(September 14) Francois Labourie: Anosov flows and representations of surface groups — The purpose of this talk is to explain how representations of surface groups in SL(n,R) are associated to Anosov flows. Moreover, I will explain that a certain "moduli space" of Anosov flows is related to representations of surface groups in an extension of the group of diffeomorphisms of the circle. This moduli space contains all the moduli spaces of representations of surface groups in SL(n,R), as well as the space of negatively curved metrics on the surface.

(September 21) Andy Sanders: Geometrization of 3-manifolds and long time behavior of the Ricci flow — Since the famous recent work of Perelman, a number of groups have provided complete, detailed proofs of both the Poincare conjecture (for 3-manifolds) and Thurston's Geometrization conjecture following the initial program of Ricci flow which was conceived of by Richard Hamilton. Concerning the existence and uniqueness of a Ricci flow with surgery on a compact, oriented 3-manifold, the existing proofs are conceptually similar; but the analysis of the Ricci flow for large time is treated a number of different ways. A recent manuscript of Bessières, Besson, Boilieu, Maillot and Porti provides a new approach to the final step in the proof of Geometrization, which uses previous work of Thurston, Gromov and many others. In this talk, I will attempt to quickly review the large scale scheme of the Ricci flow approach to Geometrization, and then explain the results of the above mentioned authors which lead to a proof of the Geometrization conjecture.

(September 24) Bill Goldman: Three-dimensional affine space forms and geodesic flows of noncompact hyperbolic surfaces — The classification of 3-manifold quotients of R³ by discrete groups of affine transformations naturally leads to deformations of hyperbolic-geometry structures on noncompact surfaces. The deformations of particular interest are those in which the lengths of geodesics (specifically, measured geodesic laminations) uniformaly increase. The quotients have natural geodesically complete flat Lorentzian metrics. This represents joint work with Charette, Drumm, Labourie, Margulis and Minsky.

(September 28) Thomas Koberda: Homological representation theory of the mapping class group — I will talk about various properties of mapping classes that can be detected from looking at their actions on the homology of finite covers of surfaces. In addition, I will illustrate an explicit method of obtaining the regular representations of the Galois group on the holomorphic forms on a finite cover of Riemann surfaces, which is an important base case for understanding the action of the entire mapping class group on the homology of a cover. I will discuss some connections to the study of 3-manifolds.

(October 8) Moon Duchin: The space of flat metrics — What billiard trajectories can occur on a rational table? One approach to this classical problem is to first develop the table to obtain a (singular) flat metric on a hyperbolic surface S, and study the space of flat metrics under the action of SL(2,R). These flat structures also arise as metrics induced on S by quadratic differentials, where the diagonal part of the SL(2,R) action can be interpreted as Teichmüller geodesic flow. I'll discuss a collection of geometric and dynamical questions about flat surfaces with emphasis on recent joint work with Leininger and Rafi on their length spectra and asymptotic geometry.

(October 12) Tullia Dymarz: Bilipschitz equivalence is not equivalent to quasi-isometric equivalence for finitely generated groups — We give an example of two finitely generated quasi-isometric groups that are not bilipschitz equivalent. The proof involves structure of quasi-isometries from rigidity theorems, analysis of bilipschitz maps of the n-adics and uniformly finite homology.

(October 19) Hisham Sati: Generalizations of Spin structures — A Spin structure is a lift of the structure group of the tangent bundle from the special orthogonal group SO to the covering group, the Spin group. We discuss how the process of killing higher homotopy groups leads to generalizations of a Spin structure called String and Fivebrane structures. Ideas from physics provide the motivation and terminology. We will also discuss some of the related geometry as time permits.

(October 26) Jonathan Block: Talk 1: General introduction to derived equivalences
Talk 2: Differential graded categories in geometry and topology — We give an introduction to differential graded categories and their homological/homotopical algebra, especially their use in derived equivalences. In the second talk, we give an introduction to differential graded categories and their homological/homotopical algebra, especially their use in derived equivalences.

(November 2) Virginie Charette: Proper affine actions on Minkowski spacetime — In joint work with Todd Drumm and William Goldman, we showed that an affine representation of a three-holed surface group is proper if and only if the Margulis invariant -- a measure of signed Lorentzian length -- on the three boundary components have the same sign. In fact, we showed that this sign condition is equivalent to the existence of a fundamental domain for the action. We proved this by looking at a certain geodesic lamination on the three-holed sphere. We will discuss this work and how it may apply to a more general class of surfaces.

(November 9) John Loftin: Surfaces, SL(3), and an equation of Tzitzeica — I will discuss the geometry related to four different elliptic versions of an equation of Tzitzeica:
Δ u ± 2e^-2u |U|² ± 2e^u - 2κ = 0
on a Riemann surface equipped with a conformal background metric with curvature κ and a holomorphic cubic differential U. Each of these equations is an integrability condition for developing a special type of surface whose geometry is governed by a real form of SL(3,C). Two of these Tzitzeica type equations lead to affine geometry, producing elliptic and hyperbolic affine spheres, which are related to singularity models for Strominger-Yau-Zaslow conjecture. Another version of Tzitzeica produces minimal Lagrangian tori in CP², and has been well-studied from the point of view of integrable systems. Finally, in joint work in progress with Ian McIntosh, we discuss minimal Lagrangians into CH². This should give us a description of a region of the moduli space of surface group representations into SU(2,1), which we expect to be similar to the quasi-Fuchsian representations as a subset of representations into SL(2,C).

(November 16) David Futer: Cusp volume of fibered 3-manifolds — Consider a 3-manifold M that fibers over the circle, with fiber a punctured surface F. I will explain how the volume of a maximal cusp of M (in the hyperbolic metric) is determined up to a bounded constant by combinatorial properties of arcs in the fiber surface F. This is joint work with Saul Schleimer.

(November 30) Dragomir Saric: Circle homeomorphisms and shears — The space of homeomorphisms Homeo(S¹) of the unit circle S¹ is a classical topological group which acts on S¹. Homeo(S¹) contains many important subgroups such as the infinite dimensional Lie group Diffeo(S¹) of diffeomorphisms of S¹, the group QS(S¹) of quasisymmetric maps of S¹, the characteristic topological group Symm(S¹) of symmetric maps of S¹, and many more. We use the shear coordinates on the Farey tesselation to parametrize the coadjoint orbit spaces Möb(S¹) \ Homeo(S¹), Möb(S¹) \ QS(S¹) and Möb(S¹) \ Symm(S¹). To our best knowledge, this gives the only known explicit parametrization of the universal Teichmüller space T(H)= Möb(S¹) \ QS(S¹).

(December 1) Andrew Putman: The Picard group of the moduli space of curves with level structures — The Picard group of an algebraic variety X is the set of complex line bundles over X. In this talk, we will describe the Picard groups of certain finite covers of the moduli space of curves. The methods we use combine ideas from algebraic geometry, finite group theory, and algebraic/geometric topology.

(December 7) Pierre Albin: Equivariant cohomology and resolution — The equivariant cohomology of a manifold with a group action is, in some sense, the cohomology of the space of orbits. I will describe joint work with Richard Melrose where we make this precise.
In fact our method of lifting the group action and the equivariant cohomology to a manifold with corners and smooth orbit space also allows us to define an `improved' equivariant cohomology extending a construction of Baum, Brylinski, and MacPherson.

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