(September 12) Stephen Halperin: Homotopy groups, loop space homology, free loop spaces and counting closed geodesics — (Joint work with Yves Felix and Jean-Claude Thomas) All connected finite dimensional spaces fall into three categories, for each of which we can give an explicit description of the behavior of the higher homotopy groups. This in turn provides good information about the rational homology of the loop space. An old conjecture asserts that the rational homology of the free loop space should show similar behavior which, by a theorem of Gromov would give lower bounds on the number of closed geodesics in a generic Riemannian manifold as a function of their length. I will also report some progress on this problem.

(September 19) John Millson: The first Betti number of a compact hyperbolic n manifold — In 1976 (Annals of Math 104, pg 235-247) I showed that for every n the standard arithmetic examples of compact n-manifolds M (there are infinitely many of these in each dimension n) had nonzero first Betti number by constructing nonseparating totally geodesic hypersurfaces. Now 35 years later Nicolas Bergeron, Colette Moeglin (of the University of Paris) and I can show that these hypersurfaces span the (n-1)-st homology of M. This is proved by combining the work I did with Steve Kudla in the 1980's where we constructed closed one forms dual to these hypersurfaces using the Weil representation and work of Jim Arthur on the trace formula which proves all the first cohomology comes from the above Weil representation construction. There are corresponding results for all orthogonal locally symmetric spaces of standard arithmetic type.

(September 26) Christian Zickert: Parametrizing representations of 3-manifold groups — We give an efficient parametrization of the set of boundary-unipotent representations of a 3-manifold group into SL(n, C), and use it to give a concrete formula for the Chern-Simons invariant. Numerical computations suggest that the imaginary part of the Chern-Simons invariant is always an integral linear combination of volumes of hyperbolic 3-manifolds. The talk will mostly deal with the case n=2, and Thurston's gluing equations.

(October 17) Gerard Freixas: An axiomatic characterization of holomorphic higher analytic torsion forms — The analytic torsion forms of Bismut-Köhler transgress the Grothendieck-Riemann-Roch theorem at the level of differential forms. A question that arises is to what extent the construction of these forms is natural. We will discuss a reasonable list of axioms to expect for a theory of analytic torsion forms, and explain how they characterize such possible theories. Time permitting, we will show that the theory of Bismut-Köhler is the unique theory for which the de Rham complex has torsion 0. This talk is based on joint work with J. Burgos and R. Litcanu.

(October 24) Tullia Dymarz: Rigidity and enveloping discrete groups by locally compact groups — If a finitely generated group G is a lattice in a locally compact group H then we say that H envelopes G. This terminology was introduced in the sixties by Furstenberg, who also proposed studying the problem of which locally compact groups can envelope which discrete groups. In the case when G is a lattice is a semisimple Lie group and H is a semisimple Lie group, the problem is solved by celebrated rigidity results of Mostow, Prasad and Margulis that show that there is only one possible envelope for each such lattice. In contrast we focus on classes of groups which are not lattices in any Lie group but do sit as lattices in the isometry groups of nice metric complexes. We show how in certain cases techniques from quasi-isometric rigidity can be used to give rigidity results.

(October 31) Richard Wentworth: Gluing determinants of Laplacians — I'll talk about certain elliptic boundary conditions for Laplace operators of framed holomorphic hermitian line bundles on Riemann surfaces with boundary. The framing is a choice of trivialization near the boundary. These boundary conditions give rise to gluing formulas for determinants on closed surfaces and a new proof of the "insertion formula" for determinants on exact sequences.

(November 7) Peter Zograf: Large genus asymptotics of intersection numbers on moduli spaces of algebraic curves — The aim of the talk is to review the latest developments in understanding the large genus behavior of intersection numbers of tautological classes on moduli spaces of (pointed) algebraic curves, with the main emphasis on the Weil-Petersson volumes (after a recent work by M. Mirzakhani and the speaker)

(November 14) Jason Behrstock: Quasi-isometric classification of 3-manifold groups — Any finitely generated group can be endowed with a natural metric which is unique up to maps of bounded distortion (quasi-isometries). A fundamental question is to classify finitely generated groups up to quasi-isometry. Considered from this point of view, fundamental groups of 3-manifolds provide a rich source of examples. Surprisingly, a concise way to describe the quasi-isometric classification of 3-manifolds is in terms of a concept in computer science called "bisimulation." We will focus on describing this classification and a geometric interpretation of bisimulation. (Joint work with Walter Neumann.)

(November 21) Adam Sikora : Character Varieties of surfaces as completely integrable systems — It is known that the trace functions of a maximal set of disjoint simple closed curves on a closed surface make its SU(2)-character variety into an (almost) completely integrable dynamical system. We prove an analogous statement for all rank 2 Lie groups. We will discuss the possible generalizations of this result to higher ranks and, if time permits, its applications to quantization of character varieties.

(November 28) Daniel Mathews: Hyperbolic cone-manifolds with prescribed holonomy — We examine the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A geometric cone-manifold structure on a surface, with all interior cone angles being integer multiples of 2?, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a geometric cone-manifold structure. Our constructions involve the Euler class of a representation, the universal covering group of the orientation-preserving isometries of the hyperbolic plane, and the action of the outer automorphism group on the character variety

(December 1) Barney Bramham: Approximating Hamiltonian systems by integrable systems using pseudo-holomorphic curves — I will talk about an approach, using pseudo-holomorphic curve techniques from symplectic geometry, to the following question in dynamical systems of Anatole Katok: "In low dimensions is every conservative dynamical system with zero topological entropy a limit of integrable systems?"

(December 5) Shawn Rafalski: The smallest Haken hyperbolic polyhedra — We determine the lowest volume hyperbolic Coxeter polyhedron whose corresponding hyperbolic polyhedral 3-orbifold contains an essential 2-suborbifold, up to a canonical decomposition along essential hyperbolic triangle 2-suborbifolds. This is joint work with Chris Atkinson (Temple University).

(January 12) Yanir Rubinstein: Einstein metrics on Kähler manifolds — The Uniformization Theorem implies that any compact Riemann surface has a constant curvature metric. Kähler-Einstein (KE) metrics are a natural generalization of such metrics, and the search for them has a long and rich history, going back to Schouten, Kähler (30's), Calabi (50's), Aubin, Yau (70's) and Tian (90's), among others. Yet, despite much progress, a complete picture is available only in complex dimension 2.

In contrast to such smooth KE metrics, in the mid 90's Tian conjectured the existence of KE metrics with conical singularities along a divisor (i.e., for which the manifold is `bent' at some angle along a complex hypersurface), motivated by applications to algebraic geometry and Calabi-Yau manifolds. More recently, Donaldson suggested a program for constructing smooth KE metrics of positive curvature out of such singular ones, and put forward several influential conjectures. In this talk we will try to give an introduction to Kähler-Einstein geometry and briefly describe some recent work mostly joint with R. Mazzeo that resolves some of these conjectures. One key ingredient is a new C^2,α a priori estimate and continuity method for the complex Monge-Ampère equation. It follows that many algebraic varieties that may not admit smooth KE metrics (e.g., Fano or minimal varieties) nevertheless admit KE metrics bent along a divisor.

(January 30) Matthias Goerner: Principal Congruence Links (part 1/2) — Thurston gave an example of an 8-component link whose complement is covered by 24 regular ideal hyperbolic tetrahedra such that the symmetries of the link complement can take any tetrahedra to any other tetrahedra in all possible 12 orientations of the tetrahedra. I show how to construct two more links with 12 respectively 20 components with this special property.
These links are examples of prinicipal congruence links, i.e., links whose complement is H³ divided by a principal congruence subgroup of PGL(2,O_D) where O_D is the ring of integers of the imaginary quadratic number field of discrimninant D=-3. The groups PGL(2,O_D), respectively, PSL(2,O_D) (Bianchi groups) are of special interest because every arithmetic cusped hyperbolic 3-manifold is commensurable with such a group for some D. The principal congruence subgroup are the easiest to define subgroups of PGL(2,O_D). They are given by ker(PGL(2,O_D) → PGL(2,O_D/I)) for some ideal I, yielding an interesting class of arithmetic cusped hyperbolic 3-manifolds with very large symmetry group.

(February 6) Matthias Goerner: Principal congruence links (2/2) — See previous abstract.

(February 13) Rodrigo Trevino: Typical behavior of non-orientable measured foliations on flat surfaces — The Kontsevich-Zorich cocycle is a dynamical system defined on the cohomology bundle of the moduli space of quadratic differentials whose projection to the moduli space is the Teichmüller flow. Using a criterion of Giovanni Forni applied to measures on the moduli space of Abelian differentials supported on points coming from quadratic differentials by a standard, double cover construction, we can prove that the Kontsevich-Zorich cocycle is non-uniformly hyperbolic for this measure. I will discuss applications to the study of deviations in homology of typical leaves of the corresponding non-orientable foliations as well as applications to the study of deviations of ergodic averages, which exhibit phenomena different from the Abelian (i.e., orientable) case.

(February 20) Scott Wolpert: Infinitesimal deformations of nodal stable curves — The germ of the coordinate axes in C² is an example of a Riemann surface with a node. The projection from C² to C given by p(z,w) = zw defines a family of hyperbolas limiting to a node, the coordinates axes. Deligne-Mumford compactified the moduli space of compact Riemann surfaces by adding points corresponding to certain noded Riemann surfaces/stable curves (compact spaces with each component of the complement of the nodes being a negative Euler characteristic Riemann surface). By Kodaira-Spencer, at a smooth compact Riemann surface, the moduli space cotangent space is the space of holomorphic quadratic differentials on the surface.

The focus of the lecture is the description of the moduli space cotangent space at noded Riemann surfaces and the accompanying geometry. The description is in terms of sections of an analytic sheaf and a residue map. A sketch will be given of the local deformation theory of noded Riemann surfaces, the moduli tangent/cotangent bundles and the plumbing construction. A change of plumbing formula and an example of plumbing an Abelian differential will be presented.

(February 27) Jason Deblois: Mutation and commensurability of link complements — I'll describe the construction of a family of two-component link complements (in S³) and describe the commensurability relation among its members (manifolds are commensurable if they have a common finite-sheeted cover). This is a convenient platform for introducing a family of questions about the geometry of knot complements and their algebraic invariants, which I'll then do.

(March 5) Jonathan Rosenberg: Dualities in field theories and the role of K-theory — It is now known (or in some cases just believed) that many quantum field theories exhibit dualities, equivalences with the same or a different theory in which things appear very different, but the overall physical implications are the same. We will discuss some of these dualities from the point of view of a mathematician, and explain how K-theory plays an important role. This will be an introductory talk; no knowledge of physics is assumed.

(March 12) Jonathan Rosenberg: String theory on elliptic curve orientifolds and KR-theory — Orientifolds are spacetime manifolds equipped with an involution. In string theory on such a spacetime, charges take their values in the KR-theory of Atiyah ("Real" K-theory, with a capital R). We will concentrate on the case where the spacetime is an elliptic curve defined over R (crossed with R⁸ with trivial involution) and discuss how T-duality matches up with the KR calculations. This is joint work with Doran and Mendez-Diez.

(March 26) Todd Drumm: Dirichlet domains in several geometries — For a group G acting properly on a geometric space X, the Dirichlet domain centered at y is the set of points closer to y than to any other elements in the G-orbit of y. We review the results for the Dirichlet domain for cyclic subgroups of Isom(H²) and Isom(H³). Finally, we look at new results for Dirichlet domains in the bidisk, H² × H².

(April 2) Karin Melnick: Geometric Structures, Cartan Geometries, and their Automorphisms — Cartan geometries are a notion encompassing essentially all classical rigid geometric structures, and they provide an ideal setting in which to study automorphisms of such structures. The starting point for the major theorems of the Gromov-Zimmer theory of automorphisms of rigid geometric structures is Gromov's difficult Frobenius theorem. I will present a version of this theorem in the setting of Cartan geometries, where it admits a substantially more appealing formulation and simpler proof.

(April 9) Christian Zickert: Thurston's gluing equations for PSL(n,C) — Thurston's gluing equations are polynomial equations invented by Thurston to explicitly compute the hyperbolic structure of a triangulated 3-manifold. The equations parametrize representations in PSL(2,C). We generalize these equations to obtain a parametrization of representations in PSL(n,C). These have applications in quantum topology, and there is an intriguing duality between the gluing equations and the Ptolemy coordinates of Garoufalidis-Thurston-Zickert.

(April 16) Andreas Arvanitoyeorgos: Recent progress on homogeneous Einstein metrics on generalized flag manifolds — A Riemannian manifold (M, g) is called Einstein if the Ricci tensor satisfies Ric(g) = λg for some λ ∈ R. A generalized flag manifold is a compact homogeneous space M = G/K = G/C(S), where G is a compact semisimple Lie group and C(S) is the centralizer of a torus in G. Equivalently, it is an orbit of the adjoint representation of G. In the first part of the present talk I will review certain known aspects about G-invariant Einstein metrics on generalized flag manifolds, stressing the fact that when the number of the irreducible summands of the isotropy representation of M = G/K increases, then explicitly finding the Ricci tensor (and moreover solving the Einstein equation) becomes a difficult task. Then, I will discuss the aspect of the classification of flag manifolds by using the painted Dynkin diagrams. Finally, I will present a new method for the construction of solutions of the Einstein equation, by using Riemannian submersions.

(April 23) Boris Botvinnik: Concordance and isotopy of metrics with positive scalar curvature — Two positive scalar curvature metrics g₀, g₁ on a manifold M are called psc-isotopic if they are homotopic through metrics of positive scalar curvature. It is well known that if two metrics g₀, g₁ of positive scalar curvature on a closed compact manifold M are psc-isotopic, then they are psc-concordant, i.e., there exists a metric g of positive scalar curvature on the cylinder M × I which extends the metrics g₀ on M × {0} and g₁ on M × {1} and is a product metric near the boundary. We sketch a proof that if psc-metrics g₀, g₁ on M are psc-concordant, then there exists a diffeomorphism φ : M × I → M × I with φ|M×{0} = Id (a pseudoisotopy) such that the metrics g₀ and (φ|M×{1})*g₁ are psc-isotopic. In particular, for a simply connected manifold M with dim M ≥ 5, psc-metrics g₀, g₁ are psc-isotopic if and only if they are psc-concordant. To prove these results, we employ a combination of relevant methods: surgery tools related to Gromov-Lawson construction, classic results on isotopy and pseudo-isotopy of diffeomorphisms, standard geometric analysis related to the conformal Laplacian, and the Ricci flow.

(April 30) Dragomir Saric: Bending by finitely additive cocycles — Pleated surfaces inside hyperbolic three-manifolds induce finitely additive transverse cocycles to their geodesic lamination pleating loci. We give a sufficient condition on finitely additive transverse cocycles that guaranties that the pleated surfaces give rise to quasiFuchsian manifolds. Our work is motivated by the proof of the surface subgroup conjecture by Kahn and Markovic. They use a geometric condition on complex Fenchel-Nielsen coordinates to guarantee that the pleated surface gives a quasiFuchsian manifold. Our approach works for arbitrary (including infinite) geodesic laminations whereas the complex Fenchel-Nielsen coordinates are by definition restricted to finite geodesic laminations. We also give an explicit numerical values for the size of the parameters which guarantees that the pleated surface is quasiFuchsian.

(May 7) Joseph Maher: Random walks on the mapping class group — The mapping class group is the group of all homeomorphisms from a surface to itself, up to isotopy. We will show that a random walk gives a pseudo-Anosov element with asymptotic probability one, using the action of the mapping class group on the complex of curves.

(May 14) Steve Balady: Microbundles — This talk will be a quick review of Milnor's theory of microbundles, with some of its applications.

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