(September 10) Dragomir Saric: The Teichmüller distance between finite index subgroups of PSL(2,Z) - For a given ε > 0, we show that there exist two finite index subgroups of PSL₂(Z) which are (1+ε)-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any ε > 0 there are two finite regular covers of the once punctured Modular torus T₀ (or just the Modular torus) and a (1+ε)-quasiconformal between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space T(S) of the punctured solenoid S under the action of the corresponding Modular group has the closure in T(S) strictly larger than the orbit and that the closure is necessarily uncountable.

(September 17) Graeme Wilkin: Morse theory for rank 2 Higgs bundles - In this talk I will describe a Morse-theoretic way to compute the equivariant cohomology of the space of semistable Higgs structures on a rank 2 vector bundle over a compact Riemann surface. The methods are in the spirit of Atiyah and Bott's approach for studying semistable holomorphic structures on vector bundles over compact Riemann surfaces, however here we need to develop the Morse theory techniques on a singular space. When the degree of the vector bundle is zero then this gives us information about the topology of the SL(2, C) character variety of the surface.

(September 24) Giovanni Forni: Rational polygonal billiards and Teichmüller geodesics I - An introductory talk on interval exchanges, Teichmüllergeometry and dynamics, and hyperbolicity properties of the Teichmüller metric. First in a series.

(October 1) Giovanni Forni: Rational polygonal billiards and Teichmüller geodesics II - The second introductory talk on interval exchanges, geometry and dynamics, and hyperbolicity properties of the Teichmüller metric.

(February 4) Jim Schafer: A curious result of Turaev's - Turaev recently proved a strange formula about the Dijkgraaf-Witten invariants of surfaces. We will explain what this formula is and why it's interesting.

(February 11) Ryan Hoban: The Siegel Upper Half Space - The Siegel Upper Half Space is the homogeneous space for the symplectic group Sp(4,R). It is in many ways a natural generalization of the hyperbolic plane to higher dimensions. We will describe some of the similarities as well as the major differences between this geometry and hyperbolic space.

(February 18) Karin Melnick: An embedding theorem for automorphism groups of Cartan geometries - Cartan geometries are a promising setting for the study of automorphisms of geometric structures. They infinitesimally model a manifold with a geometric structure on a homogeneous space. I will present an embedding theorem for Lie group actions on compact manifolds by automorphisms of a Cartan geometry, in which the acting group is related to the group of automorphisms of the model space. Consequences include general bounds on rank and nilpotence degree of automorphism groups and flatness theorems, giving sufficient conditions for the geometric manifold to be locally homogeneous. This is joint work with Uri Bader and Charles Frances.

(February 25) Jane Long: The Steenrod Algebra and Group Cohomology - The discovery of the Steenrod algebra was an extremely important development in algebraic topology. Some preliminary facts about the Steenrod operations and their applications to group cohomology will be discussed.

(March 3) Nicolas Flores-Castillo: Synge's trick - Synge's trick was used in Riemannian geometry to prove some theorems about manifolds of positive sectional curvature: those of Frankel, Weinstein-Synge and Wilking. We will present Synge's trick, the classical proofs of the theorems mentioned above, and a reformulation of Synge's trick in terms of a lower bound for the index of a special kind of geodesics.

(March 24) John Parker: Unfaithful triangle groups and the hunt for complex hyperbolic lattices - A lattice is a group of isometries of a metric space that acts properly discontinuously and for which the quotient space has finite volume. A triangle group is a group generated by reflections in the sides of a triangle. We know relatively few examples of complex hyperbolic lattices. Deligne and Mostow, using ideas that go back to Picard, gave a family of lattices which are triangle groups with extra relations. These include the first examples of non-arithmetic complex hyperbolic lattices due to Mostow. Recently Deraux constructed a new example of a complex hyperbolic lattice that is also a triangle group with extra relations. In this talk I will give an elementary account of the above constructions and then outline a programme (which is joint work with Julien Paupert) for finding other triangle groups that may be candidates for lattices.

(March 31) Lowell Abrams: Topological structure of digital images from a non-Morse height function - We present a combinatorial approach, based on the construction and analysis of three associated graphs, to the topological analysis of digital images through the use of a [very] degenerate height function. Such topological analysis of digital images has proven useful in the automated analysis of magnetic resonance images of the cerebral cortex, particularly for data correction purposes.

(May 12) Jack Morava: Classical mechanics in tree space - The space of rooted trees with n leaves has a surprisingly natural structure as a compact, smooth (though non-orientable, if n > 3) closed manifold of dimension n-2. It is moreover aspherical, with a fundamental group that shares a surprising number of features with the group of braids on n strands.

These spaces are of great practical importance in computational biology and phylogenetics, but the study of dynamical systems (i.e., of interesting flows) on such spaces is only beginning. The papers posted at arXiv:math/0507514 and arXiv:math/0702515 give some idea of the breadth and interest of the subject.