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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:
- 2024.02.28 (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)
- Title:
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- Abstract:
- venue: (IME-USP)
Future talks and events:
- 2022
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Previous talks and events:
- 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)
Previous Years (2013-2020)
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