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Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

Institutskolloquium am 30.06.2015

Wann 30.06.2015 ab 17:00 (Europe/Berlin / UTC200) iCal
Wo Berlin-Adlershof, Rudower Chaussee 25, 1.013

Kolloquium des Institutes für Mathematik
Prof. Dr. Jean-Michel B i s m u t (Université Paris-Sud, Orsay)
Euler characteristic and the trace formula

Tuesday, June 30, 2015, at 17 c.t.

Institut für Mathematik der Humboldt-Universität zu Berlin
Rudower Chaussee 25, 12489 Berlin-Adlershof
Room 013, House 1, first floor

 

 

Abstract: If X is a Riemannian manifold, its Euler characteristic is a global invariant, that can be computed 'locally' via the Chern-Gauss-Bonnet theorem. If X is a real vector space, this result specializes to the fact that 1, the dimension of the kernel of the harmonic oscillator, is just the integral of the Gaussian distribution.
If X is a compact Riemannian manifold, consider the trace of its heat kernel exp (tΔX /2). We will ask the question of whether it can be viewed as an Euler characteristic. In this context, the family of harmonic oscillators along the fibres of the tangent bundle play a key role. On locally symmetric spaces, they can be suitably coupled to ΔX, so as to produce a family of hypoelliptic operators on the total space of a vector bundle over X, that interpolates between -ΔX /2 and the generator of the geodesic flow, this deformation being essentially isospectral. Ultimately, an explicit version of Selberg's trace formula is obtained that is valid for all locally symmetric spaces, and which can be viewed as a generalized Lefschetz fixed point formula.
As a by-product of these constructions, the roles of the exterior algebra Λ. (T* X) and of the symmetric (or polynomial) algebra S. (T* X) have been put on the same footing. This is especially relevant to probability theory, since a completion of S. (T* X) can be identified with the L2 space for the Gaussian distribution.

 

About the speaker: Jean-Michel Bismut graduated from École Polytechnique in 1970 and got his PhD in 1973 at Université Paris VI, under the direction of Jacques-Louis Lions; since 1981 he is professor at Université Paris-Sud in Orsay. Bismut has published groundbreaking work in two fields, starting in probability theory and moving over to global analysis, with significant cross fertilization. He is probably best known for his analytic proofs of the Atiyah-Singer family index theorem which formed the basis for an impressive and voluminous work in search of higher invariants of manifolds and the analysis that makes them appear.

Jochen Brüning, HU Berlin