The talks will take place at Auditório Antonio Gilioli, Bloco A (Thursday) and Auditório CCSL, Bloco C (Friday). Please, find below all the titles and abstracts of the talks.

Isometry flows on orbit spaces and applications to the theory of foliations.

Speaker: Marcos Alexandrino, IME-USP

Abstract: We prove here that given a proper isometric action $K\timesM \to M$ on a complete Riemannian manifold $M$ then every continuous isometric flow on the orbit space $M/K$ is smooth,  i.e., it is the projection of an K-equivariant smooth flow on the manifold M. As a direct corollary we infer the smoothness of isometric actions on orbit spaces. Another relevant application of our result concerns Molino's conjecture, which states that the partition of a Riemannian manifold into the closures of the leaves of a singular Riemannian foliation is still a singular Riemannian foliation. We prove Molino's conjecture for the main class of foliations considered in his book, namely orbit-like foliations. This talk is based on a joint work with Dr. Marco Radeschi (wwu- Munster) and is aimed at a broad audience of students, faculties and researchers in Geometry.


Lie theory for vector bundles

Speaker: Henrique Bursztyn, IMPA

Abstract: I will discuss vector bundles in the world of  Lie algebroids and groupoids, with focus on Lie theory in this context. This is based on joint work with A. Cabrera and M. del Hoyo.

Riemannian metrics on Lie groupoids

Speaker: Rui Fernandes, University of Illinois at Urbana-Champaign

Abstract: We introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The exponential map of these metrics allow us to establish a Linearization Theorem for Riemannian groupoids, obtaining both a simpler proof and a stronger version of the Weinstein-Zung Linearization Theorem for proper Lie groupoids. This new notion of metric has a simplicial nature and should play an important role in understanding the classifying space of the Lie groupoid. Joint work with Matias del Hoyo (IMPA).

Homotopy representations and open-closed structures

Speaker: Eduardo Hoefel, Universidade Federal do Paraná

Abstract: In this talk we study the homotopy representations of associative and Lie algebras in connection with the several algebraic structures that appear in the study of the Swiss-cheese operad. We refer to this class of algebraic structures as "Open-Closed Structures", which naturally include the OCHAs discovered by Kajiura and Stasheff. This talk is based on joint work with Muriel Livernet.

Instanton flags

Speaker: Marcos Jardim, UNICAMP

Abstract: I will provide an ADHM description of the moduli space of pairs (E,F) of framed torsion free sheaves on P^2, such that E is a subsheaf of F whose quotient is a torsion sheaf of fixed length l. In the case where l=1, I show that this moduli space is a non-singular quasi-projective variety of dimension 2cr-r+1, where r and c are the rank and second Chern class of F, respectively. Finally, I will also discuss how this generalizes the classical results on nested Hilbert schemes and the problem of describing geometric structures on these varieties.

A characterization of diagonal Poisson structures

Speaker: Renan Lima, ITA

Abstract: Let $X=\lm{CP}^{2n}$ be the complex projective space of dimension $2n$ and let $\Pi\in H^0(X,\bigwedge^2T X)$ be a holomorphic log-Symplectic Structure. It is well known that the degeneracy locus of $\Pi$ is a singular hypersurface of degree $2n+1$. We prove that if the degeneracy locus is composed by smooth irreducible hypersurfaces in smooth normal crossing position, then $\Pi$ is the diagonal Poisson structure and the degeneracy locus consists of $2n+1$ hyperplanes in general position.

K3 surfaces and Poisson manifolds of strong compact type

Speaker: David Martinez Torres, PUC-RJ

Abstract: In this talk I will present a construction of a Poisson manifold whose Weinstein groupoid is compact. The construction relies on properties of the moduli of K3 surfaces. This is joint work with R.L. Fernandes and M. Crainic.

A generalization of Dirac structures

Speaker: Nicolás Martínez, IMPA

Abstract: Dirac structures unify both Poisson and symplectic structures on the context of mechanical systems. In this talk I will define a higher order Poisson structure inspired on the multisymplectic and polysymplectic geometries which have been developed in the context of Field Theories and motivate the definition of Higher Dirac structures.

O complexo BRST e variedades simpléticas de grau 1

Speaker: Ricardo Paleari, IMPA

Abstract: Nesta palestra revisitaremos a definição do complexo BRST para um espaço G-hamiltoniano, lembrando como este complexo fornece um modelo homológico para o espaço reduzido da ação. Em seguida generalizaremos esta discussão no contexto de variedades simpléticas de grau 1 com uma ação hamiltoniana de uma álgebra de Lie graduada e veremos que o novo complexo ainda recupera o espaço de "funções" do quociente simplético.

DFR-Bundle for Poisson vector bundles

Speaker: Daniel Vasques, IME-USP

Abstract: Introduced in 95 by Doplicher, Fredenhagen and Roberts, the DFR model works as a "covariant quantazation" (in the C* algebra sense) of Minkowski spacetime". In this talk we expose an abstraction of their construction, which can be applied for any Poisson vector bundle (i.e. a vector bundle with a fixed bivector field). In order to accomplish that, we first revisit Rieffel's strict deformation for Poisson vector spaces, mainly rephrasing it in the geometric language needed for our purposes, and the using this reformulation we show how to "assemble" the C* algebra, obtained by strict deformation of the fibers of our Poisson vector bundle, into a C* bundle, which we propose to call the DFR bundle. We close this talk with two examples, the case of homogeneous bundles, in which one recovers the original DFR model, and what we propose to call the covariant quantum spacetime.