Abstract:

This talk presents fast optimization algorithms from the viewpoint of continuous dynamical systems.

We begin with the classical correspondence between gradient flow and gradient descent, and then discuss how nonsmooth gradient flows naturally lead to proximal-type methods. The main focus is on inertial dynamical systems, where second-order terms introduce memory and momentum into the optimization process.

Through this perspective, we explain how continuous-time models such as the heavy-ball system and Nesterov-type dynamics provide insight into acceleration, damping, stability, and convergence of fast first-order methods.

We also discuss how these ideas can be extended beyond standard minimization problems, including recent directions for nonconvex minimax optimization.

Speaker: Raúl Tintaya (CMM, U. de Chile)

La investigación propone un método basado en tecnologías geoespaciales y datos multifuente para caracterizar metales pesados en suelos y vegetación. El estudio fue desarrollado en España y transferido a Ecuador, el cual permite identificar elementos como As, Cd, Cr, Cu, Hg, Ni, Pb y Zn mediante imágenes satelitales (Sentinel-2 y PRISMA), técnicas espectrales e índices de diferencia normalizada. Los resultados muestran precisiones prometedoras, especialmente con Sentinel-2, y ofrecen una herramienta útil para agricultores, instituciones públicas e investigadores en la gestión sostenible del territorio y la seguridad alimentaria.

El evento se realizará en el Centro de Modelamiento Matemático (CMM) de la Universidad de Chile, y se transmitirá en vivo por el canal de YouTube de CopernicusLAC Chile en este enlace: https://go.cmm.uchile.cl/seminario29mayo

El expositor será Francisco Cabrera-Torres, Especialista Geomático en la Fuerza Aérea Ecuatoriana y PhD de Ingeniería Geomática de la Universidad Politécnica de Madrid.

Abstract:
We will introduce the Young generating function and use it to characterize the law of large numbers and the central limit theorem behaviors for random partitions. As an application of these results, we present a framework to obtain conditional Gaussian Free Field fluctuations for height functions associated with random standard Young tableau. To prove these results we develop algebraic formulas for operators on the Gelfand–Tsetlin algebra of the symmetric group.

Speaker: 
Gabriel Raposo (UC Berkeley)

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https://reuna.zoom.us/j/84521834914?pwd=OTZ6Y0NWM3pYTGtTbEt3c0luTG96UT09
ID de reunión: 845 2183 4914
Código de acceso: 997973

Abstract:

Resumen: We develop a new methodology for model-based clustering. Optimizing the log-likelihood provides a principled statistical framework for clustering, with solutions found via the EM algorithm. However, because the log-likelihood is nonconvex, only convergence to stationary points can be guaranteed, and practitioners often use multiple starting points in the hope that one will converge to the global solution. We consider a new loss function based on entropic optimal transport that shares the same global optimum as the log-likelihood but has a much better-behaved landscape, thereby avoiding spurious local-optima configurations that are pervasive with the log-likelihood. Similar to the EM algorithm for the log-likelihood, this new loss can be optimized by the Sinkhorn-EM algorithm, which we show converges at a rate comparable to that of EM. By analyzing extensive numerical experiments and two real-world applications in image segmentation in C. elegans microscopy and clustering in spatial transcriptomics, we show that this new loss outperforms log-likelihood optimization, indicating that it represents a valuable clustering methodology for practitioners. We also comment on finite-sample properties of this procedure, leveraging novel convergence bounds for objects arising from entropic optimal transport.

Speaker:  Gonzalo Mena (UC Berkeley)
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https://reuna.zoom.us/j/84521834914?pwd=OTZ6Y0NWM3pYTGtTbEt3c0luTG96UT09
ID de reunión: 845 2183 4914
Código de acceso: 997973

Abstract:

Covering the vertices of edge-coloured graphs with monochromatic components has been a beloved pastime of combinatorialists for decades, largely motivated by a celebrated conjecture of Ryser. Some years ago, Bal and DeBiasio inquired about the minimum degree threshold that guarantees that the vertex set of an r-edge-coloured graph can be covered with at most r monochromatic components. This question has since led to many more related problems and accompanying results. Most notably, Girao, Letzter and Sahasrabudhe proved that every large enough n-vertex 2-edge-coloured graph with minimum degree exceeding (2n-5)/3 can be covered with at most two monochromatic components.

In a recent paper, Fernandez, Pavez-Signe and Stein worked on an analogous problem for bipartite graphs. Namely, they proved that every large enough 2n-vertex, 2-edge-coloured, balanced bipartite graph with minimum degree at least (13/16 + ε)n can be covered with at most three monochromatic components. That said, they expected the threshold to be around 2n/3. In this talk, I will prove that they were right.

This is joint work with Cesar Bispo, Marcelo Lage, Guilherme Mota and Bruno Skarmeta.

Speaker: George Kontogeorgiou (CMM)

Abstract:

We study the spectral properties of the Dirac operator L_0 obtained by linearizing the one-dimensional Soler model around standing waves with power nonlinearity f(s) = s|s|^{p-1}, p > 0. We give a sharp characterization of the spectral gap. If p ≥ 1, the gap contains no eigenvalues other than the symmetry-induced energies -2ω and 0. If 0 < p < 1, additional eigenvalues bifurcate from the thresholds of the essential spectrum and enter the gap. We further prove that the thresholds are never eigenvalues for any p > 0 and that there are no resonances for p > 1.

Speaker: Danko Aldunate (CMM, U. de Chile)

Abstract: 

Which properties hold for a generic invariant measure of a continuous map T from a compact metric space X to itself? We prove the genericity of rank one measures under a relaxed specification hypothesis (the vague specification or the approximate product property), sharpening Sigmund’s classical genericity theorem. The method applies, in particular, to certain minimal and proximal systems. As a by-product, every measure produced by the Gorodetski–Ilyashenko–Kleptsyn–Nalsky construction of non-uniformly hyperbolic ergodic invariant measures is rank one.

The key new tools are an approximation scheme for the natural extension of T, inspired by the classical approximation of a subshift by a descending sequence of shifts of finite type, and notions of convergence of invariant measures generalising Ornstein’s d-bar metric and Feldman’s f-bar metric. Outside the symbolic setting, these notions of convergence are new and allow several classical results from the theory of d-bar and f-bar to be transferred to general topological dynamical systems.

Based on joint work with M. E. Can, J. Konieczny, M. Kupsa; with D. Buldağ and B. Jacelon; and with M. Łącka and A. Trilles.

Speaker: Dominik Kwietniak (Jagiellonian University)
Venue: Room 2, Faculty of Mathematics, Pontificia Universidad Católica

Abstract:

As a special case of a theorem of Pollington, the set of numbers that are normal in base 2, but not normal in base 3 is known to have full Hausdorff dimension. An interpretation of results such as this are that in some way the notions of normality in base 2 and normality in base 3 are “independent”.

The purpose of this talk is to introduce notions from descriptive set theory that may better capture this idea of independence: that of – and -completeness. For example, the previously mentioned set of numbers normal in base 2, but not normal in base 3 is known to be -complete by a recent result of Jackson, M., and Vandehey. Difference sets that satisfy these properties will also have additional interesting properties that will be discussed

Speaker: Bill Mance (Adam Mickiewicz University)
Venue: Room 2, Faculty of Mathematics, Pontificia Universidad Católica