Michael Kapovich (UC Davis)
Discrete isometry groups of higher rank symmetric spaces
In my minicourse I will talk about geometry of nonpositively curved symmetric spaces and their discrete isometry groups, covering dynamical, topological and geometric aspects of the theory. The primary focus of my lectures will be on discrete subgroups which are not lattices. This is based on my work with Bernhard Leeb and Joan Porti.
Gil Bor (CIMAT)
Alejandro Betancourt (CIMAT)
Painlevé analysis of Ricci solitons with symmetry
The Ricci soliton equation can be reduced to a Hamiltonian system of ODE's under certain symmetry assumptions. This system can then be analyzed using techniques from dynamical systems. In particular, we show how to do a Painlevé analysis of the Ricci soliton equation and as a consequence we identify various cases where we expect to obtain explicit, closed form solutions for the metric.
Matthew Dawson (CIMAT/CONACyT)
Introduction to representation theory and harmonic analysis for direct-limit groups
Among infinite-dimensional Lie groups, those which are constructed as direct limits of finite-dimensional Lie groups are in some sense the "smallest" examples and inherit many of the properties of their finite-dimensional cousins. Nevertheless, there are also striking differences from the finite-dimensional theory: for instance, none of them have Haar measures, which makes it unclear how one should formulate a theory of harmonic analysis for these groups and their associated homogeneous spaces.
In this talk, we will discuss some interesting results from the theory of unitary representations of direct limits of semisimple Lie groups and then discuss two replacements for the missing Haar measures, which allow one to construction unitary representations using the geometry of the group and its homogeneous spaces.
Ramón Vera (IMATE, UNAM)
Poisson Structures of Near-symplectic Manifold
Abstract: In this talk we will show a connection between two singular geometric structures: near-symplectic manifolds and Poisson structures. Near-symplectic forms were originally introduced by Taubes as a way of generalizing symplectic topology in dimension 4. These singular symplectic structures are closely related to broken Lefschetz fibrations, which are known to exist on any 4-manifold and have found applications in low-dimensional topology. After introducing near-symplectic forms on any even dimensional manifold we will discuss its link to Poisson geometry and review some basic features of their Poisson cohomology. This is joint work with Panagiotis Batakidis.
Luis Alberto Wills-Toro (Univ. Nal. de Colombia, Medellín)
Mixing internal and space-time symmetries
Spinors are the most fundamental geometric objects of a Minkowski space, since all further vector and tensor representations can be composed from them. We explore further objects such that cubic compositions of them build a vector. After a brief discussion on No Go theorems, we consider graded extensions of the Poincaré algebra that include either SU(2)xU(1) or SU(3) rigid symmetries. We explore symmetries analogous to super-symmetry in that they generate space-time translations by iterations of dimensionful generators. We construct differential representations of the algebra and covariant derivatives in super-space.
Samuel Lisi (Univ. of Mississippi)
Symplectic homology for complements of smooth divisors
Symplectic Homology is a Floer homology defined for a class of non-compact symplectic manifolds including cotangent bundles and smooth affine algebraic varieties. In a sense to be made precise, this can be thought of as some kind of Morse homology of the infinite dimensional loop space. In joint work with Luis Diogo, we have developed a method for computing Symplectic Homology for the complement of a smooth divisor in a projective variety in terms of the Gromov-Witten invariants of the divisor and of the variety. I will provide some background on symplectic homology, including a discussion of some of its applications, and will then discuss some of the ingredients of the proof of our theorem.