(September 13) David Hamilton: Geometry of Meromorphic Semigroups — These generalizations of Kleinian groups/Complex Dynamics are rigid in the category of "asymptotic conformal tranformations". eg if the limit set/julia set J are contained in an asymptotically conformal curve than J is contained in a circle/line.

(September 20) Karin Melnick: Normal forms for conformal vector fields on Lorentzian manifolds — Isometries of a Riemannian or pseudo-Riemannian manifold fixing a point are conjugate to their differential via the exponential map. No such linearization exists in general for conformal transformations fixing a point. The main theorem of this talk asserts that on a real-analytic Lorentzian manifold M, any conformal vector field vanishing at a point has linearizable flow, or M is conformally flat. This result leads to a normal form for any such vector field near its singularity.

(September 27) Emily Landes: Identifying the Canonical Component for the Whitehead Link — Although character varieties have proven to be a useful tool in studying hyperbolic 3-manifolds, only recently have explicit models for the SL(2,C) character varieties of twist knot complements been constructed. As the twist knot complements can be obtained by Dehn filling one of the cusps of the Whitehead link complement, we are naturally interested in determining the canonical component of the Whitehead link character variety and studying the relationship among character varieties of manifolds obtained by Dehn surgery. In my talk I will show how the canonical component of the Whitehead link character variety is P² blown-up at 10 points and discuss the canonical components for a few other hyperbolic 2-component link complements.

(October 11) Caroline Series: Top terms of trace polynomials in Kra's plumbing construction — Kra's plumbing construction manufactures a surface S by `plumbing' together a suitable family of triply punctured spheres. This gives a natural pants decomposition of S, together with a projective structure for which the associated holonomy representation ρ depends on the `plumbing parameters' τ. In particular Trace ρ(γ), for γ in the fundamental group of S, is a polynomial in the τ. Simple curves on S can be described in terms of their Dehn-Thurston coordinates relative to the pants decomposition. After explaining the construction, we show that if γ is simple there is a remarkably easy formula relating the coefficients of the top terms of ρ(γ) and its Dehn-Thurston coordinates. The formula generalises ones previously obtained by Keen, Parker and Series for the once and twice punctured torus. The proof involves a rather interesting result on matrix products. This is joint work with Sara Maloni.

(October 18) Louis Theran: Generic rigidity of periodic frameworks — A planar periodic bar-joint framework is an infinite structure, periodic with respect to a lattice, made of fixed-length bars connected by universal joints with full rotational freedom. The allowed continuous motions are those that preserve the length and connectivity of the bars. Furthermore, the lattice is allowed to deform.

The most basic question one might ask about a periodic framework is the so-called rigidity question: are all the allowed motions Euclidean isometries? In this talk I'll describe the answer for generic periodic frameworks in the plane: generically, rigidity is a property of a finite graph with elements of Z² "coloring" the edges derived from the framework, and the combinatorial properties characterizing rigidity can be checked in polynomial time.

This is joint work with Justin Malestein.

(October 25) Jack Calcut: Extending Quillen's Quartet on the Pullback Functor — Quillen gave a quartet of algebraic equivalents for certain properties of the pullback functor on coverings. We extend this quartet by giving three new equivalences. We will also show, using a reduction to the finite group case and the theory of Burnside rings, that if f^* : Cov(Y) → Cov(X) is not essentially injective, then f^* is not essentially injective on finite component covers of Y. We will state some open problems for future study. This is joint work with John D. McCarthy and Jeremy Walthers.

(November 1) Alexander Gaifullin: The Torelli group of genus 3 is not finitely presented — The Torelli group is the subgroup of the mapping class group of an oriented closed surface consisting of all mapping classes acting freely on the homology of the surface. It is well known that the Torelli group of genus 1 is trivial, the Torelli group of genus 2 is an infinitely generated free group (Mess, 1992), and the Torelli group of genus g is finitely generated for g>2 (Johnson, 1983). We prove that the Torelli group of genus 3 is not finitely presented. The main tool is the Cartan-Leray spectral sequence for the action of the Torelli group on the complex of cycles constructed by Bestvina, Bux, and Margalit in 2007.

(November 8) Jeremy Kahn: Building immersed hyperbolic surfaces in hyperbolic 2- and 3-manifolds — We describe an approach to building a compact surface in a closed hyperbolic manifold, and building a finite-area surface in a finite-volume 3-manifold, using basic building blocks (ideal triangles and pairs of pants) and an analysis of how they are fit together.

We will then outline the proofs of the Weil-Petersson Ehrenpreis conjecture and the surface subgroup conjecture along the lines of this approach. If time permits we will also discuss the difficulties that arise in this approach to the (Teichmuller) Ehrenpreis conjecture.

This is all joint work with Vladimir Markovic.

(November 22) Zeno Huang: Counting minimal surfaces in hyperbolic three-manifolds — We describe a type of counting problems in for minimal surfaces in hyperbolic three-manifolds. We look for minimal immersions in a prescribed conformal class, with a prescribed second fundamental form. In her approach to use minimal surfaces to parameterize a space of almost Fuchsian manifolds, Uhlenbeck showed the existence of such a minimal immersion for some parameter interval. We proved a nonexistence result, and a non-uniqueness result for this counting problem. This is a joint work with Marcello Lucia.

(December 6) Aaron Magid: Hyperbolic Dehn filling and the shape of the Maskit slice — The Maskit slice is an embedding of the Teichmüller space of the punctured torus into C based on a deformation space of hyperbolic 3-manifolds. Minsky showed that the boundary of the Maskit slice is a Jordan curve, and Miyachi proved that the boundary is not a quasicircle. We reprove Miyachi's result using the hyperbolic Dehn filling theorem. More generally, we show how this filling theorem can be used to study the shape of other deformation spaces of hyperbolic 3-manifolds.

(December 14) Virginie Charette: The Crooked Plane Conjecture — The crooked plane conjecture states that any complete flat Lorentzian 3-manifold arises from a tiling of Minkowski spacetime under the action of a holonomy representation of its fundamental group. We would consequently obtain an explicit topological description of the manifold as a solid handlebody.

The problem originates in some late 20th century work of Margulis, who discovered surprising examples of free groups of isometries acting freely and properly discontinuously on 3d Minkowski spacetime. Drumm introduced crooked planes as a means to describe fundamental domains for these actions, setting the stage for some far-reaching extensions of Margulis' discovery.

In joint work with Drumm and Goldman, we proved the crooked plane conjecture for all surfaces of Euler characteristic -1. We will discuss how we proved this using ideal triangulations on these surfaces.

(January 31) Christian Zickert: Hilbert's third problem and its generalizations — In the year 1900, Hilbert presented his famous list of problems. The third problem asks whether any two 3-dimensional euclidean polyhedra with the same volume are "scissors congruent", i.e., whether one of them can be cut into finitely many smaller polyhedra, that can be reassembled into the other. We discuss this problem as well as its relations to contemporary research, including hyperbolic geometry, algebraic K-theory and Chern-Simons theory.

(February 1) David Ben McReynolds: Symmetry and realization problems — Finding manifolds, groups, covers, etc., with specific special properties is the focus of this talk. One famous, simple, and open example of a realization problem is the inverse Galois problem, namely, whether or not a given finite group G arises as the Galois group of a Galois extension of the rationals. Geometric and algebraic symmetry typically appear in realization problems as obstructions. I will discuss a few problems in geometry/analysis/algebra and explain how controlling symmetry is key in solving these example problems.

(February 7) Kate Petersen: Character Varieties of Once-Punctured Torus Bundles — I will discuss the SL(2,C) character varieties of once-punctured torus bundles with tunnel number one. The SL(2,C) character varieties of finite volume, non-compact, hyperbolic 3-manifolds encode a lot of data about the original manifolds, but are usually prohibitively difficult to compute. Therefore, connections between the algbro-geometric structure of these varieties and topological invariants of the manifolds are not well understood. I will compute defining equations for this nice family of manifolds, and compare some classical invariants of these varieties with invariants of the underlying manifolds. This is joint work with Ken Baker.

(February 14) Ser Peow Tan: A new identity for closed hyperbolic surfaces — We prove a new identity for closed hyperbolic surfaces whose terms depend on the geometry of embedded pairs of pants and one-holed tori in the surface. This is joint work with Feng Luo.

(February 21) Misha Kapovich: RAAGs in Ham — I will explain how to embed arbitrary RAAGs (Right Angled Artin Groups) in Ham (the group of hamiltonian symplectomorphisms of the 2-sphere). The proof is combination of topology, geometry and analysis: We will start with embeddings of RAAGs in the mapping class groups of hyperbolic surfaces (topology), then will promote these embeddings to faithful hamiltonian actions on the 2-sphere (hyperbolic geometry and analysis).

(February 28) Fanny Kassel: Anti-de Sitter 3-manifolds and maximally stretched laminations on hyperbolic surfaces — I will explain how understanding compact Lorentz 3-manifolds of constant negative curvature reduces to a problem of representations of surface groups. Given a pair (j,ρ) of representations of a surface group into PSL₂(R) with j Fuchsian, an important issue is to understand if there exists a (j,ρ)-equivariant map from the hyperbolic plane H² to itself with Lipschitz constant < 1. When such a map does not exist, Francois Gueritaud and I prove the existence of a geodesic lamination that is maximally stretched by all equivariant maps of H² with minimal Lipschitz constant. Our work generalizes that of Thurston, and builds on the extension theory of Lipschitz maps in spaces of bounded curvature.

(March 7) Nitu Kitchloo: Bott-Samelson resolutions and universal lifts of Kac-Moody groups — Kac-Moody groups are an interesting class of topological groups, that contains compact Lie groups, Loop groups and several other exotic examples. Given a Kac-Moody group G, we will construct a universal lift \hat{G} of G, with the property that the Schubert varieties of \hat{G} are closely related to the Bott-Samelson resolutions of G. In particular, \hat{G} encodes all the different ways one can desingularize the Schubert varieties of G. We will then explore the topology of this universal lift.

(March 14) Pierre Py: Kähler groups, real hyperbolic spaces and the Cremona group — Starting from a classical theorem of Carlson and Toledo, we will discuss actions of fundamental groups of compact Kähler manifolds on finite or infinite dimensional real hyperbolic spaces. We will see that such actions almost always (but not always) come from surface groups. We then give an application to the study of the Cremona group. The talk will take us from the infinite dimensional representation theory of PSL(2, R) to algebraic geometry. This is a joint work with Thomas Delzant.

(March 28) Walter Freyn: Kac-Moody symmetric spaces — Kac-Moody symmetric spaces are the natural infinite dimensional counterpart to finite dimensional Riemannian symmetric spaces. While they share most structure properties and have a similar classification, they are necessarily Lorentzian. In this talk, we describe the finite dimensional blueprint and explain why Kac-Moody symmetric spaces are the natural generalisation. We describe the interplay between geometric, algebraic and functional analytic structures in their construction and give some remarks about their role in infinite dimensional differential geometry and Kac-Moody geometry.

(April 04) Hossein Namazi: Combinatorial and Geometric views of hyperbolic 3-manifolds — One of the exciting trends in the study of hyperbolic 3-manifold is an attempt in finding combinatorial pictures and models that provides a "coarse" model for the geometry of hyperbolic 3-manifolds. We give an overview of various results centered around this topic and explain the major tools and features of this approach. We also explain how these descriptions can be used to understand the relationship between the geometric and topological properties of 3-manifolds.

(April 11) Florent Schaffhauser: Moduli of real and quaternionic bundles over a curve — We examine a moduli problem for real and quaternionic vector bundles over a smooth complex projective curve, and we give a gauge-theoretic construction of moduli spaces for such bundles. These spaces are irreducible subsets of real points inside a complex projective variety. We relate our point of view to previous work by Biswas, Huisman and Hurtubise, and we use this to study the Gal(C/R)-action [E] → [σ*E] on moduli varieties of semistable holomorphic bundles over a complex curve with given real structure σ. We show in particular a Harnack-type theorem, bounding the number of connected components of the fixed-point set of this action by 2^g+1, where g is the genus of the curve. In fact, taking into account all the topological invariants of a real algebraic curve, we give an exact count of the number of connected components, thus generalizing to rank r ≥ 2 the results of Gross and Harris on the Picard scheme of a real algebraic curve.

(April 18) Sarah Koch: The Deligne-Mumford compactification of the moduli space of curves — This is joint work with John H. Hubbard. We outline a proof that the Delinge-Mumford compactification of the moduli space of curves is isomorphic (as an analytic space) to the quotient of augmented Teichmueller space by the action of the mapping class group. The main difficulty is putting a complex structure on this quotient. If time allows, we relate this complex structure to one obtained by Earle and Marden.

(April 25) Jozef H. Przytycki: Homology of distributive structures — While homology theory of associative structures, such as groups and rings, has been extensively studied in the past beginning with the work of Hopf, Eilenberg, and Hochschild, the non-associative structures, such as quandles, were neglected until recently. The distributive structures have been studied for a long time and even C.S. Peirce in 1880 emphasized the importance of (right) self-distributivity in algebraic structures. However, homology for such universal algebras was introduced only fifteen years ago by Fenn, Rourke and Sanderson. I will develop this theory in the historical context and describe relations to topology and similarity with some structures in logic. I will also speculate how to define homology for Yang-Baxter operators and how to relate our work to Khovanov homology and categorification. We use here the fact that Yang Baxter equation can be thought of as a generalization of self-distributivity.

(April 28) Dani Wise: The structure of groups with a quasiconvex hierarchy — We prove that hyperbolic groups with a quasiconvex hierarchy are virtually subgroups of graph groups. Our focus is on "special cube complexes" which are nonpositively curved cube complexes that behave like "high dimensional graphs" and are closely related to graph groups. The main result illuminates the structure of a group by showing that it is "virtually special," and this yields the separability of the quasiconvex subgroups of the groups we study. As an application, we resolve Baumslag's conjecture on the residual finiteness of one-relator groups with torsion. Another application shows that generic Haken hyperbolic 3-manifolds have "virtually special" fundamental group. Since graph groups are residually finite rational solvable, combined with Agol's virtual fibering criterion, this proves that finite volume Haken hyperbolic 3-manifolds are virtually fibered.

(May 2) Rasmus Villemoes: Cohomology of mapping class groups — The mapping class group of a surface acts on moduli spaces of flat G-connections over the surface, preserving a symplectic form. Hence we get unitary representations of the mapping class group by considering the square-integrable functions. Computing the first cohomology of the mapping class group with coefficients in these modules is interesting from several viewpoints: If any of these cohomology groups is non-trivial, one has a proof that the mapping class group does not have property (T). On the other hand, the vanishing of these cohomology groups is, by work of Andersen, related to mapping class group invariant deformation quantization of the moduli spaces. In the talk, we outline these relationships, and present one approach to computing these cohomology groups, which has been successfully completed in the U(1)-case.

This is joint work with J.E. Andersen

(May 5) Dick Canary: Mapping class groups and moduli spaces of hyperbolic 3-manifolds — In two dimensions, one studies the Teichmueller space of marked hyperbolic surfaces of fixed genus. The quotient by the mapping class group of the surface is the moduli space. Our talk will focus on three-dimensional analogues of this much-studied situation. The analogue of Teichmueller space is the space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold. The quotient AI(M) of AH(M) by the action of the outer automorphism group of the fundamental group of M is the analogue of moduli space. The moduli space AI(M) can be quite pathological, often failing to even be Hausdorff. We will survey results on the outer automorphism group and the quotient moduli space. (This talk describes joint work with Darryl McCullough and Peter Storm.)

(May 9) Martin Bridgeman: Volume Identities for hyperbolic manifolds with boundary — Given a finite-volume hyperbolic n-manifold M with totally geodesic boundary, we show there is a real valued function F_n such that the volume of any finite volume hyperbolic n-manifold M with totally geodesic boundary ∂M is the sum of values of F_n on the orthogeodesic length spectrum. For n=2 the function F₂ is the Rogers L-function and the summation identities give dilogarithm identities on the Moduli space of surfaces. We will also discuss volume identities for geometrically finite Kleinian groups.

(May 23) Sara Maloni: Bers-Maskit slices of the quasifuchsian space — Given a surface S, Kra's plumbing construction endows S with a projective structure for which the associated holonomy representation f depends on the `plumbing parameters' t_i. In this talk we will describe a more general plumbing construction which gives us a group G in a particular slice of the quasi-Fuchsian space QF(S) (instead of the Maskit one given by Kra's plumbing construction). Using the complex Fenchel-Nielsen coordinates for QF(S), we can describe this slice, called the Bers-Maskit slice BM(S), as a subset of the slice where the length parameters take a fixed real value. Then one can see that, as these values tend to zero, the slices BM(S) tend to the Maskit slice M(S). The Bers-Maskit slice are also a connected component of the more general linear slices L(S). Some results about those slices will be described.

(August 3) Virginie Charette: On Margulis spacetimes arising from nonorientable surfaces — A Margulis spacetime is a complete affine 3-manifold M with nonsolvable fundamental group. Associated to every Margulis spacetime is a noncompact complete hyperbolic surface S. Surprisingly, every Margulis spacetime is orientable, even when S is nonorientable. We will talk about the case where S is a two-holed cross-surface.

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