Seminarios IMC

Espacio de encuentro y discusión en torno a preguntas matemáticas que puedan suscitar un interés general, y que puedan beneficiarse de ideas o técnicas de las diversas áreas y disciplinas presentes en el IMC.
2026-09-02
13:40hrs.
Manuel Sánchez. Instituto de Ingeniería Matemática y Computacional UC
Spectral Analysis of HDG Methods and Applications to Neural Operator-Based Local Solvers
Sala 2, Facultad de Matemáticas (Rolando Chuaqui)
Abstract:
Hybridizable Discontinuous Galerkin (HDG) methods rely on static condensation to drastically reduce globally coupled degrees of freedom. This seminar begins with a brief introduction to the HDG and hybridized mixed frameworks, followed by a rigorous spectral analysis of their statically condensed systems for second-order elliptic problems. Using novel local lifting operators and trace isometries, we establish sharp condition number estimates that explicitly capture the dependence on the mesh size $h$ and polynomial degree $k$, proving an $\mathcal{O}(h^{-2}k^2)$ scaling. Finally, we bridge this theoretical foundation with machine learning-accelerated solvers. Relying on perturbation analysis of the condensed matrix, we discuss how neural network solvers can accelerate the resolution of parametrized local solvers without compromising global stability.
 
2026-08-12
13:40hrs.
Pedro Saa. Instituto de Ingeniería Matemática y Computacional/departamento de Ingeniería Química y Bioprocesos UC
Exploración de distribuciones de flujos factibles en redes de reacciones
Sala segundo piso (DILAB), Edificio Carreras Interdisciplinarias
Abstract:
El espacio de flujos en que una red de reacciones puede operar en estacionario se representa por un politopo descrito por igualdades (balances de masa) y desigualdades (restricciones de capacidad). Dicho politopo se puede explorar eficientemente por medio de programación lineal o métodos de muestreo aleatorio en altas dimensiones. Sin embargo, distribuciones de flujo que operan en estado estacionario pueden violar leyes termodinámicas, entregando resultados que son irreales. La verificación de la factibilidad termodinámica del flujo por una reacción requiere constatar que el signo de éste sea contrario al signo de su potencial químico. Incorporación de esta última variable, y en particular, de la condición de “signos opuestos”, genera “huecos” en el politopo de flujos que hacen más difícil su exploración. Junto con presentar el problema, en esta charla esperamos discutir técnicas y herramientas matemáticas que ayuden a caracterizar mejor el espacio de flujos factibles y faciliten su exploración.
 
2026-07-08
13:40hrs.
Pablo Barceló. Instituto de Ingeniería Matemática y Computacional UC
Cuando la IA ayuda a demostrar que un problema es difícil
Sala C406 (Construcción Civil)
Abstract:
En esta charla estudiaremos la complejidad computacional del siguiente problema. Dadas dos matrices A y B cuyas entradas son variables de un conjunto X, ¿existe una asignación s de esas variables a valores booleanos (0 y 1) tal que s(A) y s(B) tengan rangos distintos?

Demostraremos que el problema es NP-completo, un resultado que obtuvimos con la asistencia de herramientas de IA. La pregunta tiene aplicaciones en lógica computacional y había permanecido abierta por casi una década.

Más allá del resultado, espero que la charla sea un espacio de reflexión sobre dos planos: la naturaleza matemática del problema y el papel de las herramientas de IA en la matemática y la ciencia de la computación.
2026-06-17
13:40hrs.
Marcelo Arenas. Departamento de Ciencia de la Computación Uc, Instituto de Ingeniería Matemática y Computacional
Contando órdenes conexos de vértices en grafos
Sala segundo piso (DILAB), Edificio Carreras Interdisciplinarias
Abstract:
Un orden de los vértices de un grafo se dice conexo si cada vértice en el orden, excepto el primero, está conectado con alguno de los vértices anteriores. Tales órdenes existen para un grafo si, y solo si, el grafo es conexo; por lo tanto, en términos de decisión, este problema puede resolverse de manera eficiente. En esta charla, mostraremos que el problema de contar el número de órdenes conexos de los vértices de un grafo es difícil. En particular, mostraremos que es #P-completo. Además, discutiremos brevemente lo que sabemos sobre la posibilidad de aproximar eficientemente este problema de conteo.
2026-06-10
13:40hrs.
Matthew Farrell. Riken (Institute of Physical and Chemical Research), Japan
How and why does the brain use multiple systems to learn individual skills?
Sala segundo piso (DILAB), Edificio Carreras Interdisciplinarias
Abstract:
It has been proposed that the brain integrates flexible, computationally expensive cortical processing with simpler, lower-cost subcortical mechanisms to achieve resource-efficient performance greater than that of either system alone. This alluring perspective is currently driving the development of theoretical frameworks that explore this hypothesis. I will present a normative framework for investigating this problem that combines three key features: (1) an environment with structure that can be leveraged to accelerate learning, (2) a nonstationary environment (i.e., changing reward contingencies), and (3) resource constraints (i.e. limited memory for the model). I will next present a simple example within this framework, showing how nonstationarity and memory constraints drive the cortical module to favor learning reward-agnostic information about the environment, and argue that this helps explain the separation of cortex and subcortical regions such as the basal ganglia, which are strongly reward-driven. Finally, I will introduce and analyze a specific mechanism by which information can be transferred from cortical to subcortical systems in the form of a Hebbian sequence-learning model.
2026-05-27
13:40hrs.
Vicente Gómez Herrera. Flatiron Research Fellow, Center for Computational Biology & Center for Computational Mathematics, Flatiron Institute
Inestabilidad de colonias de levadura sobre fluidos viscosos
Sala segundo piso (DILAB), Edificio Carreras Interdisciplinarias
Abstract:
¿Cómo cambia la forma de una colonia de levadura mientras crece sobre un fluido viscoso? En esta charla se presentará un modelo físico continuo para describir este fenómeno. Se considerará una colonia delgada que se expande sobre la superficie de un fluido y se estudiará cómo el crecimiento de los microorganismos genera fuerzas y movimientos en el líquido subyacente.
El modelo incorpora dos efectos principales: las tensiones producidas por el crecimiento de la colonia y los cambios de densidad en el fluido debidos al consumo de nutrientes. Estos mecanismos inducen flujos que pueden modificar la forma de la colonia a lo largo del tiempo.
El problema puede reformularse mediante ecuaciones integro-diferenciales definidas únicamente sobre la región ocupada por la colonia. A partir de esta formulación se analiza una solución circularmente simétrica —una colonia que crece como un disco en expansión— y se estudia cuándo esta forma resulta estable o inestable frente a pequeñas perturbaciones.
2026-05-06
13:40hrs.
Francisco Cuevas. Federico Santa María Technical University (Utfsm)
Regularized estimation for highly multivariate spatial Gaussian random fields
Sala segundo piso (DILAB), Edificio Carreras Interdisciplinarias
Abstract:
Estimating covariance parameters for multivariate spatial Gaussian random fields is computationally challenging, as the number of parameters grows rapidly with the number of variables, and likelihood evaluation requires operations of order O((np)^3). In many applications, however, not all cross-dependencies between variables are relevant, suggesting that sparse covariance structures may be both statistically advantageous and practically necessary. We propose a LASSO-penalized estimation framework that induces sparsity in the Cholesky factor of the multivariate Matérn correlation matrix, enabling automatic identification of uncorrelated variable pairs while preserving positive semidefiniteness. Estimation is carried out via a projected block coordinate descent algorithm that decomposes the optimization into tractable subproblems, with constraints enforced at each iteration through appropriate projections. Regularization parameter selection is discussed for both the likelihood and composite likelihood approaches. We conduct a simulation study demonstrating the ability of the method to recover sparse correlation structures and reduce estimation error relative to unpenalized approaches. We illustrate our procedure through an application to a geochemical dataset with p = 36 variables and n = 3998 spatial locations, showing the practical impact of the method and making spatial prediction feasible in a setting where standard approaches fail entirely.
2026-04-29
13:40hrs.
Ernesto Araya. Ludwig Maximilian University (Lmu) of Munich
Geometry, Robustness, and Dimensionality: From Spiking Networks to General Loss Landscapes
Sala segundo piso (DILAB), Edificio Carreras Interdisciplinarias
Abstract:
The rapid scaling of artificial neural networks (ANNs) has highlighted the dual need for energy-efficient architectures and a deeper theoretical understanding of how models learn. This talk addresses both challenges, moving from the specific functional properties of Spiking Neural Networks (SNNs) to broader questions regarding the geometry of neural network loss landscapes.
 
In the first part, I characterize the complexity and robustness of discrete-time Leaky Integrate-and-Fire (LIF) SNNs. We demonstrate that these networks realize piecewise constant functions on polyhedral regions and quantify how latency—a temporal dimension unique to SNNs—drives expressive power. Using Boolean function analysis, I further show that wide LIF-SNNs exhibit an inherent simplicity bias, where their Fourier spectra concentrate on low-frequency components, ensuring average-case stability under input perturbations.
 
The final part of the talk transitions to a broader investigation of neural network optimization. I will present current work-in-progress regarding the intrinsic dimension of loss landscapes in general neural networks. By analyzing the minimum subspace dimension required to reach high-quality solutions, we can better understand the structural redundancy of overparameterized models. I will conclude by discussing the implications of these geometric properties for model compression and the efficiency of the optimization process across diverse architectures.
2026-04-08
13:40hrs.
Thomas Caussade. Phd Candidate, University College London
Regularised numerical steepest descent methods for highly oscillatory integrals
Auditorio Edificio San Agustín
Abstract:
The evaluation of highly oscillatory integrals is an important topic in many areas of computational wave propagation. The Numerical Steepest Descent (NSD) method is a powerful approach to computing such integrals, which combines complex contour deformation with quadrature rules. However, for fixed frequencies NSD may lose accuracy when stationary points are close with each other, or with endpoints of the integration contour. This issue can be dealt with standard quadrature inside neighbourhoods of stationary points, in which the number of oscillations is bounded and small, combined with NSD techniques away from stationary points. In this talk, I will present a simple “black-box” interface that automates contour deformation and integration, and describe a novel framework to rigorously analyse the numerical convergence of this class of methods.
2026-01-14
15:00hrs.
Antti H. Niemi. University of Oulu
Ensuring reliability of structural simulations in the AI era
Auditorio Edificio San Agustín
Abstract:
Artificial intelligence and machine learning are increasingly shaping how engineers analyze, design, and assess structures. In structural engineering, these tools enable optimization, rapid surrogate modeling, and decision-making under complex loading and environmental conditions. As simulations become more automated and data-driven, ensuring their reliability and robustness becomes increasingly important.
 
This talk discusses how computational mathematics supports trustworthy structural simulations in the AI era. Key topics include verification and validation, uncertainty quantification, and data quality, framed within the broader context of simulation governance. The focus is on how these ideas influence practical modeling choices and interpretation of results in engineering applications.
 
The presentation will highlight examples from advanced finite element analysis and AI-enabled structural modeling, including applications to structural optimization and reliability assessment under extreme snow loads. These examples illustrate how physics-based methods and data-driven tools can be combined effectively, while emphasizing the importance of uncertainty awareness when simulations inform safety-critical decisions.
2026-01-07
13:40hrs.
David Pardo. Universidad del País Vasco/euskal Herriko Unibertsitatea (Upv/ehu)
Challenges when integrating neural networks for solving parametric PDEs
Auditorio Edificio San Agustín
Abstract:
This presentation examines the use of Physics-Informed Neural Networks (PINNs), Variational Physics-Informed Neural Networks (VPINNs), Deep Ritz methods, and First Order System Least Squares (FOSLS) combined with stochastic quadrature rules, to solve parametric partial differential equations (PDEs). It begins by introducing parametric PDEs and how these neural network techniques can be used to solve them. The presentation then delves into the challenges of solving these PDEs, including optimization, regularity, and integration. It points out that while PINNs using strong formulations may have trouble with singular solutions, they handle integration better than weak formulation methods like VPINNs or Deep Ritz. To address these integration challenges, we propose the use of unbiased high-order stochastic quadrature rules for better integration and Regularity Conforming Neural Networks to deal with complex solutions and singularities.
 
Finally, the presentation discusses the broader significance of this research for solving parametric PDE problems and suggests directions for future research, and how FOSLS and PINNs may work better than VPINNs and Deep Ritz in different cases.
2025-11-26
13:40hrs.
Federico Fuentes. Instituto de Ingeniería Matemática y Computacional UC
Global minimization of polynomial integral functionals: from calculus of variations to real algebraic geometry
Auditorio Edificio San Agustín
Abstract:
Many nonlinear integral energy functionals found in practice in continuum mechanics, such as strain energies, which we are interested in *globally* minimizing, are nonconvex, with multiple local minima and a complicated energy landscape. This makes the computation of their global minima very challenging and, except in select cases, far from guaranteed. The usual methods typically only ensure finding approximations of a local minimum through gradient descent or some version of Newton’s method on the Euler-Lagrange partial differential equations (PDEs) associated with the functional, but say nothing of whether the solution is a global minimum.
 
In this talk, for energy functionals with polynomial nonlinearity, we present an algorithm that provably converges to a global minimum and its corresponding minimizer, and can be applied to problems in nonlinear elasticity, fluid mechanics, pattern formation and PDE analysis. We do this by discretizing functions with finite element discretizations, and then leverage results from approximation theory of Sobolev spaces, calculus of variations, and most importantly, powerful representation theorems of sum-of-squares (SOS) polynomials coming from real algebraic geometry in the field of sparse polynomial optimization. More precisely, we show that as the mesh is refined and a relaxation parameter (associated to a polynomial degree) is raised, the computed results of a semidefinite program (SDP) converge to the global minimum.
 
We present numerical examples which result in excellent approximations to the global minima of different nonlinear functionals, including the pattern-forming Swift-Hohenberg free energy in two spatial dimensions, and talk about the extension to PDE-constrained minimization of such functionals, and how to use these methods in practice to produce "warm" initial guesses for Newton methods. 
2025-11-12
13:40hrs.
Javier Cembrano. Algorithms and Complexity Department, Max Planck Institute for Informatics
Impartial selection
Auditorio Edificio San Agustín
Abstract:
Impartial selection addresses the problem of choosing one or more agents from a group based on nominations by other members of the group, in such a way that no agent can influence their own chance of being selected.
 
Deterministic mechanisms for selecting a single agent face strong impossibilities. For example, Holzman and Moulin (Econometrica, 2013) showed that when each agent nominates one other agent, no impartial mechanism can simultaneously satisfy positive unanimity—that an agent nominated by everyone else is selected—and negative unanimity—that an agent with no nominations is not selected. In response to such negative results, subsequent work has explored relaxations of this setting, either by allowing randomization or by permitting the selection of a variable number of agents. In this talk, we will further motivate these relaxations by presenting a strengthening of Holzman and Moulin’s impossibility for the case where agents may nominate any number of peers, and we will cover some recent results on both relaxations. We will also discuss how the performance of impartial mechanisms can be improved if a prediction of the optimal set of agents is available.
2025-10-08
13:40hrs.
Victor Sanches Portella. Institute of Mathematics and Statistics, University of São Paulo (Ime-Usp)
Searching for Optimal Per-Coordinate Step-sizes with Multidimensional Backtracking
Presencial en Auditorio Edificio San Agustín
Abstract:
Backtracking line-search is an effective technique to automatically tune the step-size in smooth optimization, guaranteeing performance similar to what's achieved with the theoretically optimal step-size. Many approaches have been developed to instead tune per-coordinate step-sizes, also known as diagonal preconditioners, but none of the existing methods are provably competitive with the optimal per-coordinate stepsizes.
 
In this talk, I will first give an introduction to this problem and discuss many "adaptive" methods, particularly those well-known in machine learning, that construct preconditioners during the optimization process. I will then present multidimensional backtracking, an extension of backtracking line-search to find good diagonal preconditioners for smooth convex problems. Our key insight is that the hypergradients—the gradient with respect to the step-sizes—yield separating hyperplanes that allow us to search for good preconditioners using cutting-plane methods. As black-box cutting-plane approaches like the ellipsoid method are computationally prohibitive, we develop an efficient algorithm tailored to our setting. Multidimensional backtracking is provably competitive with the best diagonal preconditioner and requires no manual tuning.
2025-10-01
13:40hrs.
José Verschae. Instituto de Ingeniería Matemática y Computacional UC
Subadditive Dispatch Problem
Presencial en Auditorio Edificio San Agustín
2025-09-24
13:40hrs.
Argyrios Petras. Johann Radon Institute for Computational and Applied Mathematics (Ricam), Austrian Academy of Sciences
Computational methods for cardiac catheter ablation procedures
Presencial en Auditorio Edificio San Agustín
Abstract:
Catheter ablation has emerged as a cornerstone treatment for cardiac arrhythmias, providing a minimally invasive way to restore normal heart rhythm. Computational modeling and simulation offer powerful tools to improve both the efficacy and safety of these procedures. This presentation will review recent advances in ablation modeling, ranging from multiphysics PDE formulations to patient-specific simulations across virtual patient cohorts, with a focus on two primary ablation techniques: radiofrequency ablation and pulsed-field ablation.
2025-09-10
13:40hrs.
Víctor Verdugo. Instituto de Ingeniería Matemática y Computacional
¿ Cómo medir la calidad de un mapa distrital ?
Presencial en Auditorio Edificio San Agustín
Abstract:
Denotemos por D la familia de conjuntos compactos y conexos en R² con área y perímetro finitos. En el contexto de diseño de mapas distritales, el índice más utilizado para medir cuan bueno es un distrito es el de Polsby-Popper: para un conjunto C ∈ D el índice se calcula como 4πA(C)/P (C)², donde A(C) es el área de C y P (C) su perímetro. Este índice isoperimétrico toma valores en [0, 1] y alcanza el valor 1 cada vez que tomamos una bola en la norma l2. A pesar de su amplio uso, “to the best of my knowledge”, no hay una caracterización axiomática de este índice que permita recuperarlo a partir de propiedades buscadas en el diseñoo del índice. Por ejemplo, el índice de Polsby-Popper satisface las siguientes propiedades:

1) Es invariante bajo traslaciones, rotaciones, y escalamientos.
2) Su máximo valor se alcanza para las bolas en norma l2.

El índice Polsby-Popper no es el único que satisface estas dos propiedades: basta tomar por ejemplo el cuadrado de su valor para obtener otro que también las cumple. ¿Existe algún set extra de propiedades naturales, además de (1)-(2), que si caracterizan este índice?
2025-09-03
13:40hrs.
Cristóbal Guzmán. Instituto de Ingeniería Matemática y Computacional UC
Differentially-Private Synthetic Data Generation: A Saddle-Point Approach
Presencial en Auditorio Edificio San Agustín
Abstract:
Generating synthetic data is one of the key problems in private data analysis. In this talk, I will provide a summary of existing work, including well-established privacy attacks from the literature. After this introduction, I will present recent contributions on generating private synthetic data with relative accuracy guarantees (i.e, a mixture of additive and multiplicative error). Our main result provides (counting) error rates can be made poly-logarithmic in the sample size, data universe size, and the number of linear queries; a result which is provably unattainable in the purely-additive counterpart.
 
2025-08-27
13:40hrs.
Vera Roshchina. School of Mathematics and Statistics, University of New South Wales
Everything is possible: constructing convex sets with prescribed facial dimensions
Presencial en Auditorio Edificio San Agustín
Abstract:
Given any finite set of nonnegative integers, there exists a closed convex set whose facial dimension signature coincides with this set of integers, that is, the dimensions of its nonempty faces comprise exactly this set of integers. We show that such sets can be realised as solution sets of systems of finitely many convex quadratic inequalities, and hence are representable via second-order cone programming problems, and are, in particular, spectrahedral. It also follows that these sets are facially exposed, in contrast to earlier constructions. We obtain a lower bound on the minimum number of convex quadratic inequalities needed to represent a closed convex set with prescribed facial dimension signature, and show that our bound is tight for some special cases. We relate the question of finding efficient representations with indecomposability of integer sequences and other topics, and present a substantial number of open questions.

The talk is based on joint work with Levent Tunçel (University of Waterloo, Canada).
2025-05-28
13:40hrs.
Roberto Cominetti. Instituto de Ingeniería Matemática y Computacional (Imc) y Departamento de Ingeniería Industrial y Sistemas UC
Fixed-Point Iterations for Nonexpansive Maps
Presencial en Auditorio Edificio San Agustín
Abstract:
In this talk we present a survey of our research in the past decade on error bounds and convergence rates for deterministic as well as stochastic fixed point iterations, including the Krasnoselskii-Mann, Halpern, and similar iterative methods. 
We will emphasize the essential role played by techniques of optimal transport and Markov chains with rewards in obtaining tight error bounds. We will also discuss a few examples that illustrate how these error bounds are used to analyze the computational complexity of known algorithms in optimization and reinforcement learning for Markov decision processes, and how they can lead to design new algorithms with better complexity guarantees.