Please contact Saul Schleimer or Caroline Series if you would like to speak or suggest a speaker.
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Thursday May 22, 15:00, room MS.04 Vaibhav Gadre (Warwick) Partial sums of excursions along random geodesics |
Abstract: In the context of continued fractions, Diamond and Vaaler showed the following strong law: for almost every expansion, the sum of first n continued fraction coefficients minus the largest coefficient, then divided by $n \log n$, tends to a limit. I will generalize this to non-uniform lattices in $\SL(2,\RR)$ with excursions in cusps of the quotient hyperbolic surface generalizing continued fraction coefficients. The general theorem relies on the exponential mixing of geodesic flow, in particular on the fast decay of correlations due to Ratner. If time permits, I will state a similar theorem for the moduli space of curves. |
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Thursday May 29, 15:00, room MS.04 Clifford Earle (Cornell) Some global coordinates for Teichmüller space |
Abstract: Many years ago, Al Marden and I learned how to use Kleinian groups to simulate classical $zw = t$ plumbing and put global holomorphic coordinates on the Teichmüller space of a compact hyperbolic Riemann surface. These coordinates can be used to define a complex analytic structure on the compactified moduli space. This talk will describe some of their properties when we start with a pants decomposition of the Riemann surface. |
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Thursday June 5, 15:00, room MS.04 Grant Lakeland (UIUC) Systoles and Dehn surgery for hyperbolic |
Abstract: (Joint work with Chris Leininger.) The systole of a Riemannian manifold $M$ is its shortest closed geodesic. If $M$ is a hyperbolic, finite volume 3-manifold, Mostow rigidity means systole length is a topological invariant of $M$, and thus we may study how systole length varies under Dehn surgery. A result of Adams and Reid provides a universal bound on the systole length of any hyperbolic link complement $M - L$, provided $M$ is non-hyperbolic. In this talk, I'll discuss the relationship between systole length and volume, with the goal of generalizing their theorem to any 3-manifold $M$. |
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Thursday June 12, 15:00, room MS.04 John Mackay (Bristol) Low density random groups and conformal dimension |
Abstract: Gromov hyperbolic groups are often studied using the topology of their boundaries at infinity. The conformal dimension of a (Gromov) hyperbolic group is a quasi-isometric invariant of the group that captures some of the analytic information in the boundary, as opposed to just topological information. We will discuss a new approach to bounding the conformal dimension of a small cancellation group, and outline applications to certain random groups. |
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Thursday July 31, 15:00, room MS.04 Grace Work (UIUC) Gap distributions for saddle connections on the octagon |
Abstract: (Joint with Caglar Uyanik.) Following a strategy developed by Athreya and Cheung, we compute the gap distribution of the slopes of saddle connections on the octagon by translating the problem to a question about return times of the horocycle flow to an appropriate Poincaré Section. This same strategy was used by Athreya, Chaika, and Lelièvre to compute the gap distribution on the Golden L. The octagon is the first example of this type of computation where the Veech group has two cusps. |