He was a truly wonderful and lively person, with an effervescent and inspiring personality. He valued collaboration in Mathematics and loved intense mathematical discussions and exchanges.
He seemed to derive many of his insights and ideas in the process of talking. His many insights made him a distinguished geometer, a Fields Medalist and winner of the Abel Prize, whose work creatively combined Analysis, Differential Geometry and Topology. His major contributions were to Topological K-Theory, Index Theorem for linear elliptic operators, and mathematical aspects of gauge theory arising in theoretical physics.
He developed Topological K-Theory together with Friedrich Hirzebruch, which associates to a topological space X, a topological invariant, an abelian group K X , using vector bundles on X, in a way reminiscent of associating homology groups of X. K-Theory proved to be a remarkably powerful tool in topology, helping to solve some outstanding problems in the area and giving simpler proofs to some known results.
His best-known result is the celebrated Atiyah-Singer Index Theorem, which is a vast generalisation of famous results like the Riemann-Roch theorem and intimately related to K-theory. Given a linear elliptic operator between sections of two vector bundles on a compact smooth manifold, the index of the operator is defined to be the difference of the dimensions of the space of solutions of the operator and that of its adjoint. The index theorem gives a formula for the index of the operator in terms of topological invariants of the manifold and the vector bundles.
A major related work is the Atiyah-Patodi-Singer index theorem, which deals with elliptic operators on manifolds with boundary and non-local boundary conditions. Motivated by insights from theoretical physics, Atiyah worked extensively on gauge theory as well. From a mathematical point of view, gauge theory is the study of connections on a principal bundle, the related Yang-Mills equation satisfied by connections, a non-linear elliptic analogue of the Maxwell equations, and the geometry of the space of solutions of the Yang-Mills equation.
Atiyah and Bott made an extensive study of the Yang-Mills equation on compact Riemann surfaces and its relation to the moduli spaces of holomorphic vector bundles on the Riemann surface. This work was very influential and popularised these moduli spaces among physicists. I was fortunate enough to know Professor Atiyah well; it was easy to interact with him. I had close and fruitful contact with him over the years, and had several conversations with him on moduli problems and some aspects of the Index theorem.
Chern forms of line bundles. Curvature of connections. Positive line bundles. Kodaira vanishing theorem. Hodge structures and Hodge ltration. Kodaira's proof of Kodaira embedding. Topological applications. Calculation of Hodge numbers of varieties in examples, including complete intersections. Examples of variation of Hodge structure. Skip to main content.
Index Theory of Elliptic Operators on Spaces with Symmetries
Menu Admissions. On the index of nonlocal elliptic operators for the group of dilations more. Pure Mathematics. View on springerlink.
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We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah-Patodi-Singer problems in subspaces. The boundary We prove the Fredholm property for elliptic boundary value problems.
This class also contains nonlocal boundary value problems of math. We compute a topological obstruction similar to the Atiyah-Bott obstruction to the existence of elliptic boundary conditions for a given operator. Geometric operators with a nontrivial obstruction are given. On the index of elliptic operators for the group of dilations more. We obtain an ellipticity condition, which implies that the problem has the Fredholm property, compute the index, and study how the We obtain an ellipticity condition, which implies that the problem has the Fredholm property, compute the index, and study how the index depends on the exponent of the Sobolev space in which the problem is considered.
Bibliography: 15 titles. View on dx. On the index of elliptic operators associated with a diffeomorphism of a manifold more. On the homotopy classification of elliptic operators on manifolds with corners more. We obtain a classification of elliptic operators modulo stable homotopy on manifolds with edges this is in some sense the simplest class of manifolds with nonisolated singularities.
We show that the operators are classified by the We show that the operators are classified by the K-homology group of the manifold. To this end, we construct exact sequence in elliptic theory isomorphic to the K-homology exact sequence. The main part of the proof is the computation of the boundary map using semiclassical quantization. View on arxiv. Homotopy classification of elliptic operators on stratified manifolds more.tegsatowil.ga
We find the stable homotopy classification of elliptic operators on stratified manifolds. As a corollary, we obtain an explicit formula for the obstruction of Atiyah--Bott type to making interior elliptic operators Fredholm.
Applied Mathematics and Pure Mathematics. Defect of index in the theory of non-local problems and the -invariant more. This paper is concerned with elliptic theory on manifolds the boundary of which is a cover.
Non-local boundary value problems are considered and their indices are calculated. The Atiyah-Patodi-Singer problem is studied on such manifolds The Atiyah-Patodi-Singer problem is studied on such manifolds. For non-trivial covers the defect of the index is calculated.
Elliptic theory and noncommutative geometry. Nonlocal elliptic operators more. Eta Invariant and the Spectral Flow1 more. The index problem on manifolds with edges more. Contemporary Mathematics. View on researchgate. We consider the index problem for a wide class of nonlocal elliptic operators on a smooth closed manifold, namely differential operators with shifts induced by the action of an isometric diffeomorphism.
The key to the solution is the The key to the solution is the method of uniformization: We assign to the nonlocal problem a pseudodifferential operator with the same index, acting in sections of an infinite-dimensional. Mathematical Analysis and Indexation. The Eta-invariant and Pontryagin duality in K-theory more. The topological significance of the spectral Atiyah-Patodi-Singer eta-invariant is investigated under the parity conditions of P.
We show that twice the fractional part of the invariant is computed by the linking pairing in We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold.