Activities

Abstract: The geometric formulation of thermodynamics has a long history, from Gibbs to Carathéodory to the more recent developments in contact and information geometry. In this talk I will give a panoramic view of the latter, focusing on how differential geometry provides a natural language for understanding equilibrium and non‑equilibrium phenomena. I will begin by introducing the structures underlying information geometry, presenting several recent results along the way. The talk will be self‑contained and aims to show how geometry offers a unified framework to study thermodynamics, stochastic dynamics, and information theory.

In 1919, Besicovitch constructed a compact set in the plane with Lebesgue measure 0 that contains a unit line segment pointing in every direction. Such objects are now called measure 0 Besicovitch sets (aka Kakeya sets). By replacing a measure zero Besicovitch set by its delta-thickening, one obtains a collection of 1 x delta rectangles pointing in different directions, the sum of whose areas is 1, but whose union has very small volume. The existence of such collections of
rectangles is called the Besicovitch compression phenomenon. The Kakeya set conjecture is a quantitative statement controlling the strength of the Besicovitch compression phenomenon. In this talk, I will discuss connections between the Besicovitch compression phenomenon, the Kakeya set conjecture, and questions in harmonic analysis and PDE.

Abstract

Given an Anosov flow \(\phi\) on a 3-manifold \(M\), a periodic orbit \(\gamma\) and an integer \(m\), there is an operation, called Fried surgery, that allows to construct a new 3-manifold \(N\) endowed with an Anosov flow \(\psi\). This operation is a Dehn surgery of slope \(=m\) along the simple closed curve \(\gamma\), but adapted to the pair (Anosov flow, 3-mfld) in such a way that the new manifold is also equipped with an Anosov flow.

Starting from the suspension flow generated by the matrix $$A=\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$$ and making Fried surgeries along its periodic orbits, we can obtain many different Anosov flows, including other suspension flows, some geodesic flows or even flows on hyperbolic 3-manifolds. It is an open problem, however, to determine whether every transitive and orientable Anosov flow can be obtained by doing Fried surgeries on the suspension of $$A=\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$$.

The aim of this talk is to show how, by introducing techniques coming from affine geometry on surfaces, we can relate the previous problem with a problem of realizability of quadratic rotations by piecewise affine maps, whose derivatives and breaking points are in a field extension of the rationals by a Perron number.

• 📅 martes 16 de junio — 11:00
• 📍 Aula A25, Facultad de Ciencias
• 👤 Conferenciante: Jorge Ruiz Cases (Universidad Autónoma de Madrid)
• 📖 Título: Gradual smoothing: strong hypercontractivity and logarithmic Sobolev inequalities

📄 Resumen:

In this talk we study the possibility of a gradual improvement as time progresses of the regularity of solutions to evolution problems of parabolic type driven by Lévy-type operators, not necessarily translation invariant. In the course of our analysis we study the equivalence between general smoothing effects and a family of logarithmic Sobolev inequalities. This equivalence allows us to identify a new type of regularization, strong hypercontractivity, characterized by the existence of a time at which solutions belong to every $L^p$ space with $p$ finite. It can also be used to prove logarithmic Sobolev inequalities in a context not previously seen in the literature.

We then show that any purely nonlocal Lévy-type operator whose kernel is comparable to that of $\log(I-\Delta)$ is strongly hypercontractive, but fails to be supercontractive and, consequently, also fails to be ultracontractive. Furthermore, in the translation-invariant case, we also prove that solutions get bounded eventually and start improving in differentiability right after doing so.

Finally, we show that this behaviour only appears if the kernel defining the operator behaves as $|x-y|^{-N}$ for small interactions ($0^+$-order operators): more singular kernels yield instantaneous smoothing, while less singular ones do not regularize.

• IMAG Functional Analysis Seminar
• Title: Characterizing Jordan maps through products at a fixed element
• By Jorge J. Garcés (UNED)
• Abstract: We shall present some new and old results on linear maps that (in some sense) behave as a Jordan homomorphism or a Jordan derivation locally. More generally, by paralleling the theory of Jordan zero-product determined algebras, we study von Neumann algebras that are square-zero determined or determined by Jordan products at a fixed point and show that all properly infinite von Neumann factors are both square-zero determined and determined by Jordan zero-products at any fixed point. Finally, we shall discuss a generalisation of these results to more general classes of von Neumann algebras.Part of the results of this talk are from a joint work with Mykola Khrypchenko https://doi.org/10.1016/j.laa.2025.09.013

• IMAG Functional Analysis Seminar
• Title: Two classical concepts under the Ky Fan metric
• By Willian Franca (Federal University of Juiz de Fora, Brazil)
• Abstract: In this talk, we study measurable functions modulo equality almost everywhere via the Ky Fan metric, a standard metric that encodes convergence in measure. Two themes guide the talk. First, we present the notion of  boundedness in measure and explain how it isolates the largest linear subspace of $L_0$ that still carries a compatible topological vector space structure; we also discuss completeness and algebraic properties under pointwise multiplication. Second, on a measure space equipped with a measurable exhaustion, we consider convergence in measure at infinity. We present criteria guaranteeing that an ``asymptotic value'' (when it exists) is well-defined, and we outline some properties of the subspace of $L_0$ consisting of functions that admit a limit at infinity. In the atomic setting we describe how certain numerical invariants of the measure control when these spaces become locally convex, Fr\'echet, or even Banach, and we indicate how familiar classical function and sequence spaces fit into this metric framework.

• IMAG Functional Analysis Seminar
• Title: Polyhedral normed spaces
• By Helena del Río (UGR)
• Abstract: A normed space is said to be polyhedral if the unit ball of each of its finite dimensional subspaces is a polytope. A celebrated result by Fonf, nowadays known as the "Structural Theorem", states that the unit ball of every polyhedral Banach space can be covered by faces with non-empty topological interior. In this talk we provide an extension of this theorem to polyhedral normed spaces, as well as a weaker version considering faces with non-empty algebraic interior. Several examples show that without completeness Klee's polyhedrality is no longer sufficient to recover Fonf's result and stronger types of polyhedrality are needed, even for the algebraic version. As an application, we characterize when a normed space has a locally finite tiling by bounded bodies. The talk concludes with some renorming results which show how diverse the situation can be in the incomplete case.

The seminar will review and critically discuss the development of Multiple Frame (MF) surveys over more than six decades, from their first appearance in the sampling literature of the 1960s–1970s to the re-assessment of their role in today’s fast-evolving landscape of digital data sources and emerging data needs. An MF survey uses two or more sampling frames, usually with partial coverage and overlapping units, which together adequately cover the target population. MF surveys are especially useful when no single complete frame exists and no sufficient resources are available to construct one via record linkage. They can also be cost-effective even when a complete frame is available, which was, in fact, the practical motivation behind their initial introduction in the sampling statistics literature. Since then, MF surveys have undergone a significant and resilient evolution in their objectives, application areas, and estimation approaches. The initial cost-saving motivation has expanded to include applications to difficult-to-sample populations (e.g., undocumented immigrants), the handling of attrition in longitudinal studies, agricultural surveys combining satellite and field data, and MF-based strategies addressing long-standing challenges in sample surveys, such as rising costs and declining response rates. A parallel evolution in MF estimation has also occurred, mainly due to the inherent methodological difficulties associated with complex survey designs and notational cumbersomeness. This initially led to several seemingly disconnected estimation approaches, which can now be simplified and unified under the Multiplicity approach and the Generalized Multiplicity-Adjusted Horvitz–Thompson framework. The unprecedented availability of new digital and Big Data sources, together with the recent surge of interest in combining information from probability and non-probability samples to address budget constraints, the demand for timely estimates, and lack of respondent cooperation, calls for a re-assessment of MF practices and outlines directions for future research.

Día: 26 de mayo

Hora: 12:00

Aula: Seminario Gauss (1 planta)

Seminario GOYA

Conferenciante: Federico Nudo, Università Della Calabria

Título: Bivariate polynomial histopolation techniques

Abstract:

This work explores the reconstruction of a real-valued function $f$ defined over a domain $\Omega \subset \mathbb{R}^2$ using bivariate polynomials that satisfy triangular histopolation conditions. More precisely, we assume that only the averages of $f$ over a given triangulation $\mathcal{T}_N$ of $\Omega$ are available and seek a bivariate polynomial that approximates $f$ using a histopolation approach, potentially flanked by an additional regression technique. This methodology relies on the selection of a subset of triangles $\mathcal{T}_M \subset \mathcal{T}_N$ for histopolation, ensuring both the solvability and the well-conditioning of the problem. The remaining triangles can potentially be used to enhance the accuracy of the polynomial approximation through a simultaneous regression. We will introduce histopolation and combined histopolation-regression methods using the Padua points, discrete Leja sequences, and approximate Fekete nodes. The proposed algorithms are implemented and evaluated through numerical experiments that demonstrate their effectiveness in function approximation.

📅 martes 19 de mayo — 11:00

📍 Aula A13, Facultad de Ciencias

👤 Conferenciante: Zineb Hassainia (Universidad de Granada)

📖 Título: Leapfrogging vortex rings for 3D Euler equations

📄 Resumen: In this talk, I will present a rigorous construction of time-periodic leapfrogging vortex rings for the three-dimensional incompressible Euler equations. More precisely, we prove the existence of solutions in which two coaxial vortex rings periodically exchange positions, as observed in experiments and numerical simulations. The construction relies on a desingularization of two interacting vortex filaments within the contour dynamics formulation, leading to a Hamiltonian description of nearly concentric vortex rings. A central difficulty arises from a singular small-divisor problem in the linearized dynamics, where the effective time scale degenerates with the ring thickness parameter. We overcome this issue by combining a degenerate KAM-type analysis with pseudo-differential techniques and a Nash-Moser iteration scheme. This approach yields families of nontrivial time-periodic solutions in an almost uniformly translating frame and thereby provides a rigorous mathematical construction of classical leapfrogging motion for axisymmetric Euler flows without swirl, with no restriction on the time interval of existence. The talk is based on a joint work with Claudia García and Taoufik Hmidi.