Purpose:

There is currently great interest in the study of the geometric structure spaces of a surface (moduli spaces) via the representation and character varieties of its fundamental group, and in the separate study of the mapping class group of the surface.  The action of the latter on the former forms a discrete dynamical system which links the topology of the surface to its geometry in a highly structured fashion.  Thurston's classification of the mapping class group of a surface is based on its action on an appropriate closure of Teichmuller space, a component of the PSL(2,R)-character variety of the surface group. And in "Ergodic theory on moduli spaces," (Ann. Of Math 1997), Goldman takes the initial steps in understanding the general dynamical system. In this special session, we highlight this interaction as a way to gain information on both the acting mapping class group, and on the moduli spaces. This session hopes to assess the current focus and future directions of this subject among some of the core group currently studying the field. Since this topic brings together researchers from many different fields (topological dynamics, ergodic theory, low-dimensional topology, and representation theory, to name a few), this session will facilitate the sharing of their ideas and on developing a common theme and shared goals.

Participants:

Tara E. Brendle, Cornell University

Richard Douglas Canary, University of Michigan

Moon Duchin, University of Chicago

Charles Frohman, University of Iowa

William M. Goldman, University of Maryland

Elmas Irmak, University of Michigan

Rinat Kashaev, Université de Genève

Nikolai A. Krylov, International University Bremen

Dan Margalit, University of Utah

Walter Neumann, Barnard College, Columbia University

Joseph P. Previte, Pennsylvania State University at Erie

George Stantchev, University of Maryland

Peter Storm, Stanford University

Ser Peow Tan, National University of Singapore

Richard A. Wentworth, Johns Hopkins University

Schedule:

• Friday January 7, 2005, 8:10 a.m.-10:50 a.m.
  

  • 8:10 a.m.
    Deformation spaces of surface group representations.
    William M. Goldman*, University of Maryland
    (1003-22-653)
  • 9:00 a.m.
    The pentagonal equation and representations of the mapping class group of a punctured surface.
    R Kashaev*, Université de Genève
    (1003-58-1305)
  • 9:30 a.m.
    Commensurations of the Johnson kernel.
    Tara E. Brendle*, Cornell University
    Dan Margalit, University of Utah
    (1003-51-1162)
  • 10:00 a.m.
    Proper actions of the mapping class group and the energy of harmonic maps.
    William M. Goldman, University of Maryland
    Richard A Wentworth*, Johns Hopkins University
    (1003-57-1117)
  • 10:30 a.m.
    Integrating on the SU(3)-character variety of a surface group.
    Charles Frohman*, The Univerisity of Iowa
    Kathy Zhong, Sacramento State University
    (1003-57-909)

• Saturday January 8, 2005, 1:00 p.m.-5:50 p.m.
  

  • 1:00 p.m.
    Relative mapping class group of S^p\times D^q$.
    Nikolai A. Krylov*, International University Bremen
    (1003-55-767)
  • 1:30 p.m.
    Automorphisms of the Hatcher-Thurston Complex.
    E Irmak*, University of Michigan
    M Korkmaz, Middle East Technical University
    (1003-58-726)
  • 2:00 p.m.
    The Mod(S)-action on the variety of PSl(2,C)-characters.
    Juan Souto, CNRS
    Peter Storm*, Stanford
    (1003-53-720)
  • 2:30 p.m.
    Dynamics of the Automorphism Group of the $GL(2,\mathbb{R})$-Characters of a Rank Two Free Group.
    George Stantchev*, University of Maryland at College Park
    William Goldman, University of Maryland at College Park
    (1003-53-1018)
  • 3:00 p.m.
    Dynamics of the mapping class group on the moduli of a punctured sphere with rational holonomy.
    Joseph P Previte*, Penn State Erie
    Eugene Z Xia, National Cheng-Kung University Taiwan
    (1003-57-680)
  • 3:30 p.m.
    Weil-Petersson isometries.
    Dan Margalit*, University of Utah
    (1003-20-576) Talk slides: Postscript, PDF
  • 4:00 p.m.
    Homological action of the modular group on some cubic moduli spaces.
    Walter D Neumann*, Barnard College, Columbia University
    William M Goldman, University of Maryland
    (1003-57-552)
  • 4:30 p.m.
    Generalized Markoff Maps and McShane's Identity.
    Ser Peow Tan*, National University of Singapore
    Yan Loi Wong, National University of Singapore
    Ying Zhang, National University of Singapore
    (1003-22-398)
  • 5:00 p.m.
    Deformation theory of hyperbolic 3-manifolds.
    Richard Douglas Canary*, University of Michigan
    (1003-57-1686)
  • 5:30 p.m.
    An Ergodic Theorem for Random Walks on the Mapping Class Group.
    Moon Duchin*, University of Chicago
    (1003-37-1722)

Other recent work in the field: