(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.

(January 25) Alex Nabutovsky: Length of geodesics and related problems — A famous theorem proven by J.P. Serre asserts that for every pair of points on a closed Riemannian manifold there exist infinitely many distinct geodesics connecting these points. We prove that for every m there exist at least m distinct geodesics of length < 4nm^2d connecting these points, where n is the dimension of the manifold, and d is its diameter. Our proof provides some new information about Morse landscapes of the length functional on loop spaces. (Joint work with Regina Rotman).

(February 1) Mehdi Khorami: Generalized Thom spectra and twisted K-theory — In this talk, we quickly recall the construction of the generalized Thom spectra and use it to define twisted K-theory and twisted spin bordism for any space X equipped with a three dimensional integral cohomology class. Hopkins and Hovey proved that the (untwisted) complex K-theory of X is related to the Spin^c bordism of X via an isomorphism of Conner-Floyd type. We investigate the analogous question for the twisted theories. This investigation leads to a clarification formula for twisted K-theory.

(February 15) Todd Drumm: Lorentzian Parabolic Transformations — Groups of Lorentzian (2+1) transformations of rank >1 which act freely and properly on 3 space were not originally thought to contain parabolic transformations. We will investigate how the existence of these groups was shown and how other tools, originally created for hyperbolic transformations, can be adapted to parabolic transformations. Beware: Crooked planes will exist in this talk.

(February 22) Charles Frances: Large essential singularities for higher dimensional conformal maps — Conformal immersions between Riemannian manifolds of same dimension at least 3 are a natural higher dimensional analogue of holomorphic maps. The aim of the talk is to investigate what classical 2-dimensional results such as Picard's or Caratheodory's theorems become in this context.

(February 23) Thierry Barbot: Anosov actions of R^k — A locally free action of R^k (resp. Z^k) on a closed manifold is Anosov if the flow induced by some one-parameter subgroup is partially hyperbolic with central direction tangent to the orbits of R^k (resp. if some element of Z^k) acts as an Anosov diffeomorphism. Such an action is codimension one if the unstable direction of some Anosov element has dimension one.

The ultimate goal of our work is to prove the following:

(Generalized Verjovsky conjecture): Any irreducible codimension one action of R^k (k ≥ 2) on a closed manifold of dimension n+k is topologically conjugate to the suspension of an action of Z^k on the torus by linear automorphisms.

I will present the "state of art" on this conjecture. (joint work with C. Maquera)

(March 1) Irine Peng: Multiplicative sets in solvable groups — A multiplicative set is a pair (A, G) where A is a finite set in a (multiplicative) group G. The doubling constant of A is defined to be the ratio |A²|/|A| where A²= {a₁. a₂: a₁, a₂ &isin A}.

When the group is abelian (additive), Freiman's theorem roughly says there is a subgroup H < G such that A is contained in a (finite) subset of a coset of H whose size is not too much bigger than A (where the "not too much bigger" is in terms of the doubling constant of A).

There has been generalizations of Freiman's theorem to a few specific non-abelian groups such as SL₂(C) and SL₂(F_p). In this talk I will discuss some recent works of Breuillard-Green, Tao, and Fisher-Katz-Peng on generalizing Freiman's theorem to nilpotent and solvable groups.

(March 8) Melissa Macasieb: Commensurability of Knot Complements — Let K be a hyperbolic (-2, 3, n) pretzel knot and M = S³ - K its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of M. Indeed, if n is not 7, we show that M is the unique knot complement in its class.

(March 22) Chris Atkinson: Two-sided combinatorial volume bounds for hyperbolic Coxeter polyhedra — I will describe how to compute two-sided volume bounds for certain hyperbolic polyhedra in terms of the combinatorics of their 1-skeleta. The bounds are an application of, among other things, Andreev's theorem, Schlaefli's formula, a combinatorial description of the geometric decomposition of polyhedral orbifolds, and a classification of Seifert-fibered polyhedral orbifolds.

(April 5) Ian Biringer: Geometric consequences of algebraic rank in hyperbolic 3-manifolds — Mostow's rigidity theorem states that a closed hyperbolic $3$-manifold M is determined up to isometry by the algebra of its fundamental group. We will discuss how the geometry of M is constrained by the minimal number of elements needed to generate its fundamental group; this invariant is called the (algebraic) rank of M. In particular, we will explain how M can be decomposed into a number of geometric building blocks such that the complexities of the blocks and of the decomposition depend only on M's algebraic rank and on a lower bound for M's injectivity radius.

Our work links rank and injectivity radius to a number of other geometric invariants, including Heegaard genus, the Cheeger constant and the first eigenvalue of the Laplacian. One can also use the techniques involved to prove a finiteness statement for the number of commensurability classes of arithmetic closed hyperbolic 3-manifolds with bounded rank and injectivity radius.

Joint with Juan Souto.

(April 12) Darren Long: Commensurators of infinite volume Kleinian groups — There is a celebrated theorem of Margulis which describes the commensurator of a lattice in a semi-simple Lie group. This has motivated much study of the commensurator of various classes of groups, together with its role in geometry and topology. We will discuss the case of infinite co-volume Kleinian groups which are not free groups. The commensurator is shown to be discrete and a lattice in the case of a virtual fibre group

(April 21) Shinpei Baba: CANCELLED — A (complex) projective structure is a certain geometric structure on a (closed) surface, and this structure corresponds to a (holonomy) representation of the surface group into PSL(2,C). However, this correspondence is not one-to-one.

(2,π)grafting is a certain surgery operation on a projective structure that produces a different projective structure with the same representation. Using grafting, Goldman gave a characterization of projective structures with a fixed fuchsian representation (1987). More recently, Gallo-Kapovich-Marden asked whether, given two projective structures with a fixed (general) representation, there is a sequence of graftings and inverse-graftings that transforms one to the other (2000).

We answer this question in the affirmative for all purely loxodromic representations, which are generic in the character variety.

(April 26) Jonathan Rosenberg: Metrics of positive scalar curvature on the 3-sphere — I will talk about work in progress with Boris Botvinnik, in which we use the Hamilton-Perelman theory of Ricci flow with surgery for metrics of positive scalar curvature ("psc") on S³ to show that the space of psc metrics is contractible. (The result is rather surprising, since the space of metrics of positive scalar curvature on S^4k+3 is not even connected when k > 0.) The idea of the proof is to use an induction to deform the space of psc metrics requiring at most n surgery steps in the Ricci flow down to the subspace of psc metrics requiring at most n-1 surgery steps.

(May 3) Ser Peow Tan: Dynamics of the Out(F₂) action on the SL(2,C) character variety of F₂, the free group on two generators. — We investigate the dynamics of the Out(F₂) action on the SL(2,C) character variety of F₂, the free group on two generators. We will pose several questions, and give some partial results.

(May 10) David Constantine: Group actions and compact forms of homogeneous spaces — Given a homogeneous space J\H, does there exist a discrete subgroup Γ in H such that J\H/Γ is a compact manifold? These compact forms of homogeneous spaces turn out to be rare outside of a few natural cases. Their existence has been studied by a very wide range of techniques, one of which is via the action of the centralizer of J in H. In this talk I'll show that no compact form exists when H is a simple Lie group, J is reductive and the acting group is higher-rank and semisimple. The proof uses cocycle superrigidity, Ratner's theorem and techniques from partially hyperbolic dynamics.

(May 26) Talia Fernós: Reduced 1-cohomology and relative property (T) — The celebrated theorems of Delorme (1977) and Guichardet (1972) establish the equivalence between property (T) and the vanishing of 1-cohomology, where the coefficients are taken in a unitary representation. In 2000 Shalom proved that the (a priori) weaker condition of the vanishing of reduced 1-cohomology is in fact equivalent to property (T) for the class of compactly generated groups. In 2005-2006 de Cornulier, Jolissaint, and Fernos independently showed that the vanishing of the restriction map on 1-cohomology is equivalent to relative property (T). One may ask if the relative version of Shalom's theorem is true. In a joint work with Valette we exhibit a large class of non-compact amenable group-pairs where the restriction map on reduced 1-cohomology always vanishes. Since amenable groups can not have relative property (T) with respect to non-compact subgroups, our result gives a strong negative answer to the above question.

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