Title: Gray stability for contact groupoids
Abstract: A Jacobi structure is a Lie bracket on the sections of a line bundle. These brackets encode time-dependent mechanics in the same way Poisson brackets encode mechanics. Contact groupoids are finite-dimensional models for the “integrations” of these infinite-dimensional Lie algebras. In this talk, we explain how, under a certain compactness hypothesis, one can adapt the argument of Gray-Moser to these multiplicative contact structures and point out some applications. This is joint work with María Amelia Salazar and Daniele Sepe.
Title: SYMMETRIC KILLING TENSORS ON RIEMANNIAN 2-STEP NILPOTENT LIE GROUPS
Abstract: A symmetric Killing tensor on a Riemannian manifold is a symmetric tensor field such that the symmetrized part of its covariant derivative with respect to the Levi-Civita connection vanishes. This concept generalizes that of Killing vector fields on Riemannian manifolds. These tensors define first integrals of the equation of motion and, as such, they define constant functions along the geodesics. Parallel symmetric tensors, symmetric products of Killing vector fields and linear combinations of those give examples of symmetric Killing tensors on the manifold. These tensors are called ’decomposable’ and any symmetric tensor which is not of this form is called ’indecomposable’. In general, it is not easy to determine whether a Riemannian manifold admits indecomposable symmetric Killing tensors.
In this talk, I will discuss recent results on the existence of indecomposable left-invariant symmetric Killing 2-tensors on 2-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. After a basic introduction of the geometrical features of such Riemannian manifolds, we will introduce the tools that allowed us to show that for 2-step nilpotent Lie groups of dimension less than or equal to 7, every symmetric Killing tensor is decomposable. Also, it will be shown that for every dimension greater than 7, there are Lie groups admitting indecomposable symmetric Killing 2-tensors. The present talk is based on joint work with Andrei Moroianu.
Title: Toroidal integer homology spheres have irreducible SU(2)-representations.
Abstract: The fundamental group is one of the most powerful invariants to distinguish closed three-manifolds. One measure of the non-triviality of a three-manifold is the existence of non-trivial SU(2)-representations. In this talk I will show that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)-representations. This is joint work with Tye Lidman and Raphael Zentner.
Title: On extensions of Lie algebras and Lie groups
Abstract: In this talk, I will deal with the theory of Lie group and Lie algebra extensions. I will discuss the construction of semi-direct products, its relation with cohomology and non-abelian cohomology, and the behavior under differentiation and integration. I will also describe Lie group extensions from the viewpoint of fibered categories, mention some generalizations to groupoids and algebroids, and propose related problems.
Title: Nonperiodic leaves of codimension one foliations
Abstract: In this talk, we shall review some well known facts about the end topology of leaves of codimension one foliations and its interconnection with one dimensional dynamics (Denjoy, Kopell and Duminy's theorems). Under some natural assumptions (proper) leaves of codimension one foliations lie on one of the following two classes: leaves with periodic ends or leaves with infinitely many ends. This is well known to be the case for codimension one foliations of class 2. It will be shown the construction of an example of a codimension one foliation of class 1 that admits a proper and nonperiodic leaf with just two ends. Our method of construction allows to realize nonperiodic ends that are almost periodic leaving as an open question the non almost periodic case.
Title: Infinite staircases in symplectic embeddings
Abstract: A classic result due to McDuff and Schlenk asserts that the function that encodes when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball has a remarkable structure: the function has infinitely many corners, determined by the odd-index Fibonacci numbers, that fit together to form an infinite staircase.
I will discuss a general framework for analyzing the question of when the ellipsoid embedding function for other symplectic 4-manifolds is partly described by an infinite staircase, in particular by giving an obstruction to the existence of an infinite staircase that experimentally seems strong. We will then look at the special case of rational closed symplectic toric manifolds, where the targets with infinite staircases seem to be exactly those whose moment polygon is reflexive. Finally I will mention some results about the non-rational case, where there are a whole lot more infinite staircases to be found.
This talk is based on joint work with Dan Cristofaro-Gardiner, Tara Holm and Alessia Mandini, and also on joint work with Maria Bertozzi, Tara Holm, Emily Maw, Dusa McDuff, Grace Mwakyoma, and Morgan Weiler.
Título do minicurso: Geometria associada às representações de grupos de Lie compactos.
Resumo do minicurso: O intuito deste minicurso é descrever algumas conexões entre geometria e representações de grupos de Lie compactos (conexos). O objetivo principal é tentar convencer de que representações irredutíveis de tais grupos estão em correspondência com certas órbitas do dual de suas álgebras de Lie. Ao fazer isso, veremos como a geometria simplética e complexa (Kahler) desempenham um papel central. Devemos também apontar para uma conexão clássica da teoria da representação com a análise harmônica.