Seminars
Contact: If you have any suggestion or you would like to give a talk, please contact the organizers Cristián Ortiz (cortiz@ime.usp.br) and Ivan Struchiner (ivanstru@ime.usp.br)
Schedule: During the first semester of 2018 we will have a working group on “Deformation Theory of Lie groupoids”. More information soon.
Past Seminars
2017
The activities of first semester consisted on a Working Group on Simplicial Techniques in Symplectic Geometry. For more information, please visit the following website.
2016
The first semester was devoted to the study of the paper “Principal actions of stacky Lie groupoids”, which can be found here.
June 06
10:00
Title: Principal actions of stacky Lie groupoids II
Speaker: Camilo Angulo
May 30
10:00
Title: Principal actions of stacky Lie groupoids I
Speaker: Camilo Angulo
May 09
10:00
Title: Conditions for a pre-quotient to be a stack
Speaker: Camilo Angulo
May 02
10:00
Title: Pre-quotient of actions of stacky Lie groupoids
Speaker: Cristian Ortiz
April 11
10:00
Title: Stacky principal bundles
Speaker: Cristian Ortiz
April 04
10:00
Title: Actions of stacky Lie groupoids
Speaker: Camilo Angulo
March 28
10:00
Title: Stacky Lie groupoids
Speaker: Camilo Angulo
March 14
10:00
Title: Differentiable stacks II
Speaker: Camilo Angulo
March 07
10:00
Title: Differentiable stacks I
Speaker: Camilo Angulo
2015
December 10
15:00
Title: Lie algebroids, classifying stacks and Maurer-Cartan forms
Speaker: James Waldron (Newcastle University)
Abstract: A Lie algebroid is a smooth vector bundle equipped with some extra structure: a Lie bracket on its space of sections, and a map to the tangent bundle of the underlying manifold, satisfying some conditions. Basic examples include tangent bundles, Lie algebras, and integrable distributions. Other examples arise from actions of Lie algebras, Poisson structures, and principal bundles.
I will explain how one can construct a stack which classifies Lie algebroids. This means that for a manifold X, Lie algebroids over X are equivalent to maps from X to the classifying stack. This perspective gives a new way of looking at certain constructions involving Lie algebroids, and several new ideas arise naturally.
On the other hand, the classifying stack allows one to define Lie algebroids over an arbitrary stack. For example, one can understand Lie algebroids over orbifolds, quotient stacks, or monodromy stacks of foliations. A particularly interesting example is that of the classifying stack of a Lie group, where Maurer-Cartan forms play a role.
I will not assume any knowledge of Lie algebroids or stacks.
November 19
15:00
Title: Transverse measures and densities: from leaf spaces to differentiable stacks
Speaker: João Nuno Mestre (Utrecht University)
Abstract: Differentiable stacks let us treat singular spaces (such as leaf spaces of foliations, or orbit spaces of Lie group actions) by looking at non-singular ones endowed with extra-structure—Lie groupoids. We explain how an extension of Haefliger's approach to transverse integration for foliations allows us to define and study measures and geometric measures (densities) on differentiable stacks. The abstract theory works for any differentiable stack, but it becomes very
concrete for proper stacks (orbifolds, orbit spaces of compact Lie group actions, ...)—in this case we obtain Weyl-type integration formulas.
The first hour will be aimed at a general audience: I will explain how to set the framework of transverse measures, through examples of leaf spaces and orbit spaces. In the second hour I will discuss applications and properties of transverse measures. The talk is based on joint work with
Marius Crainic."
November 05
15:00
Title: Principal connections and Chern-Weil homomorphism - Part IV
Speaker: Genaro Zamudio (IME-USP)
Abstract: In this talk we will introduce principal connections in principal group-bundles. Principal connections permit us to relate vector fields and differential forms on the total space to the respective ones on the base space of the bundle. We will also introduce the Chern–Weil homomorphism which associates to every principal bundle with a principal connection some closed differential forms on the base space and as we will prove, the corresponding classes in de Rham cohomology do not depend on the connection but only on the isomorphism class of the principal bundle.
October 29 (Double session)
14:00
Title: An equivalence of categories in Poisson Geometry (and a funny cobordism)
Speaker: Pedro Frejlich (PUC-RJ)
Abstract: We revisit Lerman's symplectic cut construction in the Poisson realm and reinterpret it as an equivalence between the category of Poisson cylinders and that of Poisson Doppelg\"{a}ngers, which both conceptualizes and generalizes cut-and-paste operations. An interesting byproduct of our approach, which relies heavily on new normal form results, is that we are able to realize cut-and-paste as a cobordism within the Poisson category.
15:00
Title: Integração de VB-algebroides e representações 2-homotópicas
Speaker: Olivier Brahic (UFPR)
Abstract: VB-algebroides são estruturas que combinam algebroides de Lie e fibrados vetoriais de maneira compatível. O objetivo dessa palestra é o de tratar o problema de integração de tais estruturas, imitando a construção do grupoide de Weinstein na categoria de fibrados vetoriais.
October 15
15:00
Title: Principal connections and Chern-Weil homomorphism - Part III
Speaker: Genaro Zamudio (IME-USP)
Abstract: In this talk we will introduce principal connections in principal group-bundles. Principal connections permit us to relate vector fields and differential forms on the total space to the respective ones on the base space of the bundle. We will also introduce the Chern–Weil homomorphism which associates to every principal bundle with a principal connection some closed differential forms on the base space and as we will prove, the corresponding classes in de Rham cohomology do not depend on the connection but only on the isomorphism class of the principal bundle.
October 08
15:00
Title: Principal connections and Chern-Weil homomorphism - Part II
Speaker: Genaro Zamudio (IME-USP)
Abstract: In this talk we will introduce principal connections in principal group-bundles. Principal connections permit us to relate vector fields and differential forms on the total space to the respective ones on the base space of the bundle. We will also introduce the Chern–Weil homomorphism which associates to every principal bundle with a principal connection some closed differential forms on the base space and as we will prove, the corresponding classes in de Rham cohomology do not depend on the connection but only on the isomorphism class of the principal bundle.
October 01
15:00
Title: Principal connections and Chern-Weil homomorphism - Part I
Speaker: Genaro Zamudio (IME-USP)
Abstract: In this talk we will introduce principal connections in principal group-bundles. Principal connections permit us to relate vector fields and differential forms on the total space to the respective ones on the base space of the bundle. We will also introduce the Chern–Weil homomorphism which associates to every principal bundle with a principal connection some closed differential forms on the base space and as we will prove, the corresponding classes in de Rham cohomology do not depend on the connection but only on the isomorphism class of the principal bundle.
September 17
15:00
Title: Cohomology of compact Lie groups
Speaker: Cristian Cárdenas (IME-USP)
Abstract: In this talk, I will discuss the de Rham cohomology ring, H*(G), of compact Lie groups. This will be made in two ways, by using the idea of H-spaces and Hopf algebras which gives us the algebraic structure of H*(G). And then by working directly with the differential forms on G and its multiplicative structure. This will give us a subcomplex of the de Rham complex which, we will see, induces the same cohomology.
September 03
15:00
Title: Basic properties of equivariant cohomology
Speaker: Jeffrey Carlson (IME-USP)
Abstract: Having justified and proven the existence of equivariant cohomology, we proceed to develop some of its fundamental properties, including: its values on individual orbits, the axioms for an equivariant cohomology theory, the equivariant Mayer–Vietoris sequence, the equivariant Künneth theorem, and the relation between $G$-equivariant cohomology for $G$ a compact Lie group and T-equivariant cohomology for T a maximal torus in G,
with the proviso that some of what we do will be dependent on application of the Serre spectral sequence and classical facts about the cohomology of flag varieties to be established later in the seminar.
Depending on time constraints, and assuming knowledge of H^*(BG)---to be developed in the upcoming weeks---we may also establish the Borel localization theorem (probably due in the general case to Atiyah) and the K-equivariant cohomology of a homogeneous space.
August 27
15:00
Title: Actions of Lie groups, principal bundles, and the universal bundle
Speaker: Jeffrey Carlson (IME-USP)
Abstract: In the previous lecture, we alluded to the existence of a fiber bundle called $EG \to BG$, the ``universal principal $G$-bundle.'' This object having been produced, we can begin to discuss equivariant cohomology, a contravariant functor from the category of $G$-spaces and $G$-maps to the category of commutative graded $H^*(BG)$-algebras.
To understand what this object tells us and how it can be applied, it will be necessary first to assemble some generalities about the former category, including orbit structure, equivariant tubular neighborhoods, and $G$-CW structures. The lecture will be pitched to a hypothetical audience member who has seen algebraic and differential topology, but not continuous group actions.
August 20
15:00
Title: Equivariant cohomology: motivation and overview.
Speaker: Jeffrey Carlson (IME-USP)
Abstract: (Borel) equivariant cohomology is a ring-valued topological invariant originally introduced in 1950 and deployed starting in the 50s as a way to understand smooth actions of compact Lie groups. While it has had much success in this regard, it was later noticed in the 1980s that from it one can recover integration formulas admits of many applications in symplectic and algebraic geometry, theoretical physics and even combinatorics. In this introductory talk, necessarily without going into overwhelming technical detail, I will discuss the kinds of questions equivariant cohomology was designed to deal with and some of the topological and geometric uses it has
been put to since.
June 25
15:00
Title: Bisubmersions and atlases for singular subalgebroids.
Speaker: Marco Zambon (UK Leuven, Belgium)
Abstract: This talk will discuss in some detail the building blocks of the holonomy groupoids of singular subalgebroids, namely bisubmersions. Unlike the spaces of Lie algebroid paths used in the construction of the Weinstein groupoid of a Lie algebroid, bisubmersions are finite dimensional objects. We will also discuss atlases of bisubmersions, and if time permits show how using them one can obtain explicit descriptions of certain holonomy groupoids.
June 11
15:00
Title: Codimension 1 analytic foliations, Part II
Speaker: Camilo Angulo (IME-USP)
Abstract:
May 28
15:00
Title: Codimension 1 analytic foliations, Part I
Speaker: Camilo Angulo (IME-USP)
Abstract:
16:15
Title: TBA
Speaker: Bruno Suzuki (IMECC-UNICAMP)
Abstract: A ideia deste seminário é apresentar a cohomologia orbifold, como definida por Chen/Ruan e calculá-la para os espaços projetivos com pesos. Para isso começamos com uma introdução à teoria de orbifolds, definimos a cohomologia e em seguida apresentamos um resultado que caracteriza a cohomologia destes espaços, fazendo o cálculo para alguns casos específicos.
May 14
15:00
Title: Teorema de classificação de \Gamma-folheações - Parte 5
Speaker: Genaro Zamudio (IME-USP)
Abstract:
May 07
15:00
Title: Teorema de classificação de \Gamma-folheações - Parte 4
Speaker: Genaro Zamudio (IME-USP)
Abstract:
April 30
15:00
Title: Teorema de classificação de \Gamma-folheações - Parte 3
Speaker: Genaro Zamudio (IME-USP)
Abstract:
April 23
15:00
Title: Teorema de classificação de \Gamma-folheações - Parte 2
Speaker: Genaro Zamudio (IME-USP)
Abstract:
16:15
Title: Ações homotópicas de espaços $A_{infty}$
Speaker: Eduardo Hoefel (UFPR)
Abstract: Falaremos inicialmente sobre espaços A_\infty e seus espaços classificantes. Em seguida mostraremos como os espaços classificantes também podem ser construídos para ações homotópicas de espaços A_\infty. Tal construção nos permite obter resultados interessantes para tais ações em termos de espaços de laços relativos.
April 09
15:00
Title: Teorema de classificação de \Gamma-folheações
Speaker: Genaro Zamudio (IME-USP)
Abstract:
March 26
15:00
Title: O espaço classificante de um grupo(ide) topológico - Parte II
Speaker: Fernando Studzinski (IME-USP)
Abstract: Nesta segunda parte do seminário, terminamos a demonstração de que o fibrado da construção de Milnor é de fato um fibrado universal. Além disso, veremos um exemplo de um grupoide de Lie G e dois G-fibrados principais, tais que esses fibrados são homotópicos mas não isomorfos.
March 19
15:00
Title: O espaço classificante de um grupo(ide) topológico - Parte I
Speaker: Fernando Studzinski (IME-USP)
Abstract: Neste seminário veremos a construção devida a J. Milnor para o espaço classificante de um grupo topológico e veremos como esta construção pode ser estendida para o caso de grupoides topológicos.
March 12
15:00
Title: Homotopia e Integrabilidade - Introdução
Speaker: Genaro Zamudio (IME-USP)
Abstract: Neste seminario apresentaremos aos pseudogrupos $\Gamma$ e as $\Gamma$-estruturas. Exemplos de $\Gamma$-estruturas são folheações e campos tensoriais com modelo local, entre outras estruturas geométricas; assim podemos usar $\Gamma$-estruturas para estudar estruturas geométricas, em particular deformações delas. Faremos uma revisão dos conceptos de grupoide de Lie e fibrados principais com grupoide de estructura. Mostraremos que $\Gamma$-estruturas podem ser pensadas como fibrados principais deste tipo, com a escolha certa do grupoide de estrutura; e portanto podemos usar a teoria de homotopia de fibrados principais para estudar deformações de estruturas geométricas.
2014
October 27
16:00
Title: Linearization of Riemannian groupoids
Speaker: Camilo Angulo (IME-USP)
Abstract: Along our series of seminars, we have studied Riemannian groupoids, i.e. Lie groupoids that admit a 2-metric. After having proved that every proper Lie groupoids admits such a Riemannian structure, we proceed to apply this theory to the Linearization problem.
In this talk, we are going to recall what the linearization problem is about, along with examples of special cases where there are theorems giving positive answers. Then, we are going to define different variations of being linearizable, and finally we are going to work out the proof of how the metric helps us in constructing one type of linearization.
October 20
16:00
Title: Grupo fundamental de orbifolds via grupoides de Lie
Speaker: Fernando Studzinski (UFPR)
Abstract: Neste seminário apresentaremos o conceito de grupoide fundamental de um grupoide de Lie G. E definiremos o grupo fundamental do grupoide G, via isotropias do grupoide fundamental. Veremos que essa noção de grupo fundamental é um invariante de Morita, e em particular, se G é próprio e étale, teremos que o grupo fundamental de G é um invariante da classe de isomorfismo do orbifold que ele representa.
Reading seminar on Riemannian groupoids. We will follow the paper “Riemannian metrics on Lie groupoids” by M. del Hoyo and R. Fernandes. The paper is available here.
October 13
16:00
Title: Exemplos de grupoides Riemannianos - Parte II
Speaker: Cristian Cardenas (IME-USP)
Abstract: O objetivo é ver alguns exemplos de grupoides Riemannianos e tratar o problema da extensao de métricas Riemannianas em grupoides.
October 06
16:00
Title: Exemplos de grupoides Riemannianos
Speaker: Cristian Cardenas (IME-USP)
Abstract: O objetivo é ver alguns exemplos de grupoides Riemannianos e tratar o problema da extensao de métricas Riemannianas em grupoides.
October 01
17:00
Title: Grupoides Riemannianos - Part II
Speaker: Genaro Zamudio (IME-USP)
Abstract: Neste seminario revisaremos as definições das 0, 1 e 2-métricas sobre um grupoide de Lie e apresentaremos exemplos destas métricas. Mostraremos que uma 0-métrica sobre um grupoide étale é equivalente a uma métrica na qual as biseções agem por isometrias. Também mostraremos que para uma classe de grupoides, chamados de "foliation groupoid", toda 0-métrica pode ser extendida para uma 1-métrica.
September 22, 16:00
Title: Grupoides Riemannianos
Speaker: Genaro Zamudio (IME-USP)
Abstract: Neste seminario revisaremos as definições das 0, 1 e 2-métricas sobre um grupoide de Lie e apresentaremos exemplos destas métricas. Mostraremos que uma 0-métrica sobre um grupoide étale é equivalente a uma métrica na qual as biseções agem por isometrias. Também mostraremos que para uma classe de grupoides, chamados de "foliation groupoid", toda 0-métrica pode ser extendida para uma 1-métrica.
September 15, 16:00
Title: Métricas invariantes - Parte 2
Speaker: Jackeline Conrado (IME-USP)
Abstract: Neste seminário revisaremos a noção de submersão Riemanniana e definiremos o conceito de métrica (transversalmente) invariante pela ação de um grupoide de Lie. A partir desse conceito é possível colocar uma métrica no quociente de uma ação própria e livre que torna a projeção canônica uma submersão Riemanniana.
September 08, 16:00
Title: Métricas invariantes
Speaker: Jackeline Conrado (IME-USP)
Abstract: Neste seminário revisaremos a noção de submersão Riemanniana e definiremos o conceito de métrica (transversalmente) invariante pela ação de um grupoide de Lie. A partir desse conceito é possível colocar uma métrica no quociente de uma ação própria e livre que torna a projeção canônica uma submersão Riemanniana.
September 01, 16:00
Title: Sobre cohomologia de grupos e resoluções
Speaker: Cristian Cárdenas (IME-USP)
Abstract: Definiremos a cohomologia de grupos e o conceito de resolução (livre) de módulos.
Veremos a relação que existe entre a cohomologia de grupos e a cohomologia vinda da chamada resolução Bar
August 25, 16:00
Title: What is a differentiable stack?
Speaker: Camilo Angulo (IME-USP)
Abstract: We start by defining properties of morphisms inside the category of stacks and differentiable stacks. Then we are going to show some examples. Finally, we are going to outline the proof of the equivalence of categories between differentiable stacks and Lie groupoids with the so-called generalized morphisms. If time permits, we are going to present a particular appearance of differentiable stacks in the problem of integration of Lie groupoids and Poisson manifolds.
August 18, 16:00
Title: What is a stack?
Speaker: Camilo Angulo (IME-USP)
Abstract: During the first lecture we are going to define what a general stack is. A general stack is, roughly speaking, a generalization of a sheaf that takes values in categories rather than sets. In so, we ease the conditions of a sheaf by allowing isomorphisms instead of equalities. We are going to define the necessary ingredients to define a stack; namely, fibered categories, Grothendieck topologies and Descent categories. Thereafter, we are going to consider examples that show in what sense stacks are generalized spaces, and point out that certain stacks seem to have additional structure.