RESUMEN: Un conjunto de números naturales A se dice b-automático si es posible determinar si un número n pertenece a A examinando su expansión en base b mediante un autómata finito (un tipo muy sencillo de "computador"). Uno de los teoremas centrales de la teoría de conjuntos automáticos es el teorema de Cobham, el cual señala que un conjunto que es automático en varias bases independientes necesariamente es muy parecido a una progresión aritmética (es "casi periódico"). Este teorema ha sido generalizado en una gran variedad de contextos, incluyendo el punto de vista lógico y el de los sistemas dinámicos.

Entre estas posibles vías de generalización, se ha dedicado mucho trabajo a la búsqueda de teoremas análogos en otros (semi-)anillos que admiten sistemas de numeración. Intentos previos de encontrar este tipo de generalizaciones comenzaban desde la hipótesis de que este tipo de teoremas requerían un conocimiento fuerte de la estructura del anillo subyacente, y por ende dependían de conjeturas profundas aún no resueltas de teoría de números. Basándonos en una demostración alternativa del teorema original propuesta por Thijmen Krebs, en esta charla discutiremos una generalización completa del teorema de Cobham al anillo Z[i] de los enteros Gaussianos, la cual se basa en argumentos relativamente elementales. Esto sugiere que dichas consideraciones de teoría de números podrían no ser necesarias en general. Este trabajo ha sido realizado en conjunto con Reem Yassawi y Robbert Fokkink.

EXPOSITOR: Álvaro Bustos Gajardo (Universidad de Chile)

Abstract: Idania Briceño is a geographer from the University of Los Andes in Venezuela. She holds a master’s degree in Remote Sensing from the Universidad Mayor of Chile, where she is a full-time faculty member, and is a Ph.D. candidate in Geomatics at the Polytechnic University of Valencia. In this seminar, she will present the MONCOSTA project, which has developed a set of technological and scientific solutions that strengthen integrated coastal management and improve adaptive capacity in the face of current and future environmental challenges.

Speaker: Idania Briceño, geographer and Ph.D. candidate in geomatics

Live stream on YouTube: https://go.cmm.uchile.cl/seminario10julio

Abstract: The Ernst equation arises from the dimensional reduction of the vacuum Einstein equations in the presence of two spacelike Killing vector fields and plays an important role in the description of stationary axisymmetric spacetimes. Originally introduced as a simplified scheme for constructing the Kerr metric, it has become a fundamental tool in the study of stationary vacuum solutions. More generally, its integrability provides a powerful mechanism for constructing large families of physically relevant solutions, including the classical black hole spacetimes.

In this work, we introduce a profile–phase decomposition of the Ernst equation that naturally induces a system of nonlinear differential operators encoding its intrinsic analytic structure. Rather than relying on a prescribed solution-generating ansatz, the induced operator system determines a broad class of admissible solutions through nonlinear differential constraints, providing an intrinsic description of the corresponding solution space. Within this framework, we establish concrete characterizations of black hole solution classes in terms of the induced nonlinear differential system. In particular, the Schwarzschild, Kerr, and Kerr–NUT spacetimes arise naturally as particular instances of the resulting solution class. This is ongoing work carried out jointly with Felipe Poblete.

Speaker: Jessica Trespalacios (Universidad Austral de Chile)

Abstract: 

Joint work with Juhan Aru, Christophe Garban, and Aleksandra Krozhenkova.

In this presentation, we will discuss the relationship between the Gaussian free field (GFF) and the spin O(N) model. In particular, we will see how correlation functions can be expressed as a percolation problem, and how we can utilize techniques developed for the GFF to study the spin O(N) model, and vice versa.

Speaker: Avelio Sepúlveda (Universidad de Chile/CMM)

The Zoom link to join the seminar is:
https://reuna.zoom.us/j/84521834914?pwd=OTZ6Y0NWM3pYTGtTbEt3c0luTG96UT09

Meeting ID: 845 2183 4914
Passcode: 997973

Abstract:  We propose a simple mechanism by which scaling laws emerge from feature learning in multi-layer networks. We study a high-dimensional hierarchical target that is a globally high-degree function, but that can be represented by a combination of latent compositional features whose weights decrease as a power law. We show that a layer-wise spectral algorithm adapted to this compositional structure achieves improved scaling relative to shallow, non-adaptive methods, and recovers the latent directions sequentially: strong features become detectable at small sample sizes, while weaker features require more data.

We prove sharp feature-wise recovery thresholds and show that aggregating these transitions yields an explicit power-law decay of the prediction error. Technically, the analysis relies on random matrix methods and a resolvent-based perturbation argument, which gives matching upper and lower bounds for individual eigenvector recovery beyond what standard gap-based perturbation bounds provide.

Numerical experiments confirm the predicted sequential recovery, finite-size smoothing of the thresholds, and separation from non-hierarchical kernel baselines. Together, these results show how smooth scaling laws can emerge from a cascade of sharp feature-learning transitions.                                         

Speaker: Arie Wortsman Zurich, Center for Data Science, ENS Paris

Abstract: 

Dynamical systems are an area of mathematics that studies how processes evolve over time. One of the main perspectives within this field is Ergodic Theory, which studies dynamical systems from the point of view of measure theory. Within ergodic theory, one of the main elements of study currently is the Lyapunov exponents.

Lyapunov exponents quantify the dependence of the system on its initial conditions. One of the central problems regarding Lyapunov exponents is the study of their continuity, that is, to understand how much the Lyapunov exponents change under small perturbations of the system’s parameters. This question is important because it allows us to better understand the behavior of dynamical systems with close parameters.

In this talk, we will introduce the concepts of Lyapunov exponents, dynamical systems, and linear cocycles. We will also discuss the possibility of obtaining continuity results and moduli of continuity for Lyapunov exponents in certain classes of dynamical systems.

Speaker: Aline Melo (PUCV)

Resumen: La extensión natural es una construcción clásica en teoría ergódica que asocia a cada sistema no-invertible un sistema invertible que lo extiende. En esta charla formalizaremos la noción de extensión invertible para el caso general de acciones de semigrupos que preservan una medida de probabilidad. Luego estudiaremos el problema de la existencia de extensiones invertibles, y analizaremos casos en los que se tiene y no se tiene unicidad de éstas.

Expositor: Miguel Donoso-Echenique (Universität Münster)

Abstract: In recent years, the integration of optimization techniques with machine learning paradigms has led to significant advances in solving inverse problems, particularly through the optimal selection of parameters or observation strategies. A rigorous and systematic approach to tackle such problems is provided by bilevel optimization, where the lower-level problem corresponds to a model-based inverse problem—often governed by a partial differential equation (PDE)—and the upper-level objective encodes a data-driven loss functional, typically defined over a training set. Unlike classical variational models, the presence of PDE constraints introduces substantial analytical and computational challenges. In this talk, I will present a bilevel optimization framework for PDE-constrained inverse problems, and discuss recent results on the existence of optimal parameters. Furthermore, I will explore the possibility of reformulating the bilevel problem as a single-level optimization problem under suitable assumptions. Finally, I will derive and discuss first-order optimality conditions for the resulting problem, highlighting key mathematical difficulties and future directions.

Speaker: Juan Carlos De Los Reyes (Centro de Modelización Matemática MODEMAT y Escuela Politécnica Nacional, Quito).

The link to join the seminar is:
https://uchile.zoom.us/j/98721784149?pwd=df7YyCDGwMXlNuDX0kwKvob08edgk7.1

Los invitamos a participar en el webinar del Plan Nacional Sumo Primero: “Estudio de la Geometría en Sumo Primero”, junto al profesor Masami Isoda, académico de la Universidad de Tsukuba (Japón) y autor de la colección de textos Sumo Primero.

Durante este encuentro profundizaremos en la propuesta para la enseñanza de la geometría presente en la colección, revisando la progresión de los aprendizajes y orientaciones para su implementación en el aula.

Inscríbete aquí: https://uchile.zoom.us/webinar/register/WN_Sun7Pe5wT8mMediwUAjOww#/registration

Speaker: Mathieu Dedenon, University of Geneva, Swizerland