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Spring 2010

Tuesday 12:30-1:30 pm
Fine 322

Lunch will be provided!


Friday, Feb. 5
12:30 pm
Different room
Marc Burger
On Ulam stability

Abstract: We report on recent progress concerning the Ulam stability problem, namely to determine under which conditions an-epsilon-homomorphism with values in  a group with a distance, is uniformly close to an actual homomorphism. This is joint work with A.Thom and N.Ozawa
Mar 9 Tomasz Zamojski
University of Chicago
Counting Rational Matrices of a given Characteristic Polynomial

Abstract: Under the assumption that the polynomial is rational and irreducible, we compute the asymptotic number of rational matrices of the given characteristic polynomial, thus solving a new case of Manin's Conjecture. The method of proof is inspired by the counting lattice point theorem of Eskin-Mozes-Shah. Here, the above rational matrices are a single orbit under the rational points of $PGL_n$, which is a lattice in the adelic points of $PGL_n$. However, we do not make use of unipotent flows and Ratner's theorems. Instead, we will prove an equidistribution theorem for an average over periodic torus orbits, relying on the measure rigidity theorem of Einsiedler-Katok-Lindenstrauss.
Mar 23 Kevin Wortman
University of Utah
Non-nonpositive curvature of some non-cocompact arithmetic groups

Abstract: I'll explain why arithmetic subgroups of semisimple groups of relative Q-type A_n, B_n, C_n, D_n, E_6, or E_7 have an exponential lower bound to their isoperimetric inequality in the dimension that is 1 less than the real rank of the semisimple group.
Apr 6 David Constantine
University of Chicago
Compact forms of homogeneous spaces and group actions

Abstract: Given a homogeneous space J\H, does there exist a discrete subgroup \Gamma in H such that J\H/Gamma is a compact manifold? These compact forms of homogeneous spaces turn out to be rare outside of a few natural cases. Their existence has been studied by a very wide range of techniques, one of which is via the action of the centralizer of J in H. In this talk I'll show that no compact form exists when H is a simple Lie group, J is reductive and the acting group is higher-rank and semisimple. The proof uses cocycle superrigidity, Ratner's theorem and techniques from partially hyperbolic dynamics.
Apr 13 Daryl Cooper
UC, Santa Barbara
The marked length spectrum of a projective manifold or orbifold.

Abstract: A strictly convex real projective orbifold is equipped with a natural Finsler metric called the Hilbert metric. In the case that the projective structure is hyperbolic, the Hilbert metric and the hyperbolic metric coincide. We prove that the marked Hilbert length spectrum determines the projective structure only up to projective duality. This result is essentially due to Inkang Kim, although a gap in his argument caused him to miss the issue concerning dual structures. A corollary is the existence of non-isometric diffeomorphic strictly convex projective manifolds (and orbifolds) that are isospectral. This is joint work with Kelly Delp.
Apr 20 Nir Avni
Harvard University
Counting Representations of Arithmetic Lattices

Abstract: I will talk about the the number of representations of dimension d of an arithmetic lattice when d tends to infinity. For higher rank lattices, this sequence grows polynomially, with some mysterious exponent. I will talk about some known values of this exponent, and the conjectural relations between the representation growths of different lattices.
Apr 27 Andrzej Zuk
Université Paris 7
L2 Betti numbers of closed manifolds

Abstract: We present results concerning possible values of L2 Betti numbers of closed manifolds.
May 4 Yves Cornulier
CNRS, Université de Rennes 1
On the large-scale geometry of Lie groups

Abstract: I will describe recent results about the class of Lie groups with polynomial Dehn function, i.e. in which loops of length r have area asymptotically bounded by r^c for some constant c. This is joint work with Romain Tessera.

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