April 11 - 13, 2008
University of Michigan
Ann Arbor, MI








Ann Arbor Information

University of Michigan

Mathematics Department

Dick Canary
Lizhen Ji
Ralf Spatzier


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Friday, April 11 - B844 East Hall
  1:30 - 2:30 PM   Howard Masur, UIC  

Quasiconformal maps, extremal problems, Grotzsch's Theorem, quadratic differentials
I will introduce quasiconformal mapping, state the main extremal problem, present the Grotzsch solution in the case of rectangles and begin the discussion of the problem on Riemann surfaces by introducing quadratic differentials.

  3:00 - 4:00 PM   Jeff Brock, Brown University  

Introduction to Weil-Petersson metric, the CAT(0)
geometry of its completion, and the space of geodesic rays

I will discuss basic results about Weil-Petersson metric,
including its non-completeness, geodesic convexity, and the global structure of the completion. I'll further discuss the idea of the visual sphere and the density of rays that terminate in the completion. This talk will focus on examples, and in particular on the one-dimensional case.

  4:30 - 5:30 PM   Alex Eskin, University of Chicago   The Teichmuller geodesic flow: definitions and basic

I will define the flow from the point of view of
quasi-conformal maps, and also from the point of view of flat structures on surfaces. I will also define the Kontsevich-Zorich cocycle.
Saturday, April 12 - B844 East Hall
  9:30 - 10:30 AM   Howard Masur, UIC   Teichmuller's theorem, Teichmuller space and its compactification
I will state Teichmuller's theorem, define Teichmuller space, and discuss the Thurston compactification of Teichmuller space.
  11:00 AM-12:00 PM   Moon Duchin, UC Davis  

The curve complex
I'll introduce complexes of curves and arcs associated to a surface. I'll discuss some of theorems about their geometry and topology due to Masur-Minsky, Harer, and others, and will aim to give a sense of how they are used.

  1:30 - 2:30 PM   Jeff Brock, Brown University  

The Weil-Petersson geodesic flow via ending laminations and recurrence
I'll discuss a new asymptotic invariant for a WP geodesic ray, the ending lamination. Such laminations are complete invariants for the asymptote classes of rays that recur to the thick part of Teichmüller space (and are not complete invariants in general). We use the resulting embedding of the recurrent rays into the boundary of the curve complex to show the density of closed orbits for the geodesic flow, as well as the existence of a dense geodesic.

  3:00 - 4:00 PM   Alex Eskin, University of Chicago   The Hodge norm and hyperbolic behaviour of the Teichmuller geodesic flow
I will define the Hodge norm, and state some key estimates of Forni on the growth rate of the Hodge norm. I will then discuss some attempts to apply these estimates to help understand the behaviour of the flow.
  4:30 - 5:30 PM   Howard Masur, UIC   The mapping class group, Thurston-Bers theorem, and the geodesic flow on the quotient moduli space.
I will define the mapping class group, discuss its action on Teichmuller space, and sketch Bers' proof of Thurston's classification of mapping class elements. If there is time I will define the Teichmuller geodesic flow on the quotient moduli space.
Sunday, April 13 - B844 East Hall
  9:00-10:00 AM  

Moon Duchin, UC Davis


  Dynamics and the Thurston boundary
I'll review some results about convergence in the Thurston compactification. Then I'll introduce Karlsson's "stars at infinity," and describe how to use the Teichmuller metric to impose a structure on the Thurston boundary which one can use to study the dynamics of actions by isometries. This produces a description of the curve complex in terms of the Teichmuller metric.
  10:30-11:30 PM   Jeff Brock, Brown University  

Bounded geometry and compact flow-invariant subsets
I will discuss combinatorial invariants associated to the
ending laminations for Weil-Petersson geodesics and prove the equivalence of bounded combinatorics and bounded geometry for WP geodesic segments, rays and lines. The combinatorics suggest general conjectures about the behavior of WP geodesics globally. Time
permitting, I will use these results this to give asymptotic estimates for the growth of closed orbits in compact invariant subsets for the flow as well as applications to estimating topological entropy.

  12:00 - 1:00 PM   Alex Eskin, University of Chicago  

Recurrence to compact sets and applications
I will discuss random walks and other techniques for showing that the Teicmuller trajectories, on average, spend most of the time in the thick part of Teichmuller space. I will then present some applications to counting problems and volumes of balls.






National Science Foundation via our RTG grant in Geometry, Topology and Dynamics and the Department of Mathematics at the University of Michigan.




Department of Mathematics  |  2074 East Hall  | 530 Church Street  
Ann Arbor, MI 48109-1043
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