(September 18) Jonathan Rosenberg: A progress report on positive scalar curvature - We discuss the current status of the problem of determining what [closed] manifolds admit metrics of positive scalar curvature.

(October 9) Leonid Chekhov: Quantum Teichmüller theory of bordered surfaces - Using the graph technique, we define and quantize Teichmüller spaces of bordered surfaces. We relate this to the mapping class group and geodesic algebras.

(November 6) Scott Wolpert: An update on the Weil-Petersson geometry of Teichmueller space - We present new results for the Weil-Petersson (WP) metric for Teichmueller space and for geodesic-length functions. The augmented Teichmueller space aT with the WP metric is a complete CAT(0) metric space. The length of a geodesic is a basic invariant of a marked hyperbolic structure. Formulas for the WP gradient and Hessian of a geodesic-length function will be presented.

For the Fenchel-Nielsen twist-length coordinates a comparison and expansion for the WP metric will be presented in terms of an elementary model metric. The comparison for the WP Kaehler form and Levi-Civita connection will also be presented. For points of aT the WP Alexandrov tangent cone will be described. The WP tangent cone is described as a cone in an inner product space.

(November 13) Calder Daenzer: A groupoid approach to noncommutative T-duality - Begining with the data of an equivariant gerbe on a principal bundle, I will construct a groupoid which is a presentation of the gerbe. This groupoid presentation of a gerbe on a principle bundle allows for two constructions: The first is a Fourier type isomorphism from the groupoid algebra of the equivariant gerbe to another interesting groupoid algebra. Here the principal bundle should be a bundle of abelian groups. The second construction is a deformed version of the important Morita equivalence called Mackey-Rieffel imprimitivity. A certain composition of these two constructions, when applied to a gerbe on a principal torus bundle, reveals itself as the T-duality that has been much studied by Mathai, Raeburn, Rosenberg, Williams and others!

(November 20) Julien Paupert: Discrete complex reflection groups in PU(2,1) - The group PU(n,1) of holomorphic isometries of complex hyperbolic space is one of the two occurrences (with PO(n,1)) of a simple real Lie group of rank 1 where Margulis superrigidity does not hold. The only known examples of nonarithmetic lattices in PU(2,1) were constructed by Mostow in the 1980's. We will recall the construction of these lattices, which are generated by complex reflections, and we will show how to find new examples of the same kind in a family of configuration polygons. This is joint work with John Parker (Durham).

(November 27) Chris Truman: Reidemeister Torsion in the Khovanov Complex - To calculate the Khovanov homology of a knot, one obtains a complex with distinguished bases (the smoothings of a diagram of the knot). One can compute the Reidemeister torsion of this complex to obtain a volume form on the Khovanov homology that is an invariant of the knot. I'll explain how one can check invariance under the Reidemeister moves by studying the torsion of the maps on the complexes induced by the moves. This is joint work with Juan Ariel Ortiz-Navarro at the University of Iowa.

(December 4) Stavros Garoufalidis: Resurgent functions and asymptotic expansions in quantum topology - A resurgent function (due to Ecalle) is a holomorphic function originally convergent in a small disk, with endless analytic continuation in the complex plane, minus a countable set of singularities. Examples of resurgent functions are meromorphic functions, and algebraic functions, or more generally, formal solutions to differential equations (linear or not). The position and shape of singularities are important invariants of resurgent functions. The Taylor series of a resurgent function (say at the origin) have asymptotic expansions with exponentially small terms included. What do resurgent functions have to do with quantum topology? Quantum topology associates formal power series to knotted objects. These series are important knot invariants whose asymptotic expansions (and their relation to the geometry of the knotted object) is a prized problem, a cpecial case of which is the Volume Conjecture.

In the talk we will formulate a general resurgence conjecture for power series associated to knotted objects. In particular, the position of the singularities of these resurgent functions are expected to be determined by the i vol + CS invariant of parabolic SL(2,C) representations of the knotted objects. As a nontrivial test case, we will give a proof of our resurgence conjecture for the simplest knot: the trefoil, and for a simple 3-manifold: the Poincaré homology sphere. Time permitting, we will indicate a proof of our resurgence conjecture for the simplest hyperbolic knot, namely the 4₁ knot. This is joint work with Ovidiu Costin.

(February 19) Steve Halperin: Exponential growth and an asymptotic formula for the ranks of homotopy groups of a finite 1-connected complex - Let X be an n-dimensional, finite, simply connected CW complex. Then either the homotopy groups are finite torsion groups in degrees > 2n-1 or else the sum of the ranks of these groups in degrees from k + 2 to k + n grows exponentially in k according to an explicit asymptotic formula.

(March 5) Guoliang Yu: Higher index theory of elliptic operators and geometry of groups - In this talk, I will give an introduction to higher index theory of elliptic differential operators and discuss its applications to geometry/topology of manifolds and its connection to geometric group theory. This talk should be accessible to nonexperts including graduate students.

(March 26) Piotr Grinevich: Circuiar motion on Riemann surfaces. - We will describe joint work with Paolo Santini. We study motion with constant geodesic curvature on Riemann surfaces with almost everywhere flat Riemann metrics. Straight-line motion on such Riemann surfaces corresponds to 2-d Hamiltonian dynamics with multivalued Hamiltonians, and has been seriously studied recently. In contrast with this, wery little is known about circular motion. Results of resent computer experiments will be discussed. We will present a conjecture about when all trajectories are periodic.

(April 30) Rebecca Goldin: A generalized Schubert calculus for symplectic circle manifolds - In this talk, we will show how a Hamiltonian $S1$ action on a symplectic manifold (with isolated fixed points) is enough information to determine its $S1$-equivariant cohomology ring. The ring is generated by 'canonical' classes associated to each fixed point. Understanding how these classes restrict to fixed points in equivariant cohomology is equivalent to understanding how to multiply in them in this special basis, though positivity is not translatable from one setting to the other.

We give a formula for the restriction of these classes to fixed points. This was done for the flag varieties, though our formula is different even in that case. This is joint work with Susan Tolman at University of Illinois, Champaign-Urbana.