UIUC-WUSTL   Joint Seminar in Symplectic and Poisson Geometry

Title: Differential geometry over C∞-rings.

Abstract: In a recent book Joyce detailed foundations for a theory of algebraic geometry over C∞-rings.  But what about differential geometry, i.e., things like vector fields, flows, differential forms and Cartan calculus?  I will describe an attempt to fill the gap in literature, including my work with Yael Karshon on flows.

Title: Duistermaat-Heckman measures for Hamiltonian groupoid actions

Abstract: Hamiltonian actions of symplectic groupoids generalize the classical notion of Hamiltonian group actions as well as several related ones. We will go over the basics and discuss the generalization of a classical Duistermaat-Heckman result on polynomial measures to this setting, using the theory of Poisson manifolds of compact type.

Title: Modular classes in Jacobi geometry

Abstract: There has recently been a variety of development and applications of modular classes of Poisson manifolds and Lie algebroids. The goal of this talk is to introduce modular classes in the more general setting of Jacobi geometry. We will first give a brief review of modular classes in Poisson geometry, then we will discuss Jacobi manifolds, Jacobi algebroids, and Gerstenhaber Jacobi algebras. Finally, we will introduce modular classes of Jacobi algebroids. This is an ongoing joint work with M.-L.  Diallo.

Title: On the Poisson cohomology of broken Lefschetz fibrations.

Abstract: Broken Lefschetz fibrations (BLfs) were introduced by Auroux, Donaldson and Katzarkov in the study of 4-manifolds. A BLf is a smooth map f between an oriented closed 4-manifold M and 2-manifold S, respectively,  which has two types of singularites: Lefschetz singularities and circles of indefinite fold singularities.  

Garcia-Naranjo, Suarez-Serrato & Vera defined a Poisson structure associated to a BLf by comparing the pullback of the area form on S under f to the volume form on M. In this talk I will report on work in progress with L. Toussaint in which we attempt to compute the Poisson cohomology associated to such Poisson structures.

Title: Homotopical Foundations of Quantum Spin Systems

Abstract: In the talk, an algebraic topological framework for studying 

state spaces of  quantum lattice spin systems is presented, using the

framework of algebraic quantum mechanics. We first provide some old and

new results about the state space of the quasi-local algebra of a

quantum lattice spin system when endowed with either the natural metric

topology or the weak* topology. Switching to the algebraic topological

side we then determine the homotopy groups of the unitary group of a

UHF algebra and then show that the pure state space of any UHF algebra

is simply connected. We finally indicate how these and related results

may lead to a framework for constructing Kitaev's loop-spectrum of

bosonic invertible gapped phases of matter. The talk is on joint work

with A. Beaudry, M. Hermele, J. Moreno, M. Qi and D. Spiegel.