Abstracts

Speakers

• Luca Accornero (USP, São Paulo) - 05/11 ,  3:00 PM.
  Principal groupoid bundles, Cartan connections, and the Chern-Weil construction
  I am going to present some constructions of cohomological invariants for principal groupoid bundles that generalize Chern-Weil theory. The talk is based on joint works with Marius Crainic and with Mateus de Melo and Ivan Struchiner.

• Emilia Alves (UFF, Niterói, Brazil) - 06/11, 2:30 PM.  
  Intersections of two real Bruhat Cells
  Bruhat (or Schubert) cell decompositions are closely related to several areas, including Singularity Theory, Representation Theory, Poisson Geometry and Topology. We investigate the intersections of a top dimensional cell with another cell (for another basis). Such an intersection is naturally identified with a subset of  the group of real unipotent lower triangular matrices. We are particularly interested in the homotopy type of these intersections. To achieve this goal we define a stratification of such intersections. As a consequence, we obtain a finite CW complex which is homotopically equivalent to the intersection. We include many examples and perform explicit computations to illustrate our methods. We will discuss new topological results and ongoing work in higher dimensions. This includes joint work with J. Lambert (UFF), G. Leal (PUC-Rio), N. Saldanha (PUC-Rio), B. Shapiro (Stockholm University), and M. Shapiro (MSU).

• Olivier Brahic (UFPR, Curitiba, Brazil) - 05/11 , 9:30 AM.
  Cohomological tranposrt along Morita fibrations
  A key result of Crainic concerning the notion of Morita equivalence in the differentiable setting establishes the invariance of the differentiable cohomology under pull backs. Concretely, the statement goes as follows: given a Lie groupoid $G\tto M$ together with a surjective submersion $\phi: X\to M$, the map: $$\Phi^*:H_{\mathrm{dif}}(G)\to H_{\mathrm{dif}}(G[X])$$ induced in cohomology by the natural projection $\Phi:G[X]\to G$ from the pull back groupoid $G [X]\tto X$ to $G\tto M$, is an isomorphism. This result constitutes a  pivotal point in the theory since it allows to transport classes -- and therefore geometric structures- - from one Lie groupoid to a Morita equivalent one. And while it is desirable to make such cohomological transport as concrete as possible, the details of the proof are unfortunately left to the reader, ultimately referring to a "Mayer-Vietoris argument" that is not made explicit, and has been further echoed in the literature without more details, at least to our knowledge. The main contribution of this joint work with Pedro Frejlich is to fill this gap.

• Emma Cupitra (USP, São Paulo) - 07/11,  10:30 AM. 
  Exact Twisted CA-groupoids
  We investigate geometric structures defined on an exact twisted CA-groupoid and analyze the conditions under which these structures are invariant. Motivated by the work of Ortiz and Waldron on the category of multiplicative sections of VB-groupoids and LA-groupoids, we define the underlying algebraic structure in the category of multiplicative sections of a twisted CA-groupoid, where the identified geometric structure is indeed invariant under Morita equivalence.

• Pedro Frejlich (UFRGS, Porto Alegre, Brazil) - 06/11,  9:00 AM.
  On the Kaehler condition for Poisson manifolds
  (joint w/ David Martínez Torres)
  In this talk, I will report on work in progress on a genuinely contravariant approach to the Kaehler condition for Poisson manifolds, and compare it to previous proposals. The coolest family of examples takes place in a certain complex Grassmannians, and was found in the wild by Sasha Anan'in in his work on classical geometries. I will finish by sketching a generalization of Sasha's family to certain other flag manifolds.

• Sebastian Herrera (UFPR, Curitiba, Brazil) - 05/11, 2:00 PM.
  Some Remarks on the Lie 2-Algebra of Multiplicative Vector Fields
  In this talk, we will review the Lie 2-algebra of multiplicative vector fields on Lie groupoids, accompanied by some illustrative examples. We will then see that this Lie 2-algebra admits a canonical representation up to homotopy on the 2-term complex of multiplicative functions associated with the Lie groupoid. Furthermore, we will establish how its cohomology naturally inherits the structure of a graded Lie–Rinehart algebra. Finally, we will focus on the case of proper Lie groupoids and investigate the intrinsic relationship between their multiplicative vector fields and the stratified vector fields on the associated orbispace.

• Hudson Lima (UFAM, Manaus, Brazil) -  05/11,  4:30 PM.
  Book Poisson cohomology for open sets
  In this talk I will explain how the linearization in a maximal domain and the knowledge of some the Poisson cohomology of some special open subsets in the book Poisson structure allows to find the Poisson cohomology of the Lu-Weinstein Lie-Poisson on SU(2). This is part of an ongoing project in collaboration with Henrique Bursztyn.

• Daniel Lopez (UFF, Niterói, Brazil) -  07/11,  11:00  AM.
  Equivariant cohomology of stacky Lie group actions
  When studying orbit spaces, one often encounters non-smooth manifolds. Equivariant cohomology provides a tool for analyzing the cohomology of differential forms on such orbit spaces. A natural generalization arises in the study of stacks, where the geometry can be understood via Lie groupoids and Morita equivalences. In this talk, we explore a generalization of equivariant cohomology to the context of stacky Lie groups acting on stacks. Stacky Lie groups can be modeled by Lie 2-groups; thus, we propose a Borel model defined through a complex associated with the action of a 2-group on a groupoid. Our first main result shows that this complex is invariant under Morita equivalence, thereby defining a cohomology theory at the level of stacks.

• Ana Carolina Mançur (USP, São Paulo) -  07/11,  9:00 AM. 
  From multiplicative Courant algebroids to crossed modules
  In this talk, we present an extension of the characterization of Lie bialgebroids via Manin triples to the context of double structures over Lie groupoids. We consider Lie bialgebroid groupoids, given by LA-groupoids in duality, and establish their correspondence with multiplicative Manin triples, i.e., CA-groupoids equipped with a pair of complementary multiplicative Dirac structures. Considering special cases, such as multiplicative Courant algebroids over the unit groupoid, this correspondence recovers the Manin triple description of Lie bialgebroid crossed modules, providing a new conceptual understanding of results originally established without invoking double structures. We conclude with some comments on the infinitesimal picture and ongoing work.

• Dimitri Panyushev (Ind. Uni. of Moscow, Russia) -  06/11 ,  10:30 AM. 
  TBA
  TBA

• Jonas Schnitzer (Uni. of Pavia, Italy) -  06/11,  4:00 PM.
  Quantization of the Momentum Map
  If a Lie group acts on a Poisson manifold by Hamiltonian symmetries there is a well-understood way to get rid of unnecessary degrees of freedom and pass to a Poisson manifold of a lower dimension. This procedure is known as Poisson-Hamiltonian reduction. There is a similar construction for invariant star products admitting a quantum momentum map, which leads to a deformation quantization of the Poisson-Hamiltonian reduction of the classical limit. The existence of quantum momentum maps is only known in very few cases, like linear Poisson structures and symplectic manifolds. The aim of this talk is to fill this gap and show that there is a universal way to find quantized momentum maps using so-called adapted formality morphisms which exist, if one considers nice enough Lie group actions. This talk is based on \url{https://arxiv.org/abs/2502.18295} with Chiara Esposito, Ryszard Nest and Boris Tsygan.

• Oksana  Yakimova (Uni. of Jena, German) - 05/11 ,  11:00  AM.
  A bi-Hamiltonian nature of the Gaudin algebras
  Let h be a direct sum of n copies of a simple Lie algebra g. In 1994, Feigin,  Frenkel, and Reshetikhin constructed a large commutative subalgbera of the enveloping algebra U(h). This subalgebra, which is an image of the Feigin—Frenkel centre, contains quadratic Gaudin Hamiltonians and therefore is known as a Gaudin subalgebra. By now it has been studied from various points of view and numerous generalisations have been obtained. We look at the `classical’ version of a Gaudin algebra, i.e., at its image in the symmetric algebra S(h). This image, say C,  is Poisson-commutative and can be obtained from a suitable pair of compatible Poisson brackets on S(h) via the Lenard—Magri scheme. An advantage of the Lenard—Magri approach is a well-developed geometric machinery. For example, it allows us to show that C is algebraically closed in S(h). We will discuss also a generalisation to a non-reductive setting. 

• Shizhuo Yu (Nankai University, China) -  06/11,  13:30 PM. 
  Decorated polyubles, Poisson homogeneous spaces and multi-flag varieties
  The polyuble, as defined by Fock and Rosly, can be regarded as the n-th power of a given Manin triple. This construction plays an important role in Poisson geometry and mathematical physics. In this talk, we will introduce the construction of decorated polyubles associated with decorated Young diagrams, along with their combinatorial and Poisson-geometric interpretation.

Posters

• Lídia Charra Alves (UFES, Vitória) - 06/11 ,  3:30 PM.
  Estruturas de transposed Poisson sobre  ́álgebras e super ́álgebras de Lie
  (joint w/ Renato Fehlberg Junior)  PDF
  A álgebra de Transposed Poisson foi introduzida em 2020 como uma extensão dual à definição da álgebra de Poisson, no sentido de que a condição de compatibilidade é modificada pela troca das duas operações binárias envolvidas. A tripla $(L, \cdot, [\, ,\, ])$ é uma álgebra de transposed Poisson se $(L, \cdot)$ é uma álgebra associativa comutativa e $(L, [\, ,\, ])$ é uma álgebra de Lie que satisfaz $2z[x, y] = [zx, y] + [x, zy]$, para todos $x, y, z \in L$.A álgebra de \textit{transposed Poisson} e a álgebra de Poisson compartilham diversas propriedades.
  As superálgebras são uma generalização das álgebras tradicionais que possuem uma estrutura de paridade, permitindo a distinção entre elementos pares e ímpares. Essa distinção é útil em diversas áreas da Matemática e da Física, como na geometria diferencial. A tripla $(L, \cdot, [\, ,\, ])$ é uma superálgebra de transposed Poisson se $(L, \cdot)$ é uma superálgebra associativa supercomutativa e $(L, [\, ,\, ])$ é uma superálgebra de Lie que satisfaz $2z[x, y] = [zx, y] + (-1)^{|x||z|}[x, zy]$, para todos $x, y, z \in L_0 \cup L_1$. Os elementos pertencentes a $L_0$ são denominados pares, enquanto os elementos de $L_1$ são denominados ímpares.
  Uma estrutura de transposed Poisson sobre uma álgebra (superálgebra) de Lie $(L, [\, ,\, ])$ é uma multiplicação associativa comutativa (supercomutativa) $\cdot$ sobre $L$ que torna $(L, \cdot, [\, ,\, ])$ uma álgebra de transposed Poisson (superálgebra de transposed Poisson). Essa estrutura é chamada trivial se $x \cdot y = 0$, para todos $x, y \in L$. É possível descrever todas as estruturas de transposed Poisson associadas a uma determinada álgebra de Lie, estabelecendo uma relação entre as $\tfrac{1}{2}$-derivações de álgebras de Lie e as álgebras de transposed Poisson.

• Glauber Moreno Barbosa (UFRJ, Rio de Janeiro) -  07/11 ,  10:30 AM.
  Dirac Structures and Morita Equivalence Associated with Spaces of Flat Connections
  We apply the infinite-dimensional Dirac structure reduction techniques developed by Cabrera–Gualtieri–Meinrenken to G-connections over the circle, considering gauge transformations that are trivial at n marked points. This reduction produces a finite-dimensional Dirac structure E(n), which for n = 1, recovers the Cartan–Dirac structure. In combination with the Atiyah–Bott symplectic geometry on spaces of G-connections over the cylinder, this result establishes a Morita equivalence between two Dirac structures with different numbers of marked points, E(n) and E(m). Consequently, we obtain an equivalence between the categories of Hamiltonian spaces associated with these structures, representing generalizations of the quasi-Hamiltonian spaces of Alekseev–Malkin–Meinrenken.
  
  

• Marciel Dias (USP, São Carlos, Brazil) - 07/11 ,  10:30 AM.
  A Baker-Campbell Formula for the Most Famous Courant Algebroid
  Lie groupoids have come to appear broadly during recent years as one object of interest in differential geometry for their capacity to abstract the idea of a family of symmetries acting on a family of objects or, in our case, smooth manifolds. The "infinitesimal version" of these objects are what we call Lie algebroids, and, although they share many similarities with their one-object version (Lie algebras), they have their own particular features, one of which that we are interested in may be the construction of Manin triples for such algebroids. The naive replication of the aforementioned apparatus for these objects is not as nicely behaved as in the Lie algebras case, but we still obtain something manageable: a Courant algebroid. What we would like to present is an attempt to integrate these objects by making use of the geometry of homogeneous spaces and a non-alternative incarnation of Lie algebras, called Leibniz algebras. All this work was developed by Alan Weinstein and Michael Kinyon (2000), and it offers a partial success in obtaining a formula that imitates the Baker–Campbell–Hausdorff formula for the classical example of Courant algebroid.

• Danuzia Figueirêdo (UFBA, Salvador, Brazil) - 05/11 ,  4:00 PM.
  Classification of Dually Flat One-Dimensional Manifolds
  Recent studies on the relationship between toric geometry (from the symplectic  viewpoint) and dually flat manifolds have led to a geometric construction called torification, which is a version of the Delzant correspondence for Kahler manifolds.  In this work, we discuss the main examples of torification, which arise from Statistics and Probability, and classify the  one-dimensional connected dually flat manifolds that admit a torification.
  
  

• Luiz Felipe Villar Fushimi (USP, São Paulo) - 05/11 ,  4:00 PM.
  Cartan Structure Groupoids and Algebroids
  Cartan structure algebroids are Lie algebroids equipped with a connection form like that of a Cartan geometry. They provide a generalizing framework for the development of Lie theory of algebroids, most notably that of G-structure algebroids with connection. Aside from defining these objects and establishing their initial basic properties, we explore some developments of Lie theory of these algebroids. This poster is based on my PhD thesis, written under supervision of Prof. Ivan Struchiner.

• Hugo Ibanez (UFAM, Manaus, Brazil) - 06/11 ,  10:00 AM.
  Fibrados ancorados afins
  Será discutido sobre estruturas afins em fibrados ancorados e pullback de fibrados ancorados com conexão.

• Cecilya Lemos (UFES, Vitória) -  07/11 ,  10:30 AM.
  An Introduction to Milnor fibrations in low dimension
  In this work, we follow Milnor’s approach to the study of singularities of map germs, focusing on the topological behavior of sets of the type $V_f = f^{-1}(0)$. We use the Local Conic Structure Theorem and the Milnor Fibration to describe the neighborhood of the origin through the singularity link $K_\varepsilon$. The existence of a locally trivial smooth fibration allows the application of techniques from algebraic and differential topology. We study singularities originating from germs $G : (\mathbb{R}^m, 0) \to (\mathbb{R}^2, 0)$, with $m = 3$ or $4$, aiming for a presentation accessible to undergraduates.

• Caio Magalhães (UFRJ, Rio de Janeiro) - 06/11 ,  3:30 PM.
  TBA
  An (Ehresmann) connection on a submersion $p : E \to B$ is a smooth distribution $H\subset TE$ that is complementary to the kernel of the differential, namely $T E= H ⊕ ker dp$. In this poster we want to proof that every smooth fiber bundle admits a complete (Ehresmann) connection.
  
  

• Celestia Piccioni (UFPR, Curitiba, Brazil) - 06/11 ,  10:00 AM.
  TBA
  Neste trabalho estudamos a fórmula de Weyl para o volume de um tubo de uma subvariedade. Introduzimos a noção de tubo, que corresponde ao conjunto de pontos da variedade M tais que exista uma geodésica do ponto até encontrar a subvariedade P ortogonalmente, cujo comprimento é menor ou igual a r >= 0. A seguir, apresentamos e analisamos a fórmula de Weyl, com exemplos do cálculo do volume dos tubos para casos mais simples, como de uma curva no R² e R³,superfícies e pontos no R³. Também destacamos o fator da curvatura no meio-tubo, uma superfície em R³ e um ponto em R^n. Assim, mostramos como a fórmula para o volume encontrada de maneira direta se relaciona com a fórmula geral de Weyl.
  
  

• Andrés Rodríguez (IMPA, Rio de Janeiro) - 06/11 ,  3:30 PM.
  Morita equivalence of Nijenhuis structures
  We introduce Morita equivalence for Nijenhuis groupoids and for infinitesimal Nijenhuis structures, establishing a global-to-infinitesimal correspondence under the Lie functor. In particular, we obtain Morita equivalence of holomorphic structures. This framework is used to extend the known Morita equivalences for quasi-symplectic groupoids and Dirac structures to Morita equivalences of compatible structures.
  
  

• André Tanure (USP, São Paulo) - 05/11 ,  4:00 PM.
  Sections of simplicial vector bundles
  Let \( \mathsf{V} \to \mathsf{G} \) a simplicial vector bundle. In this work we define its simplicial set of simplicial sections \( \underline{\operatorname{Sec}}(\mathsf{G},\mathsf{V}) \) and show that it is a simplicial vector space. We also introduce the simplicial vector space of projectable sections and show that it can be obtained as a pullback. This work is part of the speaker's doctoral thesis, supervised by Prof. Cristian Ortiz.
  
  

• José Luis Tavares dos Santos (UFAM, Manaus, Brazil) - 06/11 ,  10:00 AM.
  TBA
  Lance D. Drager, Jeffrey M. Lee, Efton Park, and Ken Richardson investigate issues related to the finite generation of generalized subbundles of vector bundles in the smooth category. First, the authors prove that every generalized subbundle of a vector bundle is \emph{globally finitely generated}, that is, there exist global smooth sections whose values generate the corresponding subspace at each point of the manifold. This result contradicts previous conjectures in the literature, including one by Bullo and Lewis. They then show that the module of smooth sections of a generalized subbundle may \emph{fail} to be finitely generated, even locally. To this end, they present an explicit example in which the distribution---including a tangent distribution---fails to admit a finite set of local generators. In this work, we prove that, given a smooth vector bundle $E \to M$, there always exists a smooth generalized subbundle $ G \subseteq E $ such that the set of smooth sections $\Gamma(G)$ is not finitely generated as a $C^\infty(M)$-module. We also investigate whether, given a generalized subbundle $G \subset TM$, the module \( \Gamma(G) \) is finitely generated. Furthermore, we study which smooth generalized distributions $G \subset T\mathbb{R}^n$ can be realized as the images of anchors $\rho\colon E \to T\mathbb{R}^n,$ where $(E,[\, ,\,],\rho)$ is a Lie algebroid over $\mathbb{R}^n$.
  
  

• Tianhao Ye (IMPA, Rio de Janeiro) - 06/11 ,  3:30 PM.
  Infinitesimal 1-Shifted Coisotropic Structures
  We introduce a new notion of 1-shifted infinitesimal coisotropic structures on quasi-Lie bialgebroids, defined via Courant morphisms. We demonstrate that this concept generalizes the infinitesimal part of Maxence Mayrand's 1-shifted coisotropic structures for 1-shifted symplectic Lie groupoids. Furthermore, we show that our notion encompasses interesting examples and satisfies properties analogous to its counterpart in shifted Poisson geometry.