|Purpose of the workshop
The purpose of this workshop is to gather some leading experts in the field and describe the present state of the art. There will be ample time and opportunity for discussions.
Ulrich Bunke, Modular Dirac operators and the f-invariant
Abstract: The eta invariant of the Dirac operator on an odd-dimensional manifold twisted by a certain bundle of vertex operator algebras is a real modular form up to integral ones. A partition of the manifold along a framed hypersurface gives rise to a decomposition of the associated q-expansion as a sum of the two eta invariants of the components. The deviation of these eta invariants of manifolds with framed boundary from integrality and modularity is expressed in terms of the f-invariant of the bordism class of the boundary considered here as a stable homotopy class of spheres.
Xianzhe Dai, Real analytic torsion form and Bismut-Freed connection
Abstract: We describe a joint work with Weiping Zhang in which we show how the real analytic torsion form can come out of the adiabatic limit of the Bismut-Freed connection, thus answering a question of Bismut-Lott.
Sebastian Goette, Bismut-Lott torsion and Igusa-Klein torsion
Abstract: As shown by Bismut and Zhang, the Witten deformation can be used to compare analytic torsion invariants with their topological counterparts. We study the Witten deformation of the fibrewise Hodge-Laplacian of a family of compact manifolds in the presence of birth-death singularities. Our results can be used to compare Bismut-Lott and Igusa-Klein torsion.
Xiaonan Ma, Superconnection and family Bergman kernels
Abstract: We establish an asymptotic expansion for families of Bergman kernels. The key idea is to use the superconnection formalism as in the local family index theorem. This is a joint work with W. Zhang.
Werner Müller, Hyperbolic manifolds, dynamical zeta functions, and analytic torsion
Abstract: We consider the twisted Ruelle zeta function of a compact hyperbolic manifold of odd dimension. This is a dynamical zeta function associated to the geodesic flow. A general conjecture of D. Fried claims that its value at zero is related to the analytic torsion. We will discuss this conjecture for a special class of representations of the fundamental group.
Weiping Zhang, Geometric quantization for proper actions
Abstract: We describe a recent joint work with Mathai, in which we establish a generalization of the Guillemin-Sternberg geometric quantization conjecture to the noncompact case of proper action of a non-compact group acting on a non-compact space with compact quotient. This essentially solves a conjecture of Landsman and Hochs.
Ken-Ichi Yoshikawa, Analytic torsion and automorphic forms
Abstract: Some years ago, we introduced an invariant of K3 surfaces with involution, which we constructed using equivariant analytic torsion. This invariant, which is a function on the moduli space, is expressed as the Petersson norm of an automorphic form. We would like to talk about the structure of this automorphic form. In many cases, this automorphic form is expressed as the tensor product of a Borcherds lift and Igusa's modular form. If time permits, we will talk about the relation between this automorphic form and the BCOV invariant of some Borcea-Voisin threefolds. The BCOV conjecture suggests that the elliptic modular form appearing in the Borcherds lift should be equivalent to the elliptic Gromov-Witten invariants of some Calabi-Yau threefolds.
Guofang Wang, Transverse Kähler geometry of Sasakian manifolds
Abstract: By exploiting the transverse Kähler geometry of Sasakian manifolds we introduced a Sasaki-Ricci flow, which converges (if it does) to an eta-Einstein metric (a joint work with K. Smozcyk and Yongbing Zhang). In the interesting case (the positive case) an eta-Einstein metric can be deformed homothetically to a Sasaki-Einstein metric. In general there is no convergence and one can only at most expect to obtain a Sasaki-Ricci soliton. In a joint work with A. Futaki and H. Ono we proved the existence of Sasaki-Ricci solitons on toric Sasakian manifolds, one of them is in fact a Sasakian-Einstein metric.
Thursday, 5.06.08 Seminarraum 2, Mathematisches Institut, Weyertal 86-90.
9:40-10:30 U. Bunke
10:50-11.40 G. Wang
12:00-14:00 Lunch at Haus Brecher
14:00-14:50 X. Ma
15:10-16:00 X. Dai
Friday, 6.06.08 Seminarraum 3, Mathematisches Institut, Gyrhofstraße.
10:00-10:50 W. Zhang
11:10-12:00 S. Goette
12:00-14:00 Lunch at Loup de Mer
14:00-14:50 K. Yoshikawa
15:10-16:00 W. Müller
last modified on
4th of June 2008