(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 3) Christian Zickert: Parametrizing representations of 3-manifold groups — see previous week

(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.

(March 26) Todd Drumm: TBA — TBA

(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 the 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 the Einstein equation, by using Riemannian submersions.

(April 23) Boris Botvinnik: TBA — TBA

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