Palestras:
Symmetries in Algebraic Geometry
Data: 07/01/21
Hora: 16:00
Palestrante: Carolina Araujo - Instituto de Matemática Pura e Aplicada -IMPA.
Chairman: Aron Simis - Universidade Federal de Pernambuco (UFPE).
Resumo: In this talk I will discuss symmetries of complex algebraic varieties. When studying a projective variety $X$, one usually wants to understand its symmetries. Conversely, the structure of the group of automorphisms of $X$ encodes relevant geometric properties of $X$. After describing some examples of automorphism groups of projective varieties, I will discuss why the notion of automorphism is too rigid in the scope of birational geometry. We are then led to consider another class of symmetries of $X$, its birational self-maps. Birational self-maps of the projective space $\mathbb{P}^n$ are called Cremona transformations. Describing the structure of the group of Cremona transformations of the plane is a classical problem that goes back to the 19th century. In higher dimensions, not so much is known, and a natural problem is to construct interesting subgroups of the Cremona group. I will end by discussing a recent work with Alessio Corti and Alex Massarenti, where we investigate subgroups of the Cremona group consisting of symmetries preserving some special meromorphic volume forms.
A 3D Phase-field Model for Solidification Under a Magnetic Field Effect
Data: 14/01/21
Hora: 16:00
Palestrante: Gabriela Del Valle Planas - Universidade Estadual de Campinas.
Chairman: Cilon Perusato - Universidade Federal de Pernambuco (UFPE).
Resumo: We present a mathematical analysis to a 3D isothermal model of solidification for a binary alloy with melt convection and under a magnetic field effect. The model consists of a highly non-linear system of partial differential equations for the state variables: the velocity field, the pressure, the potential function of the electrical field, the phase-field which represents the solid/liquid phase of the alloy, and the concentration. The well-posedness of the model is discussed. Moreover, the existence of solutions when the diffusion coefficient of the concentration equation vanishes for some values of the phase-field is investigated.
Geometric structures in 2D Navier-Stokes flows
Data: 21/01/21
Hora: 16:00
Palestrante: Lorenzo Brandolese - Institut Camille Jordan ( Université Lyon 1)
Chairman: César Niche - Universidade Federal do Rio de Janeiro (UFRJ).
Resumo: Geometric structures naturally appear in fluid motions. One of the best known examples is Saturn’s Hexagon, the huge cloud pattern at the level of Saturn’s north pole, remarkable both for the regularity of its shape and its stability during the past decades. In this paper we will address the spontaneous formation of hexagonal structures in planar viscous flows, in the classical setting of Leray’s solutions of the Navier–Stokes equations. Our analysis also makes evidence of the isotropic character of the energy density of the fluid for sufficiently localized 2D flows in the far field: it implies, in particular, that fluid particles of such flows are nowhere at rest at large distances.
Einstein-type elliptic systems
Data: 28/01/21
Hora: 16:00
Palestrante: Jorge Herbert Soares de Lira - Universidade Federal do Ceará.
Chairman: Fábio Reis dos Santos - Universidade Federal de Pernambuco (UFPE).
Resumo: We will discuss a type of semi-linear systems of partial differential equations which are motivated by the conformal formulation of the Einstein constraint equations coupled with realistic physical fields on asymptotically flat manifolds. In particular, electromagnetic fields give rise to this kind of systems. In this context, under suitable conditions, we prove a general existence theorem for such systems, and, in particular, under smallness assumptions on the free parameters of the problem, we prove existence of far from CMC (near CMC) Yamabe positive (Yamabe non-positive) solutions for charged dust coupled to the Einstein equations, satisfying a trapped surface condition on the boundary. As a bypass, we prove a Helmholtz decomposition on asymptotically flat manifolds with boundary, which extends and clarifies previously known results.
On the asymptotic Dirichlet problem for some elliptic PDE on hyperbolic spaces
Data: 04/02/21
Hora: 16:00
Palestrante: Leonardo Bonorino - Universidade Federal do Rio Grande do Sul.
Chairman: João Marcos Bezerra do Ó - Universidade Federal da Paraíba (UFPB).
Resumo: We study the Dirichlet problem for the following mean curvature PDE. We prove the existence and uniqueness of solution in C(Hn) ∩ C2(Hn), where Hn is the hyperbolic space. For that, we show the existence of solutions for the equation in compact domains and we construct barriers in the hyper- bolic space which resemble the Scherk type solutions of the minimal surface PDE. These Scherk type graphs allow us to prove also the non existence of isolated asymptotic boundary singularities for global solutions. We show that for other elliptic PDE’s, like the p-Laplace equation, the solutions can have these singularities. Some of these results are extended to some Hadamard manifolds.
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Existence of solution for repulsive chemotaxis models
Data: 11/02/21
Hora: 16:00
Palestrante: María Ángeles Rodríguez Bellido - Universidad de Sevilla.
Chairman: Pablo Braz e SIlva - Universidade Federal de Pernambuco (UFPE).
Resumo: Chemotaxis is understood as the biological process of the movement of living organisms in response to a chemical stimulus which can be given towards a higher (attractive) or lower (repulsive) concentration of a chemical substance. At the same time, the presence of living organisms can produce or consume chemical substance.
We will show different models for repulsive chemotaxis and analyze the methods and difficulties encountered in order to prove existence, uniqueness, regularity and positivity of solutions for such models.
Sobre os espectros de Markov e Lagrange
Data: 18/02/21
Hora: 16:00
Palestrante: Carlos Gustavo Tamm de Araújo Moreira ( Gugu ) - Instituto de Matemática Pura e Aplicada -IMPA.
Chairman: Ricardo Bortolotti - Universidade Federal de Pernambuco (UFPE).
Resumo: Vamos apresentar resultados clássicos e recentes sobre os espectros de Markov e Lagrange, que são conjuntos de números reais que aparecem naturalmente no estudo das melhores aproximações de números irracionais por racionais, em particular alguns resultados recentes, em colaboração com Carlos Matheus, Davi Lima e Sandoel Vieira, sobre a diferença entre esses conjuntos.
Discutiremos também a relação desses resultados com Sistemas Dinâmicos.
Nonlinear elliptic systems with local and nonlocal terms
Data: 25/02/21
Hora: 16:00
Palestrante: Liliane de Almeida Maia - Universidade de Brasília.
Chairman: José Carlos de Albuquerque Melo Júnior - Universidade Federal de Pernambuco (UFPE).
Resumo: We revisit weakly coupled nonlinear Schrodinger systems (NLS) with cubic or higher order power terms of local type and present some recent work on nonexistence, existence and asymptotic behaviour with respect to a parameter of ground state solutions for a class of systems with local and nonlocal nonlinear terms. Particularly, we illustrate with systems of this type which model Bose-Einstein condensates and Hartree-Fock-Slater methods in Physics or population dynamics in Biology.
This is work in collaboration with Gaetano Siciliano (USP/BR) and Pietro D'Avenia (Politecnico di Bari/IT).