index theory

index theory
Name	Index theory
Field	Mathematics
Introduced	1960s
Notable	Atiyah–Singer index theorem, Michael Atiyah, Isadore Singer

index theory

Index theory studies numerical invariants associated to linear operators on manifolds and relations between analytical, topological, and geometric data. It connects spectral properties of elliptic operators with characteristic classes, K-theory, and cobordism, and has ties to representation theory, mathematical physics, and global analysis. Origins involve work by Atiyah–Singer index theorem, Atiyah–Bott, Grothendieck, and developments at institutions like Institute for Advanced Study and Princeton University.

Overview

Index theory links spectra of operators such as the Dirac operator, Hodge Laplacian, and signature operator to topological invariants from K-theory, cohomology, and characteristic classes. Key contributors include Michael Atiyah, Isadore Singer, Raoul Bott, Jean-Michel Bismut, Murray Gell-Mann, Edward Witten, Daniel Quillen, Alexander Grothendieck, Sergei Novikov, Sir Michael Berry, John Milnor, René Thom, William Fulton, David Mumford, Alain Connes, Jean-Pierre Serre, Hermann Weyl, Emil Artin, and Atle Selberg. Institutional hubs for advances include Harvard University, Cambridge University, University of Oxford, University of Chicago, Massachusetts Institute of Technology, Courant Institute, University of California, Berkeley, University of Bonn, Institut des Hautes Études Scientifiques, Max Planck Institute for Mathematics, and National Academy of Sciences.

Analytical Foundations

Analytical foundations rest on elliptic operator theory, functional analysis, and partial differential equations developed by figures like Lars Hörmander, Peter Lax, Mark Kac, Eberhard Hopf, Frigyes Riesz, John von Neumann, Stefan Banach, Marcel Riesz, Norbert Wiener, Ralph Fox, Shmuel Agmon, Louis Nirenberg, and Semyon Sobolev. The Fredholm index, spectral flow, and heat kernel methods draw on contributions from Fritz Noether, Ilya Prigogine, Israel Gelfand, George D. Birkhoff, Charles Ehresmann, Alfred Tarski, Kurt Friedrichs, Gábor Szegő, Mark Kac, and Atle Selberg. Functional calculi and microlocal analysis owe much to Joseph J. Kohn, Richard Melrose, Michael Taylor, Bernard Helffer, Lars Hörmander, and Victor Havin.

Topological and Geometric Index Theorems

Topological formulations use K-theory, cobordism, and characteristic classes from pioneers like Raoul Bott, John Milnor, René Thom, Alexander Grothendieck, Daniel Quillen, Jean-Pierre Serre, Hirzebruch, Friedrich Hirzebruch, Raoul Bott, Frank Adams, J. H. C. Whitehead, Henri Cartan, André Weil, Kurt Gödel, Michael Atiyah, Isadore Singer, Shaw, and Jean-Louis Koszul. The Atiyah–Singer result unites the analytic Fredholm index with topological K-theory invariants like the Todd class, Â-genus, and Chern character; development involved Hirzebruch–Riemann–Roch theorem, Grothendieck–Riemann–Roch theorem, Lefschetz fixed-point theorem, Thom isomorphism, Pontryagin classes, Chern classes, Stiefel–Whitney classes, and Euler class. Geometric refinements include families index theorems by Michael Atiyah and Raoul Bott, and local index formulas by Berline–Getzler–Vergne, Jean-Michel Bismut, Nicole Berline, Ezra Getzler, and Michèle Vergne.

Examples and Applications

Classic examples involve the de Rham complex, signature operator, and Dirac operator on spin manifolds, with applications linking to Donaldson theory, Seiberg–Witten theory, Chern–Simons theory, and aspects of string theory and quantum field theory. Index theory impacts invariants in low-dimensional topology studied by William Thurston, Ciprian Manolescu, Simon Donaldson, Edward Witten, Andrew Wiles, Grigori Perelman, Frederick Almgren, Michael Freedman, and Vladimir Arnold. In representation theory and number theory it connects to the Selberg trace formula, characters of Lie groups, and the Arthur–Selberg theory involving Atle Selberg, Harish-Chandra, Robert Langlands, James Arthur, Peter Sarnak, Andrew Wiles, Gérard Laumon, Pierre Deligne, and Igor Shafarevich. Index computations inform spectral geometry problems like "Can one hear the shape of a drum?" originally framed by Mark Kac, and analytical torsion developed by Daniel B. Ray and Isadore Singer relates to Reidemeister torsion from John H. Conway and Kurt Reidemeister.

Extensions and Generalizations

Extensions include equivariant index theory, index theory on manifolds with boundary, and noncommutative index theory pioneered by Alain Connes and Nigel Higson. Equivariant results involve George Lusztig, David Kazhdan, G. Lusztig, Robert Bott, Victor Kac, William Fulton, and Shrawan Kumar. Boundary and corner generalizations use Atiyah–Patodi–Singer eta invariants and contributions from M. F. Atiyah, V. K. Patodi, I. M. Singer, Melrose, Richard Melrose, Boris Lerner, Luc Boutet de Monvel, and Paul Loya. Analogs in noncommutative geometry and K-homology tie to Alain Connes, John Roe, Nigel Higson, Paul Baum, Gennadi Kasparov, Higson–Roe, and Gennadi Kasparov's KK-theory, with applications to the Novikov conjecture studied by Sergei Novikov, Pierre Baum, Boris Feigin, Fedor Bogomolov, Misha Gromov, and Grigori Perelman.

Computational and Analytical Methods

Computational methods for index calculations employ heat kernel asymptotics, zeta regularization, and spectral flow techniques developed by Shubin, Seeley, Wodzicki, Ray–Singer, Elmar Schrohe, Peter Gilkey, Paul Gilkey, Bernard Booß-Bavnbek, Krzysztof P. Wojciechowski, Nicole Berline, Ezra Getzler, Jean-Michel Bismut, and Scott Axelrod. Analytic tools intersect with numerical analysis, spectral theory, and operator algebras from researchers at SIAM, American Mathematical Society, European Mathematical Society, Institute of Mathematics, CNRS, Deutsche Forschungsgemeinschaft, and National Science Foundation. Computational topology packages and symbolic algebra systems at Wolfram Research, SageMath, GNU Octave, and research codes in universities support explicit computations of indices, eta invariants, and heat coefficients inspired by work of Dmitri Pavlov, Vladimir Voevodsky, David Kazhdan, Jim Simons, and Serge Lang.

Category:Mathematics