XVIIIth Oporto Meeting on Geometry, Topology and Physics

Courses | Introductory courses | Invited talks | talks | 1 | 2 | 3 |

Main Speakers and Mini courses

The main speakers will each give a minicourse of three one-hour lectures.

Paul Biran (Tel-Aviv University, Israel) "Recent developments in Lagrangian topology"

Abstract: In this series of talks we shall survey recent developments in symplectic topology of Lagrangian submanifolds, in particular in Lagrangian Floer theory. The focus will be on the homological theory coming from the so called "pearl complex", that was recently developed by Biran and Cornea. This theory combines classical Morse theory with the theory of holomorphic disks. It produces an intermediate object which links in a very explicit way Floer homology and classical Morse homology. We shall explain how this theory is constructed and how it leads to a variety of applications in the following directions:
1) Study of the topology of Lagrangian submanifolds.
2) Questions from symplectic topology such as Lagrangian intersections, spectral numbers and symplectic packing.
3) Questions about counting of holomorphic curves with boundaries on Lagrangian submanifolds.
The talks will be essentially self contained and no serious knowledge of symplectic geometry will be assumed.

Henrique Bursztyn (IMPA, Brasil) "Geometry around Dirac Structures"

Abstract: Dirac structures were introduced by Courant and Weinstein around 20 years ago, motivated by the study of mechanical systems with constraints. Examples of Dirac structures naturally associated with symplectic geometry include closed 2-forms (e.g. the restriction of a symplectic form to a constraint submanifold) and Poisson manifolds (e.g. the quotient of a symplectic manifold by a Lie group acting freely and properly via symplectomorphisms). A key ingredient in the theory of Dirac structures is the so-called Courant bracket, which gives a unified way to view many known integrability conditions in geometry. The first of these lectures will provide an introduction to Dirac structures, with basic definitions, properties and examples. The remaining lectures will discuss recent developments and applications of the theory.
Topics should include, if time permits, Lie theoretic aspects of Dirac geometry (e.g. integration of Dirac structures and equivariant cohomology), connections with generalized momentum map theories (e.g. quasi-hamiltonian and quasi-Poisson spaces), as well as generalized complex structures and supergeometry.

Yael Karshon (University of Toronto) "Hamiltonian Group Actions"

Abstract: These talks are about symplectic manifolds with compact group actions that are generated by moment(um) maps. Such group actions model symmetries in classical mechanics and often arise in purely mathematical contexts. The moment map encodes manifold information into polytopes and graphs. The purpose of these talks is to illustrate how to use a moment map to read information about the underlying symplectic manifold. The talks will contain a sample of tools, examples, and results, old and new.

Alan Weinstein (U.C. Berkeley) "Poisson brackets, Grupoids and General Relativity"

Abstract: The solutions of Dirac's constraint equations in the 3+1 formulation of Einstein's equations in general relativity form a coisotropic subvariety in the cotangent bundle of a space of metrics on a 3-dimensional manifold. This situation resembles that for the zero set of a momentum map for a hamiltonian action, but the formalism does not work when one tries to use the group Diff(M) of diffeomorphisms of a space-time M as the symmetry group. What seems to be more relevant for this problem is the groupoid DH(M) of diffeomorphisms between all pairs of hypersurfaces in M. Christian Blohmann (Regensburg), Marco Cezar Fernandes (Brasilia), and I have found several groupoids and Lie algebroids related to DH(M) which reproduce the Poisson brackets between Dirac's constraint functions. In these lectures, I will give introductions to the variational and hamiltonian formulations of the Einstein equations and to the theory of Lie algebroids and Lie groupoids, after which I will describe the use of symmetry groupoids and their Lie algebroids in relativity. (you can see a provisional version here).