Activities

Superficies mínimas: Introducción a la teoría estructural de Colding y Minicozzi.

Superficies mínimas: Introducción a la teoría estructural de Colding y Minicozzi.

Abstract: The dynamics of spatially-structured networks of N interacting stochastic neurons can be described by deterministic population equations in the mean-field limit. While this is known, a general question has remained unanswered: does synaptic weight scaling suffice, by itself, to guarantee the convergence of network dynamics to a deterministic population equation, even when networks are not assumed to be homogeneous or spatially structured? In this work, we consider networks of stochastic integrate-and-fire neurons with arbitrary synaptic weights satisfying a O(1/N) scaling condition. Borrowing results from the theory of dense graph limits, or graphons, we prove that, as N tends to infinity, and up to the extraction of a subsequence, the empirical measure of the neurons' membrane potentials converges to the solution of a spatially-extended mean-field partial differential equation (PDE). Our proof requires analytical techniques that go beyond standard propagation of chaos methods. In particular, we introduce a weak metric that depends on the dense graph limit kernel and we show how the weak convergence of the initial data can be obtained by propagating the regularity of the limit kernel along the dual-backward equation associated with the spatially-extended mean-field PDE. Overall, this result invites us to reinterpret spatially-extended population equations as universal mean-field limits of networks of neurons with O(1/N) synaptic weight scaling. This work was done in collaboration with Pierre-Emmanuel Jabin (Penn State) and Datong Zhou (Sorbonne Université).

Abstract: The De Giorgi-Nash-Moser framework provides a powerful regularity theory for elliptic and parabolic equations. Applying these techniques to evolutionary partial integro-differential equations presents major hurdles, as nonlocal operators do not readily align with local estimates. In this talk, we present a weak Harnack inequality for supersolutions to a class of evolutionary partial integro-differential equations. Because the problem lacks a natural scaling, identifying the appropriate geometry constitutes a primary challenge. Furthermore, we address recent results on the strong Harnack inequality for globally positive local solutions to a general class of time-nonlocal subdiffusion equations. While the classical Harnack inequality holds for globally nonnegative local solutions in purely time-dependent settings, it is known to fail in spatial dimensions greater than one. Our results settle this question for the one-dimensional spatial case, filling a critical gap in the literature and illustrating the decisive role that spatial dimension plays in the validity of the Harnack estimate. It is a joint work with Adam Kubica and Rico Zacher.

Abstract: A shape monotonicity problem asks which shape yields a better value for a given functional. As a model case, we consider eccentric annuli of fixed radii and study the monotonicity of the first Dirichlet eigenvalue with respect to the distance between the centers. We further investigate shape monotonicity problems for eccentric annuli with mixed boundary conditions. To study these problems, we analyze the associated heat kernels and develop probabilistic methods.

• Date: Thursday, September 24.
• Time: 18:30h.
• Place: Aula A-10, Facultad de Ciencias, UGR.
• Area: GOYA Seminar.
• Language: English.
• Speakers: Cristina Rodríguez Perales (Universidad de Almería, Spain).
• Title: Nevai-Sobolev orthogonal polynomials: zeros, local asymptotics and differential properties.
• Abstract: In this talk, we consider the orthogonal polynomials \(\{q_n\}_{n\geq 0}\) with respect to a Sobolev inner product involving the Freud weight \(\exp(-x^4)\). This weight is a particular case of the Freud weights and, due to its historical relevance, we will refer to it as Nevai weight. So that, we consider the inner product
  \[
  (p,q)_S=\int_{-\infty}^\infty p(x)q(x)\,\exp(-x^4)\,dx+\lambda\int_{-\infty}^\infty p'(x)q'(x)\exp(-x^4)\,dx,\quad \lambda>0.
  \]
  Our first objective is to study the local asymptotics known as Mehler--Heine asymptotics for this family of polynomials. Moreover, this result together with the well-known Hurwitz's Theorem allows us to establish the asymptotic behavior of the zeros of these polynomials. Motivated by this consequence, we also address the problem of computing the zeros of the polynomials \(\{q_n\}_{n\geq 0}\). With that purpose, we construct a generalized eigenvalue problem from the five-term recurrence relation satisfied by these polynomials, so that the computation of the zeros is equivalent to computing the generalized eigenvalues. We also present numerical experiments illustrating the computation of the zeros for different values of \(\lambda\). Finally, if we have time we will deduce the fourth-order differential equation satisfied by these polynomials.

  Joint work with Juan F. Mañas-Mañas and Juan J. Moreno-Balcázar.

• Date: Thursday September 29th.
• Time: 12:30h - 13:30h.
• Place: Gauss Seminar, IMAG, UGR.
• Area: Geometry Seminar.
• Speaker: Ebru Yanik (Ankara University, Turkey).
• Title: Elliptic Helicoidal Translators in Minkowski 3-space.
• Abstract: Abstract will be available soon.

• Date: Thursday, September 24.
• Time: 17:30h.
• Place: Aula A-10, Facultad de Ciencias, UGR.
• Area: GOYA Seminar.
• Language: English.
• Speakers: Misael E. Marriaga (Universidad Rey Juan Carlos, Madrid, Spain).
• Title: Generalized symmetric classical orthogonal polynomials in two variables.
• Abstract: In this talk I will discuss bivariate symmetric orthogonal polynomials governed by reflection-type interaction weights. In several variables, the geometry of the domain and the symmetry of the weight determine not only the orthogonal basis, but also the natural differential operators acting on it. I will focus on weights with singular interaction factors along reflection walls and on the corresponding orthogonality problems on fundamental chambers.

  The main theme is an operator-theoretic construction of these systems. For product domains associated with the classical Hermite, Laguerre and Jacobi weights, for the unit disk through generalized Zernike polynomials, and for the unit triangle with Jacobi-type weights, the symmetric orthogonal families arise as eigenfunctions of self-adjoint second-order differential operators.

  I will explain how lowering and raising operators shift degrees and parameters, and how their compositions produce additional commuting operators. The passage to elementary symmetric variables plays a central role: apparent rational singularities disappear and the operators acquire polynomial coefficients. This leads to explicit descriptions of the differential-operator algebras for which the transformed polynomials form a joint eigenbasis, revealing a common Koornwinder-type structure behind these bivariate models with interaction terms.

  Join work with Miguel A. Piñar and Gema Alhama (Universidad de Granada, Granada, Spain).

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Conferenciante: Plamen Iliev