Title: Trisections of groups
Abstract: We will give a group-theoretic interpretation of some of the Gay-Kirby results on trisections of 4-manifolds.

Title: Counting curve types
Abstract: For S a closed surface and k a natural number, let N(k, S) denote the number of closed curves on S with self intersection number at most k, up to the action of the mapping class group. We obtain upper and lower bounds on N(k,S) which grow exponentially in the square root of k. While previously known estimates for N(k,S) were proved using largely combinatorial methods, we obtain these improved bounds with geometric tools, specifically: Lalley's theory of random geodesics on hyperbolic surfaces, and the geometry of Thurston's Lipschitz metric on Teichmuller space. This represents joint work with Juan Souto.

Title: Convex co-compactness subgroups of the mapping class group
Abstract: It is a well known question wether there are hyperbolic 4-manifolds that fiber over a surface. For any 4-manifold that fibers of the a surface the fibers will also be a surface (usually of different topological type) and the monodromy defines a homomorphism from the fundamental group of the base surface into the mapping class group of the fiber. Farb and Mosher defined a notion of convex co-compactness for subgroups of the mapping class group and showed that the monodromy map is an injection to a convex co-compact subgroup if the manifold is hyperbolic. It is therefore of great interest to understand convex co-compact subgroups of the mapping class group. Farb and Mosher also showed that any such subgroup is both purely pseudo-Anosov (every non-trivial element is pseudo-Anosov) and undistorted. We will show that that these necessary conditions are also sufficient. This is joint work with M. Bestvina, R. Kent and C. Leininger.

Title: Trisections of 4-manifolds and their diagrams
Abstract: Trisections are to 4-manifolds as Heegaard splittings are to 3-manifolds, and as in the 3-dimensional case we have (a) a variety of generalizations, rel. boundary, rel. knots, etc., and (b) diagrams in the form of collections of curves of various colors on surfaces. I'll give the basic definitions and then try to give a survey of various generalizations and their diagrams, and discuss some interesting issues that arise, related to mapping class groups, sequences of handle slides, and so forth. For the Heegaard-Floer loyalists, this work can be interpreted as a systematic exploration of the full range of implications of the fact, observed by Ozsvath and Szabo early on in the development of Heegaard-Floer invariants, that Heegaard triples describe 4-dimensional "stuff".

Title: The dynamics of classifying geometric structures
Abstract: The general theory of locally homogeneous geometric structures (flat Cartan connections) originated with Ehresmann's 1936 paper "Sur les espaces localement homogènes". Their classification leads to interesting dynamical systems. For example, classifying Euclidean geometries on the torus leads to the usual action of the SL(2,Z) on the upper half-plane. This action is dynamically trivial, with a quotient space the familiar modular curve. In contrast, the classification of other simple geometries on the torus leads to the standard linear action of SL(2,Z) on R^2, with chaotic dynamics and a pathological quotient space. This talk describes such dynamical systems, where the moduli space is described by the nonlinear symmetries of cubic equations like Markoff's equation x^2 + y^2 + z^2 = x y z. Both trivial and chaotic dynamics arise simultaneously, relating to possibly singular hyperbolic metrics on surfaces of Euler characteristic equals -1.

Title: Fundamental groups of complex projective varieties
Abstract: I will show that any finitely presented group is isomorphic to the fundamental group of an (irreducible) projective variety with normal crossing singularities. This answers a question raised by Carlos Simpson. The proof is an application of 3-dimension hyperbolic geometry.

Title: Flexibility of projective representations
Abstract: For which countable groups G, does the moduli space
X(G) = Out(G) \ Hom(G,PSL(2,R)) / Inn(PSL(2,R))
contain (uncountably many distinct equivalence classes of) dense faithful representations? Groups with such properties are called flexible. We prove combination theorems for flexible groups, and show that most Fuchsian groups and all limit groups (possibly with torsion) are flexible. The diversity of quasi-morphisms on those groups will follow. Implications for word-hyperbolic groups and mapping class groups will also be discussed. Joint work with Thomas Koberda and Mahan Mj.

Title: A Jones polynomial for curves on surfaces
Abstract: In a recent work, T. Koberda et R.Santharoubane have studied representations of surface groups with the following property: every simple curve has an image of finite order. These representations come from topological quantum field theory (TQFT) and depend on an integer k called level. I will explain that when k goes to infinity, these representations are controlled by a sort of Jones polynomial which may distinguish simple curves from the others. This is joint work with R. Santharoubane.

Title: Character varieties and mapping class groups dynamics
Abstract: Given a surface group, we can consider its SL(2 , R) or SL(2,C)-character variety. The mapping class group acts naturally on this space, and we can try to understand the dynamical decomposition of this action and its relation with a more geometric decomposition of the character variety. The focus of this talk will be the case when the surface group is the free group F_3 of rank three, as its character variety admits a relatively simple parametrization using trace coordinates. We will describe a combinatorial setting to study these actions that allow us to construct interesting domains of discontinuity in the complex and real cases. This is joint work with Sara Maloni and Ser Tan.

Title: Quotients of surface groups and homology of finite covers via quantum representation
Abstract: I will show how from Witten-Reshetikhin-Turaev TQFT, we can produce interesting representations of surface groups. The key fact is the following : these "quantum representations" of surface groups have infinite images but every simple loop acts with finite order. Using this key fact and integral TQFT, we will see how to build regular finite covers of surfaces where the integral homology is not generated by pullbacks of simple closed curves on the base. This talk represents joint work with T. Koberda.

Title: Energies of virtual endomorphisms
Abstract: A virtual endomorphism of a space X is a map from a finite cover of X back to X. Virtual endomorphisms of surfaces are closely related to the theory of rational maps. Virtual endomorphisms of graphs are closely related to the theory of self-similar groups, including most known examples of groups of intermediate growth. Self-similar groups can be "wild", but become tamed with an expansion criterion. We introduce a family of energies for virtual endomorphisms, capturing both expanding self-similar groups and when a virtual endomorphism of a surface is a rational map.

Title: Some things that can happen (with planar boundaries of relatively hyperbolic groups)
Abstract: We give some examples of interesting behavior that can occur for relatively hyperbolic groups with planar boundaries: strange groups hiding in peripheral groups, hidden cut points, and other phenomena. We also discuss groups whose Bowditch boundary is homeomorphic to a Schottky set: the complement of at least three round open discs in $S^2$, and prove a mild classification theorem about these groups. This work in progress arose from discussions with Francois Dahmani, Chris Hruska, and Luisa Paoluzzi.