We meet Tuesdays

In the past there were two different seminars and we just merged to promote
more interaction between the two groups of researchers. Past schedules for the
Differential Geometry Seminar are Fall
2004, Spring
2004, Fall
2003, Spring
2003, Fall
2002, Spring
2002, Fall
2001 and Spring
2001. Past schedules for the Lie Group Seminar are Fall 2004 and Spring
2004. This seminar has a 30 year history of meeting at the

**February 1: Sean Paul (****Columbia****).**

*Title: Kahler Einstein Metrics and K-Stability.*

abstract: The speaker will introduce the problem of finding special Kahler metrics on Kahler Manifolds, in particular the Kahler-Einstein metrics. This problem reduces to the question of finding a solution to a nonlinear PDE: the complex Monge-Ampere equation, so is at first sight an analytic problem. The is that this problem is deeply connected to algebraic geometry. The speaker will give an account of these connections. This is joint work with Gang Tian. Graduate students are encouraged to attend.**February 8: Emmanuel Breuillard (IHES).**

*Title: The asymptotic shape of metric balls in Lie groups of polynomial growth, and pointwise ergodic theorems.*

abstract: Let G be a connected Lie group of polynomial growth. We show that G has strict polynomial growth and obtain a formula for the asymptotics of the volume of large balls. This is done via the study of the asymptotic shape of metric balls. We show that large balls, after a suitable renormalization, converge to a limiting compact set, which can be interpreted geometrically as the unit ball for some Carnot-Caratheodory metric on the associated graded nilshadow. The results hold for a large class of pseudometrics including left invariant Riemannian metrics or ``word metrics'' associated to a compact generated set. This generalizes results of P. Pansu for discrete finitely generated nilpotent groups. As an application, we also derive new pointwise ergodic theorems on nilpotent Lie groups and Lie groups of polynomial growth.**February 15: Marc Burger (ETH -****Zurich****) and Alessandra Iozzi (Uni. Basel) Double Header.**

*Titles:**Bounded Cohomology in Hermitian Geometry*(Iozzi) and

*Maximal Representations of surface groups*(Burger*).**Schedule: Iozzi will talk at**2:30**and Burger at 4, with a half hour for tea in between. The talks will be in room 4419, and tea will be in the math program lounge.***February 22: Slava Kruskhal (****University****of****Virginia****).**

Title: Quantum representations of mapping class groups and the Fourier transform.

abstract: A question about the the metaplectic representation of SL(2,Z) leads to a problem in Fourier analysis on the real line. A variant of Kazhdan's property (T) - concerning the rigidity of representations of a group - implies that this problem does not have a solution. Quantum representations, given by the Jones-Witten theory, provide a a generalization of this to mapping class groups of surfaces - for which Kazhdan's property is not known to hold. (Joint work with M.Freedman)**March 1: Anders Karlsson (KTH-Stockholm and****Yale****University****).**

Title: Metric geometry and group actions: two examples.

abstract: Metrics and semicontractions/isometries arise in many places in mathematics. I will focus on two such instances. First, concerning random walks on infinite groups, I will discuss a conjectural and very general noncommutative law of large numbers. Some partial progress on this will be explained. Second, I will discuss a result on the radial variation of harmonic functions on SL_n(Z) and mapping class groups. The talk will be aimed at a general audience.**March 8: Indira Chatterji (Cornell,****Columbia****, and****Ohio****State****).**

Title: Property RD on connected Lie groups.

abstract:For a locally compact group, the property of Rapid Decay (abbreviated by property RD) gives a control on the convolution norm of any compactly supported measure in terms of the $L^2$-norm of its density and the diameter of its support. We give a complete classification of those connected Lie groups with property RD. This is joint work with Ch. Pittet and L. Saloff-Coste.*March 15:*Michelle Bucher (Yale).

Title:Finiteness properties of characteristic classes of flat bundles.

abstract: We will outline a new proof of Gromov's boundedness of primary characteristic classes of flat bundles which, in contrast to Gromov's orginal proof, does not rely on Hironaka's resolution of singularities. Moreover, we point out that a representative for these classes can be found which in fact only takes a finite set of values (as opposed to merely being bounded) on singular simplices. This will be illustrated on the Euler class with previous results of Milnor, Wood, and Sullivan.*March 22:*No talk. See Geometric Analysis conference March 19-20.

Title TBA

abstract TBA*March 29: (no classes at the CUNY GC)*

no meeting*April 5:*Werner Muller (IAS and U.**Bonn****).**

*Title: Analytic torsion of hyperbolic manifolds.*

abstract: The talk will be concerned with a refinement of the Ray-Singer analytic torsion, introduced by Braverman and Kappeler. The refined torsion is a holomorphic function on the representation variety of the fundamental group. In the talk we will study the refined torsion for hyperbolic manifolds. For special representations of the fundamental group we will show that it is given as the value at zero of a Ruelle zeta function. Then we will discuss some applications for hyperbolic 3-manifolds.*April 12:*Jose Rodrigo (**Yale****University****).**

Title: Evolution of sharp fronts for the quasi-geostrophic equation.

abstract: The talk will consist of two parts. In the first part I will describe the strong similarities between the quasi-geostrophic equation and 3D-Euler. I will describe the evolution of sharp fronts (the analogue of the evolution of a vortex line for 3-D Euler) and obtain a rigorous derivation of an equation for a front ( the equation for the evolution of a vortex line is an open problem). In the second part I will prove the local well-posedness of the equation for a sharp front (for smooth initial data) using a Nash-Moser argument.*April 19:*Ben Schmidt (**University****of****Michigan****).**

Title Weakly hyperbolic group actions.

abstract: A group action on a closed Riemannian manifold is said to be weakly hyperbolic provided that a finite family of group elements act by partially hyperbolic diffeomorphisms with stable distributions jointly spanning the tangent bundle. Weakly hyperbolic integer actions are generated by an Anosov diffeomorphism, providing motivation for our two main results: 1) ergodicity of volume preserving weakly hyperbolic actions, and 2)weak hyperbolicity is inherited by the induced action on the fundamental group for volume preserving weakly hyperbolic actions (with a fixed point) of Kazhdan property (T) groups on tori. This work stems from Zimmer's program of classifying the volume preserving ergodic actions of lattices in higher rank Lie groups.*April 26: Spring Recess*

no meeting*May 3: Cancelled*

**May 10: T. Szoke (Univ. of Budapest, Purdue University)**

Title: Complex Crowns of Projective Spaces**May 17: T. J. Hitchman (Rice University)**

Title: Vanishing Cohomology and Cocycle Superrigidity through Geometry

Abstract:

Two of the most important rigidity properties for lattices in semisimple Lie groups of higher real rank are-
1) vanishing of first cohomology with coefficients in a
suitable representation, and

2) Zimmer's measurable cocycle superrigidity theorem for ergodic measure preserving actions.The techniques build on work of Matsushima and Murakami, Raghunathan, Mok, and Korevaar and Schoen, among others, and involve Bochner formulae on locally symmetric spaces and harmonic maps with metric space targets.

This is joint work with David Fisher.

*Special Seminar on Wednesday July 6 at 2pm in 5212:*

R. Rentschler, Institut de Mathematiques de Jussieu

"On Dixmier's Problem on Hearts of Prime Quotients of Enveloping Algebras"

hosted by A. Rocha

Abstract: The answer to problem #8 of Dixmier's book "Enveloping Algebras" is negative: the speaker will present a counter-example and also speak on his recent work with W. Borhu.