The Geometry Research Group at IME USP

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 The Geometry Seminar at IME-USP is organized by Prof. Marcos Alexandrino and Prof. Paolo Piccione and  supported by Projeto Temático-Fapesp of Prof.  Paolo Piccione ( Fapesp 2016/23746-6/Fapesp: 2011/21362-2)

Next talks and events:

  • 2024.06.21  (at 15h)
  • Title:   A condição de Ricci para métrica warped
  • Speaker: Dr. Roney Santos (IME-USP)
  • Abstract:  Gostaríamos de introduzir e discutir brevemente o conceito recente de superfície de Ricci. Essas superfícies abstratas têm a propriedade de admitir imersão isométrica local no espaço Euclidiano tridimensional R^3 como superfície mínima quando sua curvatura Gaussiana é não-positiva, o que faz das superfícies de Ricci uma "maneira intrínseca" de olhar para superfícies mínimas do R^3. Discutiremos sobre a classificação de superfícies de Ricci que admitem métricas warped e sobre a relação entre superfícies de Ricci regradas e curvas de torção constante no R^3. Por fim, se ainda tivermos tempo, gostaríamos de explorar rapidamente uma definição de Lucas Ambrozio que estende o conceito de superfícies mínimas com bordo livre na bola Euclidiana unitária tridimensional para o contexto intrínseco das superfícies de Ricci.
  • venue: B 139  (IME-USP) 

  • Title:  Singular Riemannian Foliations, variational problems and Principles of Symmetric Criticalities,
  • Speaker:  Prof. Marcos Alexandrino (IME-USP) 
  • Abstract:  A singular foliation F on a complete Riemannian manifold M is called Singular Riemannian foliation (SRF for short) if its leaves are locally equidistant, e.g., the partition of M into orbits of an isometric action. In this talk, we discuss variational problems in compact Riemannian manifolds equipped with SRF with special properties, e.g. isoparametric foliations, SRF on fibers bundles with Sasaki metric, and orbit-like foliations. More precisely, we prove two results analogous to Palais' Principle of Symmetric Criticality, one is a general principle for basic symmetric operators on the Hilbert space W1,2(M), the other one is for basic symmetric integral operators on the Banach spaces W1,p(M). These results together with a basic version of Rellich Kondrachov Hebey Vaugon Embedding Theorem allow us to circumvent difficulties with Sobolev's critical exponents when considering applications of Calculus of Variations to find solutions to PDEs. We  exemplify this brielfy discussing the  existence of weak solutions to a class of variational problems which includes p-Kirschoff problems. This talk is based on a joint work with: Leonardo F. Cavenaghi, Diego Corro, Marcelo K. Inagaki.
  • venue:   (IME-USP) 

    Future talks and events:

  • Prof. Marcos Alexandrino

  • 2022
  • Title: 
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Previous talks and events:

  • 2024.06.07 (at 15 h)
  • Title:  Closures of Riemannian groupoids
  • Speaker: Prof. Mateus Moreira de Melo  (ufes)
  • Abstract:  From the Myers-Steenrod theorem, actions of closed subgroups of the isometry group are proper. Based on the previous example and the results of Molino, Salem, and Alexandrino-Radeschi, we address the problem of obtaining a proper groupoid from a Riemannian groupoid. As part of the ingredients of our proof, we will discuss the conditions for a foliation in a bundle to be a Riemannian foliation.
  • venue:  B09  (IME-USP) 

  • 2024.05.17 (at 15 h)
  • Title:  Geometry over algebras
  • Speaker: Dr Hugo Cattarucci Botós (IME-USP)
  • Abstract:  We discuss how geometric structures arise from Hermitian forms on linear spaces over real algebras.
    Traditionally, the real algebras used are R, C, and H, representing real, complex, and quaternionic numbers. Structures like real/complex/quaternionic hyperbolic spaces are obtained naturally from this approach. Fubini-Study geometry and Riemannian structures for Grassmanians as well. One nice feature of the linear algebraic approach is that we can express geometric objects in a linear algebraic fashion, which is advantageous, for instance, for computational reasons. We extend the discussion to the dual numbers D=R+ϵ R, with ϵ² = 0, the split-complex numbers SC = R+j R, with j² = 1, and the split-quaternions SH=R+iR+jR+kR, with  i² = -1, j² = 1, k² = 1, ij = k = -ji. We present the projective geometry for these algebras and their pseudo-Riemannian geometries. We express their geodesics and curvatures as linear algebraic objects. We also describe the transition of geometries between projective lines over C, D, and SC. Additionally, we naturally interpret these projective lines as the configuration spaces for the oriented geodesics on the 2-sphere, the Euclidean plane, and the hyperbolic plane.  For the real algebra CxC, we also obtain a natural projective model for the hyperbolic bidisc, the product of two Poincaré discs.
  • venue:  139 B  (IME-USP) 

  • 2024.05.24  (at 15h)
  • Title:  Algebricidade de folheações Riemannianas singulares
  • Speaker: Prof. Ricardo Mendes (Oklahoma State Univ)
  • Abstract: As folheações Riemannianas singulares são certas partições de variedades Riemannianas, e as fontes tradicionais de exemplos são ações isométricas e hipersuperfícies isoparamétricas. Se a variedade ambiente é uma esfera, esses exemplos são, no sentido apropriado, algébricos, o que segue de resultados clássicos de Hilbert, Cartan, e Münzner. Em 2018, A. Lytchak e M. Radeschi mostraram algebricidade para uma foliação Riemanniana singular geral em uma esfera. Em trabalho em andamento, em colaboração com S. Lin e M. Radeschi,  generalizamos o teorema de Lytchak--Radeschi das esferas para qualquer espaço homogêneo normal compacto, uma classe que inclui todos os espaços simétricos compactos.

  • venue: 139 B   (IME-USP) 

  • 2024.02.28 (Wed.)/  2024.03.01 (Fri.) /  2024.03.04 (Mon.)/ at  14h          
  • Title:  Lectures: Metric aspects of buildings, by Linus Kramer
  • Speaker:  prof. Linus Kramer (WWU Münster)
  • Abstract:  Buildings are combinatorial metric structures. They come for example from simple algebraic groups. A fundamental result by Jacques Tits says that conversely, all spherical and euclidean buildings of higher dimension arise in this way. In my lectures I will give a manifold-style introduction
    to buildings, and highlight some of their properties.
  • venue: Auditório Antonio Gilioli,  bloco A do IME.
  • 2024.02.28 (Wed.) / 2024.02.29 (Thurs.). / 2024.03.01 (Fri..) / at 15.30h
  • Title: Lectures:   Introduction to Kähler Geometry, 
  • Speaker:  profa  Bianca Santoro  (WWU Münster)
  • Abstract: 
  • venue:  Auditório Antonio Gilioli,  bloco A do IME
  • 2024.03.04   (Mon.)  / 2024.03.05 (Tues..) / 2024.03.06 (Wed..)  / at 15:30h.
  • Title:  Lectures: A gluing construction for Kähler-Einstein metrics
  • Speaker:  Prof.  Hans-Joachim Hein (WWU Münster)
  • Abstract: 
  • venue:   Auditório Antonio Gilioli,  bloco A do IME.

  • 2023.03.08-15 h
  • Title:  Special holomorphic gradients on K\"ahler manifolds
  • Speaker: Prof. Andrzej Derdzinski (The Ohio State University)
  • Abstract:  Two main results are presented. They deal with functions $\tau$ on K\"ahler manifolds $M$ of complex dimensions $m > 1$ satisfying
    a special Ricci-Hessian equation in the sense of Maschler (2008): $\alpha \nabla d\tau$ + Ric equals a function times $g$, for some
    function $\alpha$ of the real variable $\tau$, with $\alpha \nabla d\tau$ assumed nonzero almost everywhere. Examples are provided by the
    non-Einstein cases of CEKM, GKRS and SKRP (conformally-Einstein K\"ahler metrics, gradient K\"ahler-Ricci solitons, and special K\"ahler-Ricci
    potentials). If $\tau$ also happens to be transnormal (that is, the integral curves of its holomorphic gradient $v = \nabla \tau$ are reparametrized
    geodesics), the triple $(M,g,\tau)$ must represent one of the well-understood types GKRS and SKRP. We show that, in the non-transnormal case, one must have $m = 2$ and, up to normalizations, $\alpha/2$ equals $1$, or $1/\tau$, or $\cot \tau$, or $\coth \tau$ or, finally, $\tanh \tau$. Furthermore, we prove, using the Cartan-K\"ahler theorem, that each of these five options is actually realized by a non-transnormal function $\tau$ on a K\"ahler surface $M$. For $1$ and $1/\tau$ this last fact is already known due to two classic existence theorems, with $M$ equal to the two-point blow-up of $CP^2$, where $g$ is the Wang-Zhu toric K\"ahler-Ricci soliton or, respectively, the Chen-LeBrun-Weber
    conformally-Einstein K\"ahler metric. (Joint work with Paolo Piccione.)
  • venue:   (IME-USP) 

  • 2023.08.11 (15.15 h)     
  • Title:  Revisiting Arnold's Topological Proof of the Morse Index Theorem
  • Speaker: Eduardo Ventilari Sodre   (USP)
  • Abstract:  We give an exposition of the Morse Index Theorem in the Riemannian case in terms of the Maslov Index, following and expanding upon Arnold's seminal paper. We emphasize the symplectic arguments in the proof and aim to be as self-contained as possible.
  • venue:  B07    (IME-USP) 

  • 2023.08.07 (15 h)     
  • Title:  The bundle structure of compact rank-one ECS manifolds
  • Speaker: Ivo Terek   (OSU-EUA)
  • Abstract:  The local types of essentially conformally symmetric manifolds (i.e., pseudo-Riemannian manifolds with parallel Weyl tensor which are not locally symmetric or conformally flat) have been fully described by Derdzinski and Roter in 2009. They are distinguished by the rank, always equal to 1 or 2, of a certain null parallel D distribution associated with the Weyl tensor. Compact rank-one ECS manifolds exist in all dimensions starting from 5, and topological features common to all known examples are not accidental: we prove that a compact rank-one ECS manifold, if not locally homogeneous and replaced if needed by a two-fold isometric covering, must be the total space of a fiber bundle over the circle, with D^\perp appearing as its vertical distribution. This is joint work with Andrzej Derdzinski.

  • 2023.06.02 (15 h)     
  • Title:  On the Multiplicity of the Brake Orbtis
  • Speaker: prof. Dario Corona,  (University of Camerino -Italy)
  • Abstract:  This seminar will show some recent developments in the study of the brake orbits of Hamiltonian systems. Roughly speaking, a brake orbit is a periodic solution that oscillates back and forth between two rest points, as a pendulum-like motion. In 1948, H. Seifert conjectured, under some hypotheses on the Hamiltonian function, that the number of geometrically distinct brake orbits is always greater than or equal to the degrees of freedom of the system. We show that if the Hamiltonian function is even and strictly convex with respect to the generalized momenta then the brake orbits are in one-to-one correspondence with orthogonal geodesic chords in a strongly concave Finsler manifold with boundary.Thus, the multiplicity of the brake-orbits can be obtained by appropriate refinements of mini-max methods and the Ljusternik and Schnirelmann category. The seminar is completed with a historical perspective and further developments of the subject.
  • venue:  A243  (IME-USP) 

  • 2023.05.26 (15 h)     
  • Title:  Integration of generalized Kähler structures
  • Speaker: Dr. Daniel Alvarez (University of Toronto)
  • Abstract:  A generalized Kähler (GK) structure is a pair of commuting generalized complex structures  whose composite is a generalized metric, this is Gualtieri's reformulation of the concept of bihermitian  structure introduced in mathematical physics by Gates, Hull and Rocek.  We answer the question of what is the global meaning of GK potential by using the theory of symplectic double groupoids. We will review the ideas that led us to this general result by examining the situation of a Kähler metric from the viewpoint of Poisson geometry and double structures. This is based on work in progress with M. Gualtieri and Y. Jiang.
  • venue: B143  (IME-USP) 

  • 2023.05.19 (14 h)     
  • Title:  Cofluxo do Laplaciano de G2 -estruturas cofechadas e seus solitons
  • Speaker: Dr Andres Moreno - Unicamp
  • Abstract: link
  • venue: B16  (IME-USP) 

  • 2023.05.19 (15:30 h)     
  • Title:  Funções isoparamétricas e curvatura média em variedades com navegação de Zermelo
  • Speaker: Profa Patrícia Marçal  (IME-USP)
  • Abstract: O estudo de funções isoparamétricas surgiu a partir de uma pergunta simples em óptica geométrica: quais ondas tem velocidade constante em cada frente de onda? Por sua vez, o problema da navegação de Zermelo busca os caminhos que minimizem tempo em um ambiente, modelado por uma variedade Finsler (M,F), sob a influência de vento ou correnteza, expresso por um campo vetorial W. Nosso principal objetivo é investigar a relação entre as funções isoparamétricas na variedade M com e sem a presença do vento W. Para os casos positivos-definidos, também comparamos as curvaturas médias na variedade. Neste trabalho conjunto como Dr. Benigno Oliveira Alves (UFBA), buscamos seguir uma abordagem livre de coordenadas.
  • venue: B16  (IME-USP) 

  • 2023.04.19 (16 h)     
  • Title:  Conical metrics with special holonomy.
  • Speaker: Prof. Misha Verbitsky ( IMPA)
  • Abstract: Metrics with special holonomy are well understood, thanks to Ambrose-Singer, de Rham and Berger theorems.I would present the classification of special holonomies on Riemannian cones and their correspondence to Weyl connections with special holonomy, appearing in conformal geometry. The main technical result presents a conical Riemannian metric in tensorial terms, and was used to define invariant locally conformal structures on Lie groups.
  • venue: B101 (IME-USP) 

  • 2023.01.06 (11-12)     
  • Title: The topology of compact Weyl-parallel manifolds
  • Speaker: Andrzej Derdzinski ( Ohio State University)
  • Abstract: ECS manifolds are pseudo-Riemannian manifolds of dimensions n ≥ 4 which have parallel Weyl tensor, but not for one of two obvious reasons: conformal flatness  or local symmetry. They exist for every n ≥ 4, their metrics are always indefinite,  and their local structure has been completely described. Every ECS manifold has an invariant called rank, equal to 1 or 2. Known examples of compact ECS manifolds, representing every dimension n > 4, are of rank 1, and  none of them is locally homogeneous.  We prove that a compact rank-one ECS manifold, if not locally homogeneous, replaced if necessary by a two-fold isometric covering, must be the total space  of a bundle over the circle.(joint work with Ivo Terek)

  • venue: ?? (IME-USP) 

  • 2022.11.11     
  • Title: An application of Finsler geometry in wildfire propagation modeling
  • Speaker: Hengameh Raeisidehkordi (UFABC)
  • Abstract: We will talk about some basic concepts in Finsler geometry and wave propagations. We provide some discussion about our methods in wildfire propagation modeling and, finally, see some examples showing the application of our methods.
  • venue: B03 (IME-USP) 

  • 2022.10.21
  • Title: Completeness of metrics and linearization of groupoids
  • Speaker:Prof.  Matias Luis del Hoyo (UFF)
  • Abstract: Every smooth fiber bundle admits a  complete Ehresmann connection.  I will talk about the story of this theorem and its
    relation with Riemannian submersions.  Then, after discussing some foundations of Riemannian geometry
    of Lie groupoids and stacks, I will present a generalization of the theorem into this framework,  which somehow answers an open problem on the linearization of groupoids. Talk based on collaborations with M. de Melo (USP).
  • venue: B06 (IME-USP) 

  • 2022.10.07
  • Title: A pesquisa vigente na área de curvaturas positivas e temas correlatos
  • Speaker: Dr. Leonardo Francisco Cavenaghi (Imecc Unicamp)
  • Abstract: Um fato bem conhecido em geometria consiste em seu próprio uso na compreensão de variedades como espaços topológicos. Por exemplo, teoremas como o Teorema da Esfera Diferenciável e o Programa de Geometrização de Thurston, classificam a topologia de algumas variedades Riemannianas de acordo com suas geometrias. Por outro lado, o problema inverso permanece sem solução para quase todas as variedades, sendo poucas as propriedades geométricas conhecidas que uma determinada variedade pode assumir.  Embora existam resultados como o Teorema de Preissman no cenário de variedades Riemannianas com curvatura seccional negativa, e o teorema de Bonnet-Meyers para variedades com curvatura de Ricci positiva limitada por baixo, não há teorema que distingue a classe de variedades simplesmente conectadas fechadas com curvatura seccional não negativa à de variedades simplesmente conectadas fechadas admitindo métricas com curvatura seccional positiva. De acordo com este fato, seria natural esperar que toda variedade da primeira classe mencionada admitisse uma métrica de curvatura seccional positiva. No entanto, a literatura apresenta uma enorme discrepância entre os exemplos dessas classes. Além disso, existem variedades suaves $\Sigma^n$ que são homeomórficas à esfera padrão $S^n$, mas não difeomorfas a ela. Existem também inúmeras estruturas suaves (em pares não difeomorfas) em $\mathbb{R}^4$, assim como existem toros exóticos, espaços projetivos exóticos e assim por diante. Isso naturalmente levanta a questão até que ponto a estrutura suave determina/obstrui a geometria? Mais especificamente, tais esferas exóticas admitem geometrias/dinâmicas semelhantes às geometrias padrão em $S^n$?

    Nessa palestra, iremos motivar a discussão acima por meio de trabalhos já desenvolvidos na área, discutindo também as técnicas comumente empregadas, bem como problemas atuais de relevância.

  • venue: B03 (IME-USP) 
  • 2022.09.16
  • Title: Traçando Caminhos na Teoria de Transporte Ótimo
  • Speaker:  Dr. André Gomes (Imecc Unicamp)
  • Abstract: Apresentação do estado da arte da teoria de  ações lagrangianas na teoria de transporte ótimo  de Monge e Kantorovich, ressaltando o ponto de vista geométrico e seus vínculos com a análise.
  • venue: B03 (IME-USP) 

  • 2022.08.05
  • Title: A geometric take on Kostant's Convexity Theorem
  • Speaker:  Prof. Ricardo Mendes
  • Abstract: We characterize convex subsets of R^n invariant under the linear action of a compact group G, by identifying their images in the
    orbit space R^n/G by a purely metric property. As a consequence, we obtain a version of Kostant's celebrated Convexity Theorem (1973)
    whenever the orbit space R^n/G is isometric to another orbit space R^m/H. (In the classical case G acts by the adjoint representation on
    its Lie algebra R^n, and H is the Weyl group acting on a Cartan sub-algebra R^m). Being purely metric, our results also hold when the
    group actions are replaced with submetries. 
  • venue: B139  (IME-USP) 

  • 2022.08.12
  • Title:  Projective representations of real reductive Lie groups and the gradient map'
  • Speaker:  Prof. Leonardo Billiotti
  • Abstract: Let G=Kexp(p) be a connected semisimple noncompact real reductive Lie group acting linearly on a finite dimensional vector space V over R.  We assume that there exists a K-invariant scalar product g such that K\subset SO(V,g) and p \subset Sym_o (V,g), where Sym_o (V,g) is the set of symmetric endomorphisms with trace zero.  We also assume that the G-action on V and the G^C-action on V^C are irreducible. Using G-gradient map techniques we analyze the natural projective representation of G on P(V). ( arXiv:2205.15632)
  • venue: B139  (IME-USP) 

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