seminar

Stability of extremal domains for the first Dirichlet eigenvalue of the Laplacian operator

By Ivaldo Nunes (UFMA, Brazil)

Abstract: In this talk, we discuss the concept of stable extremal domains for the first Dirichlet eigenvalue of the Laplacian operator. We classify the stable extremal domains in the 2-sphere and higher-dimensional spheres when the boundary is minimal. Additionally, we establish topological bounds for stable domains in general compact Riemannian surfaces, assuming either nonnegative total Gaussian curvature or small volume. This is a joint work with Marcos P. Cavalcante (UFAL, Brazil).

Seminario 1, IMAG, 4/10/2024 12:00

Speaker: Marc Vidal Badia (Department of Data Analysis, Ghent University.
Department of Neurology, Max Planck Institute for Human Cognitive and Brain Sciences)

Title: Functional Data Analysis for Complex Neuroscientific Data

Abstract: Neurophysiological monitoring techniques generate data that is both abundant and complex, typically reflecting brain processes across spatial and temporal dimensions. This data often spans multiple time points (longitudinal) and various experimental conditions. These characteristics naturally give rise to functional or multivariate functional data, which are becoming increasingly prevalent in modern neuroscience research. I will review various applications of functional data in neuroscience, exploring how complexity introduces intricate relationships between subsets of data that go beyond the capabilities of traditional regression models. Additionally, since brain data is embedded within geometrical structures ranging from simple to highly sophisticated forms, I will also discuss how functional data analysis can be adapted to handle these geometries, enabling more accurate modeling of brain processes across spatial and temporal scales.

STUDENTS SEMINAR

By: Marcos P. Cavalcante (UFAL, Brasil)

October 1, 2, from 12:30 to 14:00

Title: Elementary Operators in JB∗-algebras

By Gerardo M. Escolano

Seminario 1

Title: Holomorphic approximation theory

By Tjasa Vrhovnik

SEMINARIO DE GEOMETRÍA

SEMINARIO DE ANÁLISIS FUNCIONAL

Conferenciante: Prof. Bojan Kuzma (University of Primorska, Eslovenia)

Abstract: Unbounded, densely defined, closed operators are encountered in quantum physics when solving linear second-order partial differential equation with the method of variable separation (Sturm-Liouville problem) and as generators of strongly continuous one-parameter semigroups. Particularly important is their spectrum. We will present a recent result that classifies additive bijections preserving the spectrum of such operators and compare it to the classification of spectrum-preserving maps whose domain consists of all unbounded operators.

SEMINARIO DE GEOMETRÍA

Conferenciante: Benjy Firester (MIT Mathematics)

Resumen: We introduce new families of four-dimensional Ricci solitons of cohomogeneity two with collapsing ends. In a local presentation of the metric conformal to a product, we reduce the soliton equation to a degenerate Monge-Ampère equation for the conformal factor coupled with ODEs. We exhibit explicit solutions and obtain abstract existence results for complete expanding solitons and singular shrinking and steady ones. These families of Ricci solitons specialize to classical examples of Einstein and soliton metrics.

SEMINARIO DE ECUACIONES DIFERENCIALES

Conferenciante: Cihangir Özemi (Istanbul Technical University, Türkiye)

Abstract: The analysis of differential equations using Lie symmetries is an efficient method that has been used extensively in the literature, especially for nonlinear equations where the systematic tools available for analytical solutions are limited. The study of symmetry groups of differential equations reveals the symmetries of the equations originating from the physical nature of the problem. Determining the symmetry groups and algebras that leave the equations invariant, finding group-invariant exact analytical solutions, and classifying families of equations according to their symmetry algebras are some of the approaches used in this field. In this talk I will present results on Lie symmetry algebras of several equations of mathematical physics, including multidimensional Benney-Roskes system, the Rosenau equation and a higher order Boussinesq equation. Besides the invariance algebras I will mention several exact solutions, some of which have physical importance, like line solitons and lump solutions.

SEMINARIO DE ECUACIONES DIFERENCIALES

Conferenciante: Haroune Houamed (New York University Abu Dhabi)

Abstract: We prove the existence and uniqueness of Yudovich-type solutions to the Euler equations for a two-dimensional incompressible fluid coupled with the full set of Maxwell’s equations for electromagnetism. This is shown to hold under the condition that the speed of light is sufficiently large (compared to the velocity of the plasma, loosely speaking). Moreover, due to a refined analysis of the dispersive (–dissipative) properties of Maxwell’s equations, involving suitable high–low–frequency cutoff (in terms of the speed of light), the bounds on the solution are shown to be uniform with respect to the speed of light. This matter of fact allows us to derive (a simple version of) the MHD system as the speed of light tends to infinity. In our setting, the convergence of the solution in strong topologies comes with a rate and it relies on a robust stability mechanism of perturbations of the Euler equations in Yudovich’s class combined with a fine study of the behavior (evanescence) of the electric field in the non-relativistic regime. This is based on joint work with Diogo Arsenio from NYUAD.