V Simpósio de Geometria



Coordenador: Prof. Llohann Dallagnol Sperança
Período: 21 e 22 de fevereiro
Local: Sala 300 - Departamento de Matemática - 3º andar do Edíficio de Ciências Exatas

 

 

Programação

 

 

Terça-feira, 21 de fevereiro

   
9h00 - 10h45 Mini-curso: G-Structures, Structure Equations, and Cartan’s Realization Problem 
Ivan Struchiner, USP
10h45 - 11h00 Coffee-break
11h00 - 11h30 A generalization of Haefliger structures
Genaro Zamudio, USP
11h30 - 12h30 Deformations of presymplectic forms
Marco Zambon, KU Leuven
12h30 - 14h00 Almoço
14h00 - 15h00 Título: Sistemas de seções transversais na mecânica celeste
Pedro Salomão, USP
15h00 - 15h30 Coffee-break
15h30 - 16h30 Título:  T-duality on nilmanifolds with applications to mirror symmetry
Lino Grama, Unicamp
16h30 - 17h30 Título: Homotopy actions and fibrations
Olivier Brahic, UFPR

 

 

Quarta-feira, 22 de fevereiro

   
9h00 - 10h45 Mini-curso: G-Structures, Structure Equations, and Cartan’s Realization Problem 
Ivan Struchiner, USP
10h45 - 11h00 Coffee-break
11h00 - 11h30 Aspectos variacionais de geodésicas homogêneas em variedades flag generalizadas
Rafaela Prado, Unicamp
11h30 - 12h30 Título: A Modern Approach to Cartan's Structure Theory for Lie Pseudogorups
Ori Yudilevich
12h30 - 14h00 Almoço
14h00 - 15h00 Título: A journey to Finsler geometry and beyond
Carlos Durán, UFPR
15h00 - 15h30 Coffee-break
15h30 - 16h30 Título: Singularities of mean curvature flows
Detang Zhou

 

 

Resumos

 

 

Mini-curso: G-Structures, Structure Equations, and Cartan’s Realization Problem

Ivan Struchiner, USP

Abstract: The theory of G-structures gives a unified framework for treating classical differential geometric structures on manifolds via reductions of the frame bundle of a manifold. In the first day of lectures, I will explain the theory of G-structures, there prolongations and structure equations. These equations play a central role in the theory of G-structures since they encode all the informations of such a structure up to local equivalence. A natural problem then arrises: given a set of structures equations, does there exist a G-structure with precisely those equations? This problem is known as Cartan’s realization problem and it will be dealt with in the second day of lectures. We will show how the Lie theory for Lie algebroids and Lie groupoids can be used to solve the problem in the case where G is compact and connected. The first part of the lecture is by now classical. I will mainly follow the approach of Singer and Sternberg in “The infinite groups of Lie and Cartan”. The second part will be based on joint work with prof Rui Loja Fernandes (University of Illinois).

 

 

A generalization of Haefliger structures

Genaro Zamudio, USP

Abstract: In 1970 André Haefliger proved a classification theorem for regular foliations on open manifolds. Roughly speaking, the idea of Haefliger was to describe foliations of codimension k as Γ-structures, then he identified these Γ-structures with a class of principal bundles with structure groupoid. Here Γ stands for a pseu-dogroup of local diffeomorphisms of R^k. Haefliger’s theorem is obtained by using the classification theorem of principal bundles with structure groupoid and a result of Gromov. The Haefliger theory of foliations is also useful to study transversal geometry, for classical geometries like Riemannian or Symplectic. In Lie groupoid theory, the concept of pseudogroup of local diffeomorphisms generalize to pseudogroup of local bisections. In this talk we will propose an generalization of Γ-structures, for Γ a pseudogroup of local bisections, which extends the description of regular foliations to include some singular foliations, and permits other transversal geometries like Generalized geometry or Courant geometry. Then we will prove a classification theorem for these structures in the spirit of Haefliger’s theorem.

 

 

Deformations of presymplectic forms

Marco Zambon, KU Leuven

Abstract: Given a real vector space, the set of skew-symmetric bilinear form of a given rank form a smooth manifold. We will exhibit local charts inspired by Dirac geometry. We will use the above linear algebra to study the deformation theory of presymplectic forms, i.e closed 2 form of constant rank on a given manifold. Notice that the kernel of such a form integrates to a (regular) foliation, and the deformations of foliations are known to be governed by a simple kind of L-infinity algebra which has only three non-trivial brackets. We will show that the deformations of presymplectic forms are also governed by such an L-infinity algebra, and related it to foliation theory.

 

Aspectos variacionais de geodésicas homogêneas em variedades flag generalizadas.

Rafaela Prado, Unicamp

Abstract: Neste trabalho, estudamos pontos conjugados ao longo de geodésicas homogêneas em variedades flag generalizadas. Este estudo é feito a partir da análise da segunda variação da energia de tais geodésicas. Também damos um exemplo de como o fluxo de Ricci pode evoluir de maneira a produzir pontos conjugados no espaço projetivo complexo CP^{2n+1} = Sp(n+1)/(U(1)xSp(n)). Este trabalho é conjunto com o Professor Lino Grama (UNICAMP).

 

 

A journey to Finsler geometry and beyond

Carlos E. Durán, UFPR

Abstract: We will describe some motivations for studying Finsler manifolds, and the projective approach to the invariants describing the associated calculus of variations. Some extensions will be presented, among them to higher order variational problems, inverse problems and generalizations of Sobolev-type inequalities.

 

 

A Modern Approach to Cartan's Structure Theory for Lie Pseudogorups

Ori Yudilevich

Abstract: In two pioneering papers dating back to 1904-05, Élie Cartan introduced a structure theory for Lie pseudogroups, generalizing Sophus Lie's structure theory for the special class of Lie pseudogroups of finite type. While the finite case has evolved into a mature and rigorous theory -- the theory of Lie groups -- Cartan's general theory has not reached that same level of maturity. In this talk, I will present a modern formulation of the theory, encompassing Cartan's three fundamental theorems for Lie pseudogroups, the notion of prolongation and a reduction algorithm by the so-called "systatic system". Two of the key ingredients in this formulation are jet spaces and Lie groupoids/algebroids. This talk is based on joint work with Marius Crainic.

 

 

T-duality on nilmanifolds with applications to mirror symmetry

Lino Grama, Unicamp

Abstract: In this talk we will discuss the construction of T-duality between nilmanifolds. Combining this construction with generalized geometry we will provide some results about mirror symmetry.

This is a work joint with Viviana del Barco and Leonardo S. Alves.

 

 

Homotopy actions and fibrations

Olivier Brahic, UFPR

Abstract: In this talk i will explain how, when stuying fibrations in various geometric contexts, one is naturally lead to the notion of homotopy action.

 

 

 

Sistemas de seções transversais na mecânica celeste

Pedro Salomão, USP

Abstract: Seções globais para fluxos tri-dimensionais permitem estudar a dinâmica através da aplicação de primeiro retorno. Quando as seções globais não existem, pode-se ainda considerar os chamados sistemas de seções transversais: uma folheação transversal ao campo de vetores no complementar de um conjunto finito de órbitas periódicas. Tais sistemas podem fornecer importante informação sobre a dinâmica. Nessa palestra falarei sobre a existência de sistemas de seções transversais para fluxos de Reeb em somas conexas do tipo RP3 # RP3. Em particular, apresentarei condições suficientes para que um conjunto formado por três órbitas periódicas admita um sistema de seções transversais no seu complementar. Este resultado é motivado pelo problema restrito dos três corpos para energias um pouco acima do primeiro valor de Lagrange e pelo problema de Euler. É um trabalho em conjunto com N. de Paulo (IME-USP) e U. Hryniewicz (UFRJ).

 

 

Singularities of mean curvature flows

Detang Zhou, UFF

Abstract: We will discuss the singularity models for mean curvature flows. In particular we will present a recent result about translating solitons. A complete connected isometrically immersed hypersurface Σ in Rn+1 is called a translating soliton if its mean curvature vector satisfies H = w, where w ∈ Rn+1 is a unitary vector and w is the orthogonal projection of w onto the normal bundle. We will discuss some examples of translating solitons of mean curvature flows and prove rigidity results for grim hyperplanes.