* M. F. Atiyah, R. Bott, L. Garding, " Lacunas for hyperbolic differential operators with constant coefficients I ", Acta Math., 124 ( 1970 ), 109 – 189.

Atiyah, R. Bott, and L. Garding, " Lacunas for hyperbolic differential operators with constant coefficients II ", Acta Math., 131 ( 1973 ), 145 – 206.

* April 22 – Michael Atiyah, British-Lebanese mathematician

An extremely far-reaching generalization of all the above-mentioned theorems is the Atiyah – Singer index theorem.

The coefficients are very important, and are used to compute the cohomology of other spaces using the Atiyah – Hirzebruch spectral sequence.

He 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.

* Atiyah – Patodi – Singer eta invariant

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 ).

The Atiyah – Singer theorem was announced by.

The Atiyah – Singer index theorem solves this problem, and states:

So the Atiyah – Singer index theorem implies some deep integrality properties, as it implies that the topological index is integral.

The proof of this result goes through specific considerations, including the extension of Hodge theory on combinatorial and Lipschitz manifolds,, the extension of Atiyah – Singer's signature operator to Lipschitz manifolds, Kasparov's K-homology and topological cobordism.

This follows from the Atiyah – Singer index theorem applied to the following signature operator.

These have an asymptotic expansion for small positive t, which can be used to evaluate the limit as t tends to 0, giving a proof of the Atiyah – Singer index theorem.

This point of view is adopted in the proof of of the Atiyah – Patodi – Singer index theorem.

The Atiyah – Singer index is only defined on compact spaces, and vanishes when their dimension is odd.

* Free online textbook that proves the Atiyah – Singer theorem with a heat equation approach

* Rafe Mazzeo: The Atiyah – Singer Index Theorem: What it is and why you should care.

* A. J. Wassermann, Lecture notes on the Atiyah – Singer Index Theorem

This effect also occurs in a much deeper result: the Atiyah – Singer index theorem states that the index of certain differential operators can be read off the geometry of the involved spaces.

# REDIRECT Atiyah – Singer index theorem

Hirzebruch and 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 ).

To handle these operators, Atiyah, Patodi and Singer introduced global boundary conditions equivalent to attaching a cylinder to the manifold along the boundary and then restricting the domain to those sections that are square integrable along the cylinder.

* This describes the original proof of the theorem ( Atiyah and Singer never published their original proof themselves, but only improved versions of it.

* Raussen, Skau, Interview with Atiyah, Singer, Notices AMS 2005.

* conjectures about the Atiyah – Singer index theorem ;

It performed an important role, historically speaking, in developments in topology in the 1950s and early 1960s, in particular in the Hirzebruch – Riemann – Roch theorem, and in the first proofs of the Atiyah – Singer index theorem.

In topology, by applying the same construction to vector bundles, Michael Atiyah and Friedrich Hirzebruch defined K ( X ) for a topological space X in 1959, and using the Bott periodicity theorem they made it the basis of an extraordinary cohomology theory.

Weak ellipticity is nevertheless strong enough for the Fredholm alternative, Schauder estimates, and the Atiyah – Singer index theorem.

it was also a precursor of the Atiyah – Singer index theorem.

* Atiyah – Bott fixed-point theorem

The theorem has been very influential, not least for the development of the Atiyah – Singer index theorem.

In the commutative case, the lemma is a simple consequence of a generalized form of the Cayley – Hamilton theorem, an observation made by Michael Atiyah ( 1969 ).

Atiyah defines abstract elliptic operators on arbitrary metric spaces.

The local case was worked out by Atiyah and Bott, but they showed that many interesting operators ( e. g., the signature operator ) do not admit local boundary conditions.

A later paper of Adams and Michael F. Atiyah uses the Adams operations to give an extremely elegant and much faster version of the above-mentioned Hopf invariant one result.

Among the awards he has received are the Bôcher Memorial Prize ( 1969 ) and the Steele Prize for Lifetime Achievement ( 2000 ), both from the American Mathematical Society, the Eugene Wigner Medal ( 1988 ), the National Medal of Science ( 1983 ), the Abel Prize ( 2004, shared with Michael Atiyah ), and the James Rhyne Killian Faculty Achievement Award from MIT ( 2005 ).

In the 1980s the category with compact manifolds as objects and cobordisms between these as morphisms played a basic role in the Atiyah – Segal axioms for topological quantum field theory, which is an important part of quantum topology.

This does not mean that the groups can be effectively computed once one knows the cobordism theory of a point and the homology of the space X, though the Atiyah – Hirzebruch spectral sequence gives a starting point for calculations.

* Sir Michael Atiyah ( 1997-2002 ), a mathematician, was awarded the 1966 Fields Medal, for his work in developing K-theory.

In 1969-70, he was a visiting member of the Institute for Advanced Study in Princeton, where he came under the influence of Michael Atiyah.

Atiyah suggested a set of axioms for topological quantum field theory which was inspired by Segal's proposed axioms for conformal field theory, and the Witten's idea of the geometric meaning of supersymmetry,.

Anomalous theories have been studied in great detail and are often founded on the celebrated Atiyah – Singer index theorem or variations thereof ( see, for example, the chiral anomaly ).