Abstract: El problema de empaquetamiento en grafos aleatorios consiste en encontrar copias arista-disjuntas de una colección de grafos en $G(n,p)$. Ferber y Samotij demostraron que una colección de árboles de grado máximo acotado y tamaño a lo más $(1-\alpha)n$ se empaqueta en $G(n,p)$ con alta probabilidad, bajo condiciones adecuadas sobre $p$ y el  grado máximo. En esta presentación se extiende dicho resultado al contexto de hiperárboles tight $k-$uniformes. Se considera el hipergrafo aleatorio $G^{(k)}(n,p)$, en el cual cada $k-$subconjunto de $[n]$ forma una hiperarista de manera independiente con probabilidad $p$ y se estudia el empaquetamiento de hiperárboles tight $k-$uniformes, esto es, hiperárboles que admiten un orden de sus vértices en el que cada nuevo vértice se incorpora junto con un conjunto padre de $k-1$ vértices ya presentes, contenido en alguna hiperarista previa.

El resultado principal establece lo siguiente. Sea  \(k \geqslant 2\) un entero fijo, $\alpha, \varepsilon, p \in (0, 1/2)$ y $n$ un entero positivo que satisfacen $\frac{20k^2 (\log n)^2}{\alpha \varepsilon n} \leqslant p \leqslant \frac{\varepsilon \alpha^{2k+1}}{120k^2}$ y $\Delta \leqslant \min\left\{ \alpha, \frac{\varepsilon}{\log(1/\alpha)} \right\} \cdot \frac{\varepsilon n p}{20k^2 \log n}$.Sean $T_1, \dots, T_N$, con $N \leqslant (1-\varepsilon)p\binom{n}{k}/n$, una colección de hiperárboles tight $k$-uniformes. Se asume que cada $T_i$ tiene a lo más $(1-\alpha)n$ vértices y que su co-grado máximo es a lo más $\Delta$. Entonces, con alta probabilidad, los hiperárboles $T_1, \dots, T_N$ se empaquetan en $G_{n,p}^{(k)}$.

Speaker: Gisela Abarca (DIM U. Chile)

Speaker: David Salas (UOH & CMM)

Abstract: Bilevel programming was first introduced in 1934 and gain a lot of interest in the 70’s due to its large modeling capability. The model has been heavily studied for the last 50 years and many generalizations have been introduced presenting new difficulties to overcome. Or that is what we thought. In 2020, B.Zeng showed that pessimistic bilevel programming, a variant we thought was way harder, can be reformulated as the usual optimistic one. Starting from this point, the community of bilevel programming has started to question the beliefs about harder variants. In this presentation we will discuss the seminal work of Zeng, the recent discussion about coupling constraints, reductions of Generalized Nash Equilibrium problems and finally the discussion of wether Single-Leader-Multi-Follower games represent a harder model or not with respect to bilevel programming.

Speaker: Andrés Carvajal (Departamento de Ingeniería Industrial, Universidad de Chile)

Abstract: We bring the notion of “satisficing” behavior to matching theory and develop the resulting solution concept, which is weaker than stability. Satisficing behavior is an alternative to maximizing behavior where, instead of always going for the best alternative, “satisficers” settle on choices that are “good enough”. We say that a matching is satisficing if every agent is matched to an achievable partner. (An agent is achievable to another if they are matched in some stable matching.)

We show that satisficing matchings have the following properties: (i) they are Pareto efficient; (ii) all women (men) weakly prefer the woman-optimal (man-optimal) stable matching to any satisficing matching; (iii) for any two distinct satisficing matchings, there exist two pairs, each formed under one of the two matchings, such that the agents in each pair have opposite preferences over the two matchings; (iv) if agents are asked to vote between a stable matching and a satisficing matching, the two matchings tie; and (v) truthtelling is a rationalizable strategy for satisficing mechanisms.

We also argue that our definition is falsifiable upon the observation of all satisficing matchings, and discuss how existing algorithms can be used for the computation of the set of satisficing matchings. Finally, using as operator an extension, via the join, of the least upper bound of the common order of one gender, we show that the set of satisficing matchings forms a semigroup one of whose ideals is the set of stable matchings.

You can also join via Zoom: 
https://uchile.zoom.us/j/98627160934?pwd=gOah5m9RbnosbtZq5sSsv3qfqFstmk.1

EXPOSITOR(A) Maximiliano Escayola (Korea Institute for Advanced Study)

RESUMEN En esta charla presentaré un resultado en colaboración con Carl-Fredrik Nyberg-Brodda, en el que construimos grupos finitamente generados de difeomorfismos de la esfera n-dimensional con regularidad crítica prescrita. Esto es, dado un número real r > 0, construimos un grupo que admite una acción fiel por difeomorfismos de clase C^r, pero que no admite ninguna acción fiel por difeomorfismos de clase C^s, para ningún s > r.

Abstract: Dramatic progress in recent years in our understanding of core-collapse supernovae has, for the first time, allowed us to connect supernova progenitors with their explosions and compact remnants. Among other insights, this framework yields unique predictions for the black-hole mass spectrum, which can be directly tested through LIGO’s gravitational-wave detections of binary black-hole mergers.

Specifically, theory predicts a peak in the black-hole mass distribution around 9 solar masses (M☉), driven by late nuclear burning phases—specifically carbon burning and the onset of neon burning—and showing only a weak dependence on metallicity. This leads to a predicted peak in the LIGO chirp-mass distribution around 8 M☉, followed by a distinct gap or deficiency of systems just beyond.

Using the binary population synthesis code COMPAS, we generated detailed predictions for observational rates. The recent release of results from the LIGO/Virgo/KAGRA O4a run brings the total number of detected black-hole mergers to 158. The observed chirp-mass distribution displays a peak-gap structure in striking agreement with our predictions, effectively ruling out several commonly used black-hole formation prescriptions.

This match confirms our foundational understanding of late stellar evolution and key aspects of the core-collapse mechanism. Furthermore, it demonstrates that forming black holes in binaries is significantly more difficult than often assumed in the literature, naturally explaining observational puzzles such as the lack of Be X-ray binaries containing black holes. This lecture will present these findings alongside broader implications for modeling black-hole binary systems.

Speaker: Prof. Philipp Podsiadlowski, research of London Centre for Stellar Astrophysics & University of Oxford

Expositor: Christopher Cabezas (CMM)

Abstract: In this talk we discuss symbolic dynamical systems over countable semigroups, where the shift maps are non-invertible. We relate how algebraic and combinatorial properties of the acting semigroup govern topological and dynamical properties on symbolic systems. In particular, we focus on the case where the shift maps are open. We provide a complete combinatorial description of such subshifts, thereby extending Parry’s classical result for one-dimensional subshifts through a notion of semigroup synchronization. We then apply this framework to several classes of semigroups, obtaining structural descriptions of the subshifts of finite type they admit and vice versa.

Speaker: Alonso Núñez (DIM UChile)

Abstract: La conjetura de Erdős–Sós, formulada en 1963, afirma que todo grafo con grado medio mayor estricto a $k-1$ contiene como subgrafo todo árbol con $k$ aristas. A pesar de la sencillez de su enunciado, el problema permaneció abierto durante más de seis décadas y dio lugar a numerosos resultados parciales, versiones aproximadas y nuevas técnicas para el estudio del embebimiento de árboles en grafos densos. En esta presentación hablaremos brevemente sobre algunos de los principales hitos en la historia de la conjetura y luego estudiamos una demostración reciente obtenida por GPT-6 Astra, que resuelve el problema general mediante un argumento de conteo que evita tanto el lema de regularidad como los métodos probabilísticos.

Hora/lugar: Jueves 8 Octubre, 10.30 sala John con Neumann

Abstract: In this talk we consider a semi-discrete forward stochastic parabolic operator with homogeneous Dirichlet conditions in arbitrary dimensions. We show the lack of null controllability for a spatial semi-discretization of a null-controllable stochastic parabolic system from any initial datum. However, by proving a new Carleman estimate for its semi-discrete backward stochastic adjoint system, we achieve a relaxed observability inequality, which is applied to derivative phi-null controllability by duality arguments.

Speaker: Ariel Pérez Contreras, Departamento de Ingeniería Matemática, Universidad de Concepción.

FORMAT: In-person with online streaming

Abstract:  Symmetry-breaking problems play a central role in the theory of distributed computing. In this talk, we consider the three classical symmetry-breaking problems graph colouring, maximal independent set (MIS), and maximal matching (MM). Motivated by the observation that many algorithms perform significantly better on real-world networks than their worst-case guarantees suggest, we design and analyse randomised algorithms for these fundamental problems on hyperbolic random graphs (HRGs), a promising model for complex real-world networks.

We present the conceivable simplest randomised algorithm that solves distributed \Delta+1 colouring in only 2 rounds (https://arxiv.org/abs/2505.19109). We also present a structural property of HRGs that separates graph colouring from MIS and MM: unlike graph colouring, which can be solved in 2 rounds, both MIS and MM require \log^{\Theta(1)}\log n rounds on
HRGs (https://arxiv.org/abs/2607.09170).

Speakers: Janosch Ruff, University of Potsdam, Alemania.

Abstract
Regularization controls model complexity, most commonly by adding a penalty to the training objective. One geometrically motivated instance is manifold regularization, which assumes that the data lies near a low-dimensional submanifold and penalizes functions that vary along it. However, this assumption is fragile. In high dimensions, the presumed manifold is difficult to estimate. Furthermore, for non-metric data, the manifold may not even exist.

We propose a regularization approach that maintains the geometric motivation while disregarding the manifold assumption. Our approach is based on the concept of intrinsic dimension, which stems from the concentration of measure phenomenon in metric measure spaces. The central tool in this approach is Gromov´s observable diameter, which measures how much of a space´s geometry remains visible under real-valued observation.

When applied layer by layer to a neural network, the observable diameter yields a computable, distribution- sensitive estimate of the effective degrees of freedom carried by each representation. This approach eliminates the need to posit or reconstruct an underlying manifold.
We use this quantity to regularize learning. It identifies the directions and components that genuinely contribute to the intrinsic dimension per layer. This gives a principled criterion for controlling complexity and pruning overparameterized models. Finally, werelate this layer-wise geometric view to the question of when and why regularization aids generalization.

Speaker: 
Tom Hanika is Co-Director of the Information Systems and Machine Learning Lab at University of Hildesheim, Research Associate at University of Kassel and Temporary Lecturer at Humboldt-Universität zu Berlin. His research bridges the mathematical foundations of machine learning with practical applications in explainable Artificial Intelligence. He specializes in the interactive extraction of knowledge from complex explicit and implicit data, ensuring that learning systems are both highly effective and structurally sound.

At the core of his foundational work is the study of semantics arising from implicational theories, with a particular emphasis on the interplay between metrics, orders and measures. He is especially driven by the mathematical challenges of learning in high dimensions, as these push the boundaries of what current machine learning models can achieve. On the applied side, he explores unsupervised learning from large text corpora and actively translates his theoretical methods into real-world solutions. He frequently collaborates to apply explanation techniques across diverse fields, including Biology, Geography, Physics, and the Digital Humanities. Beyond research, he is a senior developer for the social publication sharing system BibSonomy and maintains the Formal Concept Analysis tool conexp-clj.

En esta sesión se presentará el tema “Monitoreo satelital de humedales en Chile”. El expositor será Jonathan Ruiz Apablaza, académico investigador de la Universidad Bernardo O’Higgins (UBO) y analista de Investigación y Desarrollo del Instituto Geográfico Militar de Chile.

Wetland Monitor Chile es una plataforma de inteligencia geoespacial diseñada para la evaluación multitemporal y el monitoreo continuo de humedales a escala nacional. El sistema opera mediante la ingesta y procesamiento directo de imágenes de la constelación Sentinel-2 (L2A) a través del Copernicus Data Space Ecosystem (CDSE), calculando índices biofísicos calibrados (MNDWI, NDRE, NDCI, SAVI, FAI y WRI) con filtrado de calidad a nivel de píxel basado en la capa de clasificación de escena (SCL). La solución incorpora metodologías para la reconstrucción de series de tiempo robustas ante ruido atmosférico y nubosidad, visualización interactiva de teselas espectrales y generación automatizada de reportes técnicos orientados al diagnóstico ecológico y la conservación ambiental.

El evento se realizará en la sala Maryam Mirzakhani de Centro de Modelamiento Matemático (CMM) de la Universidad de Chile (Beauchef 851, Edificio Norte, piso 6, Santiago), y se transmitirá en vivo por el canal de YouTube de CopernicusLAC Chile en este enlace .

Abstract: Optimization models are solved as if their integrands, distributions, and constraints were exact. When they are not, that is, when expectations are approximated, probabilities estimated, or market-clearing conditions inexact, minimizers may shift discontinuously or fail to exist altogether. This talk develops an approach based on Rockafellian functions: rather than solving the nominal problem, one solves a substitute problem in which the perturbation is itself a decision variable, penalized rather than pinned at zero.

I first present stability results for the family of substitute problems generated by a Rockafellian: the relaxation, its Lagrangian, and its dual function, in the nonconvex
setting. The analysis rests on epi- and hypo-convergence rather than local sensitivity, and yields rates that give Lipschitz-type stability. Two applications follow. For discrete-time contingent claims, perturbing the probability distribution, the claim, or both produces epi-convergent primal and hypo-convergent dual approximations; dual variables converge as shadow prices, the duality gap connects to the value of perfect information, and explicit examples locate the failure boundary, where critical scenarios combine vanishing probability with unbounded impact. For exchange economies admitting no Walrasian equilibrium, slackening market clearing by a penalized nonnegative variable is well posed throughout, and as the penalty grows the residual converges to a minimum-norm vector measuring the distance to the nearest equilibrium-admitting economy.

Based on joint work with J. O. Royset, W. Breytmann, and N. Hernández

Speaker: Julio Deride  (UAI)