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SPRING SCHOOL “INDEX THEORY“

GRADUIERTENKOLLEG CURVATURE, CYCLES, AND COHOMOLOGY REGENSBURG

The Graduiertenkolleg Curvature, Cycles, and Cohomology organizes a spring school on Index theory from May 15 to May 18 2011 in Hesselberg. In this spring school we will work through the heat equation proof of the Atiyah-Singer index theorem for compatible Dirac operators. It will be based on the first few chapters of the Book Heat Kernels and Dirac Operators by Berline, Getzler, Vergne [1]. All unspecified references refer to this book. Here is the description of the talks. (1) Recap: fibre bundles and connections (45 min) [1, Sec. 1.1] [7, App. A] [6, Ch. II] • introduce G- and connections on those • introduce vector bundles and connections on those • transition between principal bundles and vector bundles and corresponding connections (Frame bundle, associated vector bundle) • explain curvature and parallel transport (2) Recap: Riemannian geometry (45min) [1, Sec 1.2], [6, IV], [3] • Riemanian metric • Levi-Civita connection • special properties of the Riemannian curvature tensor, Ricci and scalar curva- ture • Normal coordinates, expansion of metric, connection etc up to second order in normal coordinates (3) Recap: Topology (60min) [7, App. B], [4], • basic topology (Homology, Cohomology with coefficients in an abelian group) • de Rham cohomology, de Rham isomorphism, some examples (explain calcu- lations with Mayer-Vietoris) • classifying spaces for vector and principal bundles • structures (e.g. Spin) as lifting • topological obstructions against structures and classification (4) Recap: Analysis on Manifolds (45min) [9, Sec. 5], [8, Sec.7], [2, 2.4, 1.5] • smooth sections and distributions for vector bundles (use densities properly) • the scale of Sobolev spaces • elliptic operators • Rellich theorem • trace class embeddings • Fredholm index (5) Recap: spectral theory (45 min) [http://www.uni-regensburg.de/Fakultaeten/nat Fak I/Bunke/spek.dvi], [10, Ch. VI, VII] • unbounded selfadjoint operators • spectral theorem • function calculus • essential selfadjointness of symmetric elliptic operators • solution of the heat equation for elliptic self (existence of smooth integral kernel, trace class, trace as integral over local trace) (6) Characteristic classes of principal bundles. (45 min) 1 2 GRADUIERTENKOLLEG CURVATURE, CYCLES, AND COHOMOLOGY REGENSBURG

• Explain the Chern-Weyl homomorphism from invariant polynomials on the Lie algebra to characteristic forms. (Use Wikipedia page as entry) • explain the splitting principle and how formal power series give rise to char- acteristic classes for U(n) [5, 1.6] • same for even power series and O(n) • Explain Euler class Aˆ-class, Todd class and Hirzebruch Signature class [1, Sec. 1.5, 1.6] [1, 1.5,1.6] (7) Volterra series representation of the heat kernel (60 min) • we already know that the heat operator has a kernel, derive the properties stated in Def. 2.15 • show that these properties characterize the heat kernel uniquely (Prop. 2.17 (1) • define the notion of an approximate solution (conditions given in Thm. 2.20) • show the representation of the heat kernel by a Volterra series Thm 2.23 (8) Formal solution and asymptotic expansion (60 min) • Define formal solutions Def. 2.25 • show existence of formal solutions Thm. 2.26 • show existence of approximate solution Thm 2.29 (9) Clifford algebra and [7, Ch. 1], [1] • Introduce Cliffordalgebras Def. 3.1 • Introduce the Def. 3.9 • Define the representation Def. 3.20 • Introduce symbol map and show Prop. 3.21 (10) Dirac operators (45min) [7, Ch.1], [1] • Define the notion of a Dirac bundle (Clifford bundle Def. 3.32 with compatible connection) • Define Dirac operator for Dirac bundle Def. 3.36 • Introduce (Def. 3.33) and Spinor bundle as an example for a Dirac bundle • Explain the twisting construction Def. 3.23 (3) • Show ellipticity, selfadjointness, and that the square is a generalized Laplacian • Show Lichnerowicz formula Thm 3.52 (11) Mehler’s formula, Rescaling and the proof of the local index theorem (75min) • represent the index by the Mac Kean-Singer formula Thm 3.50 • explain notation and state Thm 4.1 and Thm. 4.2 • give their proof according to Sec. 4.3 (12) Examples of Dirac operators from geometry (75min) Explain the following opera- tors as twisted Dirac operators. Spezialize the index formula. Draw some topolog- ical consequences. Use Sec. 3.6 and 4.1 and additional consultation. • Spin Dirac operator (Aˆ 6= 0 as an obstruction against positive scalar curvature) • Euler operator (Chern-Gauss-Bonnet formula for the Euler characteristic) SPRING SCHOOL “INDEX THEORY“ 3 • Signature operator (Hirzebruch signatur theorem), derive some divisiblity (Rochlin from comparison with Spin-Dirac operator in 4 dimensions) (13) Dirac operators in K¨ahlergeometry (75min) • Short intro to K¨ahlergeometry (Sec. 3.6) • Dolbeault operator as twisted Dirac Prop. 3.67 • Derive Riemann-Roch (Thm 4.9) • if time permits, show Kodaira vanishing (Thm 3.72)

In the first talks (Recap’s) we collect basic material which forms the background for local index theory. These topics should not be completely new for the participanting students. The middle part of the school is devoted to the actual proof of the local index theorem based on the Book mentioned above. At the end we want to give some outlooks on further developments of local index theory. Applications should be sent to

[email protected] The Graduiertenkolleg will cover the accomodation. Applicants may ask for additional travel support.

References [1] Nicole Berline, Ezra Getzler, and Mich`eleVergne. Heat kernels and Dirac operators. Grundlehren Text Editions. Springer-Verlag, Berlin, 2004. Corrected reprint of the 1992 original. [2] B. Booss and D. D. Bleecker. Topology and analysis. Universitext. Springer-Verlag, New York, 1985. The Atiyah-Singer index formula and gauge-theoretic physics, Translated from the German by Bleecker and A. Mader. [3] Manfredo Perdig˜ao do Carmo. Riemannian geometry. Mathematics: Theory & Applications. Birkh¨auserBoston Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. [4] Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. [5] Friedrich Hirzebruch, Thomas Berger, and Rainer Jung. Manifolds and modular forms. Aspects of Mathematics, E20. Friedr. Vieweg & Sohn, Braunschweig, 1992. With appendices by Nils-Peter Sko- ruppa and by Paul Baum. [6] Shoshichi Kobayashi and Katsumi Nomizu. Foundations of differential geometry. Vol. I. Wiley Classics Library. John Wiley & Sons Inc., New York, 1996. Reprint of the 1963 original, A Wiley-Interscience Publication. [7] H. Blaine Lawson, Jr. and Marie-Louise Michelsohn. , volume 38 of Princeton Mathe- matical Series. Princeton University Press, Princeton, NJ, 1989. [8] Michael Renardy and Robert C. Rogers. An introduction to partial differential equations, volume 13 of Texts in Applied Mathematics. Springer-Verlag, New York, second edition, 2004. [9] John Roe. Elliptic operators, topology and asymptotic methods, volume 395 of Pitman Research Notes in Mathematics Series. Longman, Harlow, second edition, 1998. [10] Dirk Werner. Funktionalanalysis. Springer-Verlag, Berlin, extended edition, 2000.