Palestras de Anos Anteriores

2022

14 de Junho às 10h - Giovanni Gentili (Università di Torino, Itália) 

In-person, in Sala C116 (Bloco C do CT, UFRJ)

Título - Monge-Ampère type equations on HKT manifolds

Resumo - HKT manifolds were first introduced by physicists since they arise as internal spaces of some supersymmetric sigma model with Wess-Zumino term. From the mathematical standpoint, HKT manifolds represent in many ways the hypercomplex counterpart of Kähler manifolds. In view of this analogy and motivated by the search of "canonical metrics", Alesker and Verbitsky posed in 2010 a Calabi-Yau type problem in HKT geometry. As of today the conjecture is still open in its full generality. We will briefly overview the relevant mathematical framework and present some recent results towards the proof. This talk is based on joint works with Lucio Bedulli, Luigi Vezzoni and Jiaogen Zhang.

31 de Maio às 14h - Mauricio Godoy Molina (Univ. de La Frontera, Chile) 

Título -  On infinitesimal symmetries of distributions

Resumo -   In this talk, I will give an introduction to the notion of infinitesimal symmetries of distributions (recall that in differential geometry, a distribution is a subbundle of the tangent bundle of a manifold), with emphasis on how do they appear in some problems related to mechanics. The study of the Lie algebra of symmetries of distributions is interesting to a broad range of mathematicians since – depending on the problem – by using techniques in Lie theory and linear connections, one may address interesting problems in sub-Riemannian geometry, geometric control theory, differential operators and theoretical mechanics. The idea of the talk is to give a brief historical account starting with the seminal work of Tanaka, show some recent results, and present a few open problems.

17 de Maio às 14h - Romina M. Arroyo (Univ. Nacional de Córdoba, Arg.) 

Título -  SKT structures on nilmanifolds

Resumo -  A Hermitian metric on a complex manifold is said to be strong Kähler with torsion (SKT for short) if the torsion associated to the Bismut connection is closed. In this talk we will discuss a recent joint work in collaboration with Marina Nicolini in which we study the existence of invariant SKT structures on nilmanifolds. We will prove that an invariant SKT structure on a nilmanifold forces the nilmanifold to be at most 2-step nilpotent and we will provide examples of invariant SKT structures on 2-step nilmanifolds in arbitrary dimensions.

3 de Maio às 10h - Vamsi Pritham Pingali (Indian Institute of Science) 

Título - Numerical criteria for the solvability of some fully nonlinear PDE.

Resumo -  The Lejmi-Szekelyhidi conjecture states that if certain integrals are positive, then certain inverse Hessian equations can be solved. Motivated by G. Chen's proof of a weaker version of this conjecture in the case of the J-equation (and the deformed Hermitian Yang-Mills equation), we proved this conjecture for all generalised Monge-Ampere equations on projective manifolds. I shall describe the background and our ideas for the proof. This is joint work with Ved Datar.

26 de Abril às 14h - Julieth Saavedra (Toulouse/Unicamp) 

Título - Flows of G2-structures on 7-manifolds with symmetry

Resumo -  Flows of G2-structures have been used as tools in the study of G2-geometry. The talk will focus on some principal results of the Laplacian coflow and divergent flow of G2-structures.  We will give some  general preliminaries on Contact Calabi-Yau 7-manifolds which was used in the Laplacian coflow with the initial coclosed G2-structure  given by Habib and Vezzoni finding a singularity and show that the metric and the volume collapse at this singularity and in the case of divergent flow, we will focus on Sp(2)-invariant G2-structures on the homogeneous 7-sphere  S^7=Sp(2)/Sp(1).

(Joint work with J. Lotay, E.Loubeau, H, Sá Earp and A. Moreno)  

22 de Março às 14h - Abdelmoubine Amar Henni (UFSC) 

Título - Torsion free instanton sheaves on the blow-up of ℙ3 at a point

Resumo -  We consider an extension of the instanton bundles definition, given by Casnati-Coskun-Genk-Malaspina, for Fano threefolds, in order to include non locally-free ones on the blow-up X, of the “-projective space at a point. With the proposed definition, we prove that any reflexive instanton sheaf must be locally free, and that the strictly torsion free instanton sheaves have singularities of pure dimension 1. We construct examples and study their mu-stability. Furthermore, these sheaves will play a role in (partially) compactifying the t'Hooft component of the moduli space of instantons, on X Finally, examples of these are shown to be smooth and smoothable.     

8 de Março às 14h - Benigno Oliveira Alves (UFBA) 

Título - Folheações singulares e funções em variedades Finslerianas. 

Resumo -  As frentes de ondas de uma propagação em um meio isotrópico são níveis de uma função isoparamétrica Riemanniana e em particular são equidistantes, constituindo uma folheação Riemanniana singular. Por exemplo, as ondas em uma lagoa "calma" provocada por uma pedra. E se o meio for anisotrópico? Por exemplo, se há presença de correnteza na lagoa ou ainda a propagação de um incêndio florestal, onde há a presença de ventos. Nesse caso as frentes de ondas são níveis de uma função isoparamétrica Finsleriana, que não constituem uma folheação Finsleriana singular, pois não necessariamente são equidistantes. Essa e outras aplicações motivam nossos estudos sobre folheações Finslerianas singulares, funções transnormais e isoparamétricas em uma variedade Finsleriana. Embora tais temas em geometria Riemanniana sejam clássicos e largamente estudados, no contexto Finsleriano são recentes. Nosso objetivo é apresentar motivações, os conceitos fundamentais e os resultados recentes obtidos por nosso grupo. 

22 de Fevereiro às 14h - Thiago Pires (UFRJ/UFES) 

Título - Funcionais de Curvatura e o Funcional Norma L2 da Segunda Forma Fundamental Sem Traço

Resumo -   Funcionais que dependem das curvaturas de uma superfície são objeto de interesse não só de Geômetras, mas também de físicos, biólogos e outros. Em particular, é de interesse natural estudar os minimizantes desses funcionais bem como a estabilidade dos mesmos. Definimos um funcional dependente das curvaturas média e escalar, para hipersuperfícies de dimensão n imersas em formas espaciais de curvatura constante e calculamos suas variações, obtendo não só a equação de Euler-Lagrange a ser satisfeita pelos pontos críticos mas também um critério para definir a estabilidade destes pontos em termos de invariantes geométricas que dependem apenas da primeira e segunda forma fundamental.

Como forma de demonstrar a aplicabilidade dos resultados obtidos anteriormente estudamos o funcional de curvatura dado pela norma L2 da segunda forma fundamental sem traço. Usando a equação de Euler-Lagrange obtivemos informações sobre pontos críticos com duas curvaturas principais distintas, com uma atenção particular para as hipersuperfícies de rotação e os toros de Clifford. Além disso, estudamos a estabilidade de alguns pontos críticos conhecidos. Por fim re-interpretamos alguns teoremas de gap de modo a obter mais informações sobre os pontos críticos, dando uma ênfase maior às imersões na esfera Euclidiana.

1 de Fevereiro às 14h - Alejandro Cabrera (UFRJ) 

Título - An introduction to the Lie theory for groupoids and its applications

Resumo -  In this talk, we will first overview the Lie theory of groupoids and algebroids mentioning several applications. In the last part of the talk, we will review some recent results and their connection with Poisson and symplectic geometry.

18 de Janeiro às 14h - Daniel Fadel (Université de Bretagne Occidental) 

Título -  Asymptotics of finite energy monopoles on AC 3-manifolds

Resumo -   I will report on my recent work on sharp decay estimates for critical points of the SU(2) Yang-Mills-Higgs energy functional on asymptotically conical (AC) 3-manifolds, generalizing classical results of Taubes in the 3-dimensional Euclidean space. In particular, I will explain how we prove the quadratic decay of the covariant derivative of the Higgs field of any critical point in this general context and, with an additional hypothesis on the link, we will also sketch the proof of the quadratic decay of the curvature by combining Bochner formulas with certain refined Kato inequalities with "error terms" and standard elliptic techniques. We deduce that every irreducible critical point converges along the conical end to a limiting configuration at infinity consisting of a reducible Yang-Mills connection and a parallel Higgs field. If time permits, I will mention a few open problems and future directions in this theory.

4 de Janeiro às 14h - Jesús Nuñez Zimbron (CIMAT) 

Título - Harmonic functions on spaces with Ricci curvature bounded below

Resumo -   The so-called spaces with the Riemannian curvature-dimension conditions (RCD spaces) are metric measure spaces which are not necessarily smooth but admit a notion of “Ricci curvature bounded below and dimension bounded above”. These spaces arise naturally as Gromov-Hausdorff limits of Riemannian manifolds with these conditions and, in contrast to manifolds, RCD spaces typically have topological or metric singularities. Nevertheless a considerable amount of Riemannian geometry can be recovered for these spaces. In this talk I will present recent work joint with Guido De Phillipis, in which we show that the gradients of harmonic functions vanish at the singular points of the space. I will mention two applications of this result on smooth manifolds: it implies that there does not exist an a priori estimate on the modulus of continuity of the gradient of harmonic functions depending only on lower bounds of the sectional curvature and there is no a priori Calderón-Zygmund inequality for the Laplacian with bounds depending only on the sectional curvature.

2021