Ara Basmajian (CUNY Hunter College and Graduate Center):

Title: Half-turns and Commutators acting on Hyperbolic Space

Abstract:  We define a half-turn to be an order two (orientation preserving or reversing) isometry of hyperbolic space. In dimensions two and three it is well known  that for any two orientation preserving  isometries A and B there exist half-turns, α, β and γ, so that A = αβ and B = βγ.

 In joint work with Bernard Maskit, we consider the same situation in higher dimensions- we answer precisely the question of when an element can be  written as the product of two half-turns and further address the situation mentioned above. As a corollary of our work, every orientation preserving isometry of hyperbolic space is a commutator-this is well known in dimensions two and three.

Andre de Carvalho (Univerisy of Sao Paulo):

Title: Riemann origami and convergence of pseudo-Anosov sequences

Abstract: It is common to construct surfaces by specifying identifications on the sides of a polygon P in the plane. In this talk surfaces built in this way will be discussed where infinitely many identifications of sub-arcs on the boundary of P are allowed. In these cases, both the metric and the complex structures – which in the finite case are easily obtainable – have to be studied more carefully. A sufficient condition to ensure the quotient is a compact Riemann surface is presented, as well as a uniformity condition (a modulus of continuity) which allows one to decide whether a sequence of such constructions converges. As an application, the convergence of certain sequences of pseudo-Anosov maps is discussed. The main technical ingredients come from the geometric theory of functions, in particular, from the theory of extremal length.

Title: Completeness of Carathéodory's metric on generalized Teichmüller spaces

Abstract:  Completeness of the Carathéodory metric d_C on classical Teichmüller spaces T(S) was proved long ago. The proof used properties of the Bers embedding. In a 2006 paper, H. Miyachi extended the proof to a more general setting and showed the completeness of d_C in asymptotic Teichmüller spaces. He also obtained comparisons between d_C and the Kobayashi metric d_K on these spaces. We prove the slightly more precise inequality:

tanh d_K(a , b) ≤ 3 tanh d_C (a , b);    a and b in T(S);

Title: Teichmüller moonshine

Abstract: In their celebrated paper "Symmetric and Quasisymmetric Structures on a Closed Curve", Amer. J. of Math., 114 (1992), no. 4, 683-736, Fred Gardiner and Dennis Sullivan opened an asymptotically conformal geometry for us to study. It turns out that the asymptotically conformal geometry is an essential subject in one-dimensional dynamical systems. The Gibbs measure theory for smooth potentials is an old and beautiful subject and has many important applications in modern dynamical systems. For continuous potentials, it is impossible to have such a theory in general. However, in this talk, I will present a theory by using the asymptotically conformal geometry to extend the Gibbs measure theory for smooth potentials to a dual geometric Gibbs type measure theory for certain continuous potentials and introduce a Teichmüller structure on the space of these continuous potentials. Moreover, this Teichmüller structure is complete and is the completion of the space of smooth potentials under this Teichmüller structure. Thus our dual geometric Gibbs type theory is a completion of the Gibbs measure theory for smooth potentials. Some part of this work is a joint work with Fred Gardiner.

Abstract. I will speak about joint work with Jane Gilman. We consider non-elementary representations of two generator free groups in PSL(2 , C), not necessarily discrete or free, G = á A , B ñ. A word in A and B, W (A , B), is a palindrome if it reads the same forwards and backwards. We study the geometry of palindromes and the action of G in H³ whether or not G is discrete. We show that there is a core geodesic L in the convex hull of the limit set of G and use it to prove three results: the first is that there are well defined maps from the non-negative rationals and from the primitive elements to L; the second is that G is geometrically finite if and only if the axis of every non-parabolic palindromic word in G intersects L in a compact interval; the third is a description of the relation of the pleating locus of the convex hull boundary to the core geodesic and to palindromic elements.

Irwin Kra (SUNY at Stony Brook):

Abstract: Not much is known about the Carathéodory metric on Teichmüller space. A result that goes back to 1981 is the

THEOREM: If D Ì T(p, n) is an abelian Teichmüller disc, then c_T(p,n) |D = r.

Here T(p, n) is the Teichmüller space of surfaces of finite conformal type (p, n), c_T(p,n) its Carathéodory metric, and r is the hyperbolic metric on D.

COROLARRY: The canonical period map P of  T(p) into H²(p), the Siegel upper half-space is isometric (hence injective) on each abelian Teichmüller disc in T(p).

Title: Teichmüller density and holomorphic motions

Abstract: In a joint work with Fred Gardiner we study the connections between the Teichmüller density and holomorphic motions.  The Teicmüller density is equivalent to the Poincare density and so we obtain applications to Teichmüller theory and to conformal geometry.

 Title: Renormalizations in complex dynamics and the Teichmüller theory

Abstract: The renormalization is a way to construct a new dynamical system from a given dynamical system by taking the first return map to a certain subset. The renormalization is considered as a meta-dynamics on the space of certain class of dynamical systems, and its hyperbolicity implies the rigidity or parameter space universality of such systems.  We discuss the use of Teichmüller theory in the study of renormalizations in complex dynamics, especially those with irrationally indifferent fixed points.

Dennis Sullivan (CUNY Graduate Center and SUNY at Stony Brook):

 Title: Characters of Bundles with Connection

Title: Geodesic-length functions and Weil-Petersson geometry

Abstract: Geodesic-length functions (glfs) provide a direct understanding of Teichmüller space.  We review basic properties and then discuss recent formulas for the Weil-Petersson metric, Levi-Civita connection and Riemannian curvature in terms of glfs.  The augmented Teichmüller space T  is defined by allowing pants boundary lengths to be zero.  T  is a CAT(0) metric space.  We describe the behavior of geodesics ending at a stratum of T , and describe the Alexandrov tangent cones.  Yamada’s Coxeter complex construction and finite rank result are presented as applications. 

A write up on current understanding of Weil-Petersson geometry is available at  http://arxiv.org/abs/0801.0175 and a slide presentation is at http://www.math.umd.edu/~saw/preprints/WP_sampler.pdf