High energy limits of eigensections of the Dirac operator
Radon measures and Nef line bundles
Refined analytic torsion
Abstract: For a representation of the fundamental group of a compact oriented odd-dimensional manifold we define a refinement of the Ray-Singer torsion associated to this representation. If the representation is acyclic then our new invariant is a non-zero complex number, which can be viewed as an analytic counterpart of the refined combinatorial torsion introduced by Turaev.
The refined analytic torsion is a holomorphic function of the space of acyclic representation of the fundamental group. When the representation is unitary, the absolute value of the refined analytic torsion is equal to the Ray-Singer torsion, while its phase is determined by the eta-invariant. The fact that the Ray-Singer torsion and the eta-invariant can be combined into one holomorphic function allows to use methods of complex analysis to study both invariants. I will present several applications of this method. In particular, I will calculate the ration of the refined analytic torsion and the Turaev torsion.
(Joint work with Thomas Kappeler)
BFK-gluing formula for zeta-determinants of Laplacians.
Abstract: In early 90's Burghelea, Friedlander and Kappeler proved the gluing formula for zeta-determinants of elliptic operators on a compact closed manifold. They discussed the gluing formula for general elliptic operators and hence it is not easy to follow their argument. However, if we consider only the case of concrete Laplacians, we can reduce their argument to a much simpler form, which I'm going to present in elementary and self-contained way. Their formula contains a constant which is determined by local data and I'm going to mension this constant in some cases.
Generating spectral gaps by geometry. Riemaniann coverings and residually finite groups
In this talk we will present recent results on Riemannian coverings (X,g) -> (M,g) with a residually finite covering group \Gamma and compact base space (M,g). In particular, we will explain two procedures resulting in a family of deformed coverings (X,g_\epsilon) -> (M,g_\epsilon) such that the spectrum of the Laplacian on X has at least a prescribed (finite) number of spectral gaps provided \epsilon is small enough. One of the procedures can be applied only to the subclass of type I groups. Under a further technical assumption we can also obtain information on the nature of the spectrum and give an asymptotic estimate for the number N(t) of components of the spectrum that intersect the interval [0,t]. (This is joint work with Olaf Post).
An index for gauge-invariant operators (joint with Victor Nistor)
Abstract: In [1], we have introduced the gauge-equivariant
-theory group
of a bundle
endowed with a continuous action of a bundle
of compact Lie groups
. These groups are the
natural range for the analytic index of a family of
gauge-invariant elliptic operators (a family of elliptic
operators invariant with respect to the action of a bundle of
compact groups) and are a version of twisted -theory.
This analytical index was defined and studied in [1].
In [2], we continue our study of
gauge-equivariant -theory. In particular, we introduce and
study products, which helps us establish the Thom isomorphism in
gauge-equivariant -theory. Then we construct push-forward maps
and define the topological index of a gauge-invariant family.
In [3] the corresponding index theorem is obtained.
[1]
V. Nistor and E. Troitsky, An index for gauge-invariant operators
and the Dixmier-Douady
invariant. Trans. Am. Math. Soc. 356 (2004), no. 1, 185-218.
(math.KT/0201207)
[2]
V. Nistor and E. Troitsky, The Thom isomorphism in gauge-equivariant
-theory.
(to appear in Trends in Math., Birkhauser)
(math.KT/0506408)
[3] V. Nistor and E. Troitsky, An index theorem for gauge-invariant families: The case of compact groups, (tentative title, work in progress).
Noncommutative spectral flow and applications
Abstract: We will explain the notion of the noncommutative spectral flow for a path of unbounded selfadjoint Fredholm operators on a Hilbert C*-module. Special cases are the classical spectral flow, Dai-Zhang's family spectral flow and Melrose's divisor flow. We will discuss some applications, for example how to define the classical spectral flow in the case where the Hilbert space depends on the parameter, and how integral formulas for the classical spectral flow can be obtained from its relation with Bott periodicity.
Real Monge-Ampère equations on contact manifolds
Abstract: In parallel with the theory of the extremal (canonical) Käahler metrics we will develop a theory of extremal Sasakian metrics. Eta-Einstein and Sasaki-Einstein metrics are just particular examples. It has been recently discovered that there are examples of Sasaki-Einstein metrics for which the associated Reeb flow dos not have closed orbits. In such case the transverse space is not even Hausdorff though locally it carries a Käahler-Einstein metric. Such metrics were conjectured not to exist by Cheeger and Tian and they first emerged as testing ground for the so-called AdS/CFT Duality Conjecture. They also appear quite naturally in the sutdy of toric contact manifolds precsisely as solutions of a contact version of the complex Monge-Ampère equations or, more generally, as as extrema of the Calabi energy.
Hölder regularity of the Cauchy-Riemann equations on certain singular analytic sets
Abstract: We present a new class of weighted Hölder estimates for the well-known basic d-bar-homotopy formula on the ball. As an application, we construct a solution operator for the Cauchy-Riemann equations on certain strictly pseudoconvex domains in analytic sets and give Hölder etimates for this operator.
Neue komplexe Invarianten von Darstellungen
Abstrakt: Wir umreissen ein Programm, mit dem man zu jeder irreduziblen Darstellung einer halbeinfachen Gruppe eine komplexe Invariante zuordnen kann und erlaeutern deren Bedeutung.
The decay of cuspidal forms
Abstract: We sharpen the estimates of Langlands and explain, what they mean.