Poisson geometry, moduli spaces, and applications


Titles and abstracts for Course 2


Prof. Yuji Hirota (Azabu University)

Title: Differential operators in Poisson geometry


We shall give a talk on differential operators appearing in Poisson geometry --- specifically, the curl operators and Poisson coboundary operators. We describe explicitly those two operators on an oriented Poisson manifold in terms of a Poisson connection. Moreover, using them, we introduce a new operator for an oriented Poisson manifold, and discuss the geometric meaning of it.

Prof. Noriaki Ikeda (Ritsumeikan University)

Title: Super Poisson geometry and T-duality in string theory


Construction of a Poisson structure in terms of a Schouten-Nijenhuis bracket and a Poisson bivector field naturally leads to the concept of super Poisson geometry. I review that many geometric structures are unified as super Poisson geometries. In the second part of my talk, I discuss a recent application to T-duality in string theory. We apply super Poisson geometry to analyze the geometry of T-duality and 'double field theory' proposed by physicists. Rather curious formulations of double field theory are beautifully reformulated and generalized.

Prof. Yoshiaki Maeda (Tohoku University)

Title: Diffeomorphism groups of circle bundles over integral symplectic manifolds


Dr. Tomoya Nakamura (Waseda University)

Title: Lie algebroids and Poisson structures

Abstract: A Lie algebroid is a generalization of both a Lie algebra and the tangent bundle over a manifold. In this talk, I introduce the relation between Lie algebroids and Poisson structures.

Prof. Hiroaki Yoshimura (Waseda University)

Title: A variational formulation and Dirac structures of nonequilibrium thermodynamics


In this talk, we study a variational formulation and geometric structures of nonequilibrium thermodynamics. We first show a generalized Lagrange-d’Alembert principle for a class of nonlinear nonholonomic systems, which is a natural extension of Hamilton’s principle in mechanics. Second, we construct various induced Dirac structure over the thermodynamic phase space and then show how the associated Dirac thermodynamic systems with the energy, Lagrangian, and Hamiltonian can be developed in the context of the Dirac structures. We finally clarify the bundle picture for nonequilibrium thermodynamics. This is a joint work with Francois Gay-Balmaz of Ecole Normale Superieure in Paris.