Mathematics

Patterson-Sullivan currents, generic stretch factors and the asymmetric Lipschitz metric for Outer space

Ilya Kapovich (UIUC Math)

Abstract: We quantitatively relate the Patterson-Sullivant currents to the asymmetric Lipschitz metric on Outer space and to Guirardel's intersection number. As an application we show that for any N\ge 2 there exists a positive constant C_N>0 such that for every \phi\in Out(F_N), where F_N=F(A)=F(a_1,..,a_N) is the free group of rank N, we have $C_N\le \lambda_A(\phi)/ \Lambda_A(\phi) \le 1$. Here $\Lambda_A(\phi)= \sup_{w\ne 1} ||\phi(w)||_A/||w||_A$ is the "extremal distortion" of $\phi$, and $\lambda_A(\phi)$ is the "generic stretching factor" of $\phi$, that is, $\lambda_A(\phi)$ is the asymptotic distortion $||\phi(w)||_A/||w||_A$ where $w$ is a "long random element" in $F(A)$, as $||w||_A\to\infty$. The talk is based on joint work with Martin Lustig.

Asymptotics of equivariant syzygies

Xin Zhou (University of Michigan, Ann Arbor)

Abstract: Recent results on syzygies concentrate on their asymptotics. In this talk, I will discuss results on the asymptotics of syzygies under group actions. In particular, we study two cases. When the underlying space is the projective space, we give the asymptotic growth of syzygy modules with respect to the general linear group. When the underlying space is a toric variety, we give a sharp asymptotic description of the distribution of torus weights.

Spring Department Faculty Meeting

Abstract: The Spring Department Faculty Meeting will be held at 4 p.m. in 245 Altgeld Hall, followed by a reception in 243 Altgeld Hall.