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Mathematics Colloquium
2014 - 2015

Thursday, January 22, 2015
3:00 p.m.  //  151 Sloan
Mohammed Abouzaid (Columbia) - Family Floer cohomology and mirror symmetry

Abstract: The idea of studying Floer cohomology for families of Lagrangians was suggested by Kontsevich and Fukaya as a strategy for unifying the homological and geometric approaches to mirror symmetry.

I will explain the heuristics, the major technical problems that have prevented progress, and some recent results resolving these problems.

Finally, with generalisations to the non-commutative setting in mind, I will explain a new approach to the study of Family Floer cohomology which focuses on functions instead of points.

Tuesday, December 2, 2014

4:00 p.m.  //  151 Sloan
Richard Canary (University of Michigan) - The geometry of the Hitchin component

Abstract: If S is a closed surface, its Teichmuller space is the space of all (marked) hyperbolic structures on S.
Hitchin showed that there is a component of the space of (conjugacy classes of) representatations of the fundamental group S into PSL(n,R) which is homeomorphic to an open ball. This component contains a copy of the Teichmuller space of S which we call the Fuchsian locus.
In the first half of the talk we will introduced the Hitchin component and discuss various properties of the Hitchin component which lead one to think of it as a higher rank analogue of classical Teichmuller space. In the second half of the talk, we discuss an analytic Riemannian metric on the Hitchin component which is an analogue of the Weil-Petersson metric on Teichmuller space. In particular, it is mapping class group invariant and its restriction
to the Fuchsian locus is a constant multiple of the Weil-Petersson metric. One key tool is a metric Anosov flow associated to each Hitchin representation which is a Holder reparameterization of the geodesic flow on S whose periods encode the spectral radii of the images of the representation. (The second half of the talk describes joint work with Bridgeman, Labourie and Sambarino.)

Wednesday, November 12, 2014

4:00 p.m.  //  151 Sloan
Joseph Bernstein (University of Tel Aviv) - 
Convexity and Subconvexity bounds for automorphic periods
and representation theory

Abstract: Let Y be a compact Riemann surface with a Riemannian metric of constant curvature −1. Consider the corresponding Laplace-Beltrami operator in the space of functions on Y and fix its eigenfunction φ.  Such function is called Maass form; study of such forms plays an important role in Geometry and Number Theory.
I will introduce a general method how to bound invariants arising from Maass forms. This method is based on representation theory of the group SL(2, R).

In my talk I will discuss the following concrete problem. Fix a closed geodesic C ⊂ Y, consider the restriction f of the Maass form φ to C and decompose it into its Fourier series f = Σ an exp(2πint).  The problem is to give good bounds for Fourier coefficients an when n tends to infinity.

Time permitting; I will explain the relation of this problem with the theory of L-functions. This is an exposition of my joint works with Andre Reznikov.   



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