The Atiyah-Singer index theorem gives a topological characterization of the index of certain operators on manifolds.

Symbol of a differential operator

principal symbolsymbolleading symbol
If D is a differential operator on a Euclidean space of order n in k variables, then its symbol is the function of 2k variables

Robert Thomas Seeley

Robert SeeleySeeley, Robert Thomas
Robert Thomas Seeley (born February 26, 1932, in Bryn Mawr, Pennsylvania, United States–died November 30, 2016, in Newton, Massachusetts) was a mathematician who worked on pseudo differential operators and the heat equation approach to the Atiyah–Singer index theorem.

Chiral anomaly

Adler–Bell–Jackiw anomalyelectroweak burningchiral anomalies
Fujikawa derived this anomaly using the correspondence between functional determinants and the partition function using the Atiyah–Singer index theorem.

Raoul Bott

BottR. BottBott, Raoul
* - Personal accounts on Atiyah, Bott, Hirzebruch and Singer.
Bott made important contributions towards the index theorem, especially in formulating related fixed-point theorems, in particular the so-called 'Woods Hole fixed-point theorem', a combination of the Riemann–Roch theorem and Lefschetz fixed-point theorem (it is named after Woods Hole, Massachusetts, the site of a conference at which collective discussion formulated it).

Pseudo-differential operator

pseudodifferential operatorpseudodifferential operatorspseudo-differential operators
However, it is an elliptic pseudodifferential operator.)
They played an influential role in the second proof of the Atiyah–Singer index theorem via K-theory.

Grothendieck–Riemann–Roch theorem

Grothendieck–Hirzebruch–Riemann–Roch theoremGrothendieck-Riemann-Roch theoremRiemann–Roch theorem
The theorem has been very influential, not least for the development of the Atiyah–Singer index theorem.