Mon, 20 May 2013

12:00 - 13:00
L3

The Riemann Zeta Function and the Berry-Keating Hamiltonian

Philip Candelas
(Oxford)
Abstract
It is an old idea that the imaginary part of the nontrivial Riemann zeros s =-1/2 + iE might be related to the eigenvalues of a hermitean operator H, and so to a quantum mechanical system. Such a system has been proposed by Berry and Keating; it is a harmonic oscillator with the "wrong" signatureH=1/2(xp + px). The difficulty and interest in implementing this proposal is the need to find suitable boundary conditions, or a self adjoint extension for H, since the classical phase space orbits are hyperbolae rather than circles. I will review interesting observations of Mark Srednicki relating the ground state wave functions of the Berry Keating hamiltonian and the conventional harmonic oscillator hamiltonian to the zeta function.
Tue, 28 May 2013

15:45 - 16:45
L3

Hamiltonian reduction and t-structures in (quantum) symplectic geometry

Tom Nevins
(Illinois)
Abstract

Many interesting examples of singular symplectic algebraic varieties and their symplectic resolutions are built by Hamiltonian reduction. There is a corresponding construction of "quantum Hamiltonian reduction" which is of substantial interest to representation theorists. It starts from a twisted-equivariant D-module, an analogue of an algebraic vector bundle (or coherent sheaf) on a moment map fiber, and produces an object on the quantum analogue of the symplectic resolution. In order to understand how far apart the quantisation of the singular symplectic variety and its symplectic resolution can be, one wants to know "what gets killed by quantum Hamiltonian reduction?" I will give a precise answer to this question in terms of effective combinatorics. The answer has consequences for exactness of direct images, and thus for t-structures, which I will also explain. The beautiful geometry behind the combinatorics is that of a stratification of a GIT-unstable locus called the "Kirwan-Ness stratification." The lecture will not assume familiarity with D-modules, nor with any previous talks by the speaker or McGerty in this series. The new results are joint work with McGerty.

Mon, 27 May 2013
14:15
L3

The Pressure metric for convex Anosov representations

Martin Bridgeman
(Boston College)
Abstract

 Using thermodynamic formalism we introduce a notion of intersection for convex Anosov representations. We produce an Out-invariant Riemannian metric on the smooth points of the deformation  space of convex, irreducible representations of a word hyperbolic group G into SL(m,R) whose Zariski closure contains a generic element. In particular, we produce a mapping class group invariant Riemannian metric on Hitchin components which restricts to the Weil-Petersson metric on the Fuchsian locus. 
This is joint work with R. Canary, F. Labourie and A. Sambarino.
Thu, 09 May 2013

14:00 - 15:00
L3

Modules over Algebraic Quantizations and representation theory

Christopher Dodd
Abstract

Recently, there has been a great deal of interest in the theory of modules over algebraic quantizations of so-called symplectic
resolutions. In this talk I'll discuss some new work -joint, and very much in progress- that open the door to giving a geometric description to certain categories of such modules; generalizing classical theorems of Kashiwara and Bernstein in the case of D-modules on an algebraic variety.

Tue, 07 May 2013

15:45 - 16:45
L3

Descent for n-Bundles

Jesse Wolfson
(Northwestern)
Abstract

Given a Lie group $G$, one can construct a principal $G$-bundle on a manifold $M$ by taking a cover $U\to M$, specifying a transition cocycle on the cover, and then descending the trivialized bundle $U \times G$ along the cocycle. We demonstrate the existence of an analogous construction for local $n$-bundles for general $n$. We establish analogues for simplicial Lie groupoids of Moore's results on simplicial groups; these imply that bundles for strict Lie $n$-groupoids arise from local $n$-bundles. We conclude by constructing a simple finite dimensional model of the Lie 2-group String($n$) using cohomological data.

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