Speaker: Alvaro Almeida Gomez
Center for Mathematical Modeling, U. de Chile
Date: Monday, November 04, 2024 at 2:30 p.m. Santiago time

Abstract:  

We introduce the nonlinear dimensionality reduction problem known as Manifold Learning and present the diffusion maps algorithm (Coiffman and Lafon, 2006). Diffusion maps utilize the connectivity between data points through a diffusion process on the dataset. Additionally, we show some applications of this technique to 2D
tomography reconstruction when the angles are unknown.

 

Venue: John Von Neumann Seminar Room, CMM, Beauchef 851, North Tower, 7th Floor

Speaker: Ana Laura Trujillo
Center for Mathematical Modeling, U. de Chile
Date: Monday, October 21, 2024 at 2:30 p.m. Santiago time

Abstract:  

The tree embedding problem focuses on identifying the minimal conditions a graph $G$ must satisfy to ensure it contains all trees with $k$ edges. Here, a graph $G$ consists of a set $V$ of elements called vertices, and a set $E$ of (unordered) pairs of vertices, called edges. We say that a graph $G$ is a tree if, for any pair of vertices, there is exactly one path connecting them.

Erd\H{o}s and Sós conjectured that any graph $G$ with $n$ vertices and more than $(k-1)n/2$ edges contains every tree with $k$ edges. This conjecture has been generalized into the Antitree Conjecture by Addario-Berry et al., which states that every digraph $D$ with $n$ vertices and more than $(k-1)n$ arcs contains every antidirected tree with $k$ arcs. Here, a digraph $D$ consists of a set $V$ of vertices and a set $A$ of arcs (ordered pairs of vertices), and an antidirected tree is a tree in which the edges are directed so that each vertex has only incoming or outgoing arcs.

In this talk, we present a proof of the Antitree Conjecture for the case where the digraph $D$ does not contain certain orientations of the complete bipartite graph $K_{2,s}$, where $s = k/12$. Additionally, we explore a proof of this conjecture for antidirected caterpillars. This work is a collaboration with Maya Stein.

 

Venue: John Von Neumann Seminar Room, CMM, Beauchef 851, North Tower, 7th Floor

Speaker: Julien Boulanger
Center for Mathematical Modeling, U. de Chile
Date: Monday, October 07, 2024 at 2:30 p.m. Santiago time

Abstract:  

This talk discusses two geometric aspects of the so-called Hecke groups, defined by E.Hecke in the 1920s, and which are a generalisation of the modular group SL(2,Z) of 2×2 matrices with integer coefficients and determinant 1. Hecke groups will be used here as a pretext to talk about my research field, namely hyperbolic geometry and translation surfaces (no prior knowledge on these fields are required). More precisely, we will see that these groups are examples of lattice Fuchsian triangle groups, and that they also arise as Veech groups of translations surfaces. At the end we will explain a correspondence between these two geometries, and an application to billiard trajectories in regular polygons.

 

Venue: John Von Neumann Seminar Room, CMM, Beauchef 851, North Tower, 7th Floor

Speaker: Raúl Tintaya
Center for Mathematical Modeling, U. de Chile
Date: Monday, September 23, 2024 at 2:30 p.m. Santiago time

Abstract:  

In this talk, we examine second-order gradient dynamical systems for smooth strongly quasiconvex functions, without assuming the usual Lipschitz continuity of the gradient. We establish that these systems exhibit exponential convergence of the trajectories towards an optimal solution. Furthermore, we extend our analysis to the broader quasiconvex setting by incorporating Hessian-driven damping into the second-order dynamics. Finally, we demonstrate that explicit discretizations of these dynamical systems result in gradient-based methods, and we prove the linear convergence of these methods under appropriate parameter choices. This presentation is based on [1].

[1] F. Lara, R.T. Marcavillaca, P.T. Vuong (2024). Characterizations, Dynamical Systems, and Algorithms for Strongly Quasiconvex Functions (Submitted).

Venue: John Von Neumann Seminar Room, CMM, Beauchef 851, North Tower, 7th Floor

Speaker: Taruni Sridhar
Center for Mathematical Modeling, U. de Chile
Date: Monday, September 09, 2024 at 2:30 p.m. Santiago time

Abstract:  

Graph coloring is one of the most famous problems in graph theory. The most natural question to ask in this framework is whether or not a given family of graphs has a finite chromatic number. As graph homomorphisms generalize coloring, we study the notion of homomorphisms for (n,m)-graphs. Due to their various types of adjacencies, the (n,m)-graphs manage to capture complex relational structures and are useful for mathematical modeling. For instance, the Query Evaluation Problem (QEP) in graph databases, the immensely popular databases that are now used to handle highly interrelated data in social networks like Facebook, Twitter, etc., is modeled on homomorphisms of (n,m)-graphs.

This talk aims to introduce (n,m)-graphs and discuss some rigid sub-graphs of (n,m)-graphs in the context of coloring that are analogous to cliques in traditional vertex coloring. Bounds on the sizes of these sub-graphs for various types of graph families are obtained, which is a natural lower bound for the chromatic number of such graphs.

Venue: John Von Neumann Seminar Room, CMM, Beauchef 851, North Tower, 7th Floor

Speaker: Jorge Aguayo
Center for Mathematical Modeling, U. de Chile
Date: Monday, July 01, 2024 at 2:30 p.m. Santiago time

Abstract:  This talk presents the analysis of an elasticity equation with interface conditions as a way to understand the formation of subduction earthquakes and how it is possible to determine geophysical characteristics of some tectonic plates from surface measurements.

First, two different numerical methods based on finite elements are analyzed to solve the forward problem by comparing their algorithmic complexity and some properties necessary to solve an inverse problem. Then, the inverse problem of recovering the coseismic slip (one of the interface conditions) from surface measurements of deformations and tractions is analyzed, presenting a conditional stability result. Finally, we present an optimal control problem that allow us to recover a good approximation of the coseismic slip and the numerical analysis associated with the solution to the discretized optimal control problem. This talk will be complemented with some numerical experiments that show the efficiency of our numerical solvers and some realistic synthetic examples that simulate a subduction earthquake on the coast of Chile.

 

Venue: John Von Neumann Seminar Room, CMM, Beauchef 851, North Tower, 7th Floor

Speaker: Fernando Bolaños
Center for Mathematical Modeling, U. de Chile
Date: Monday, June 17, 2024 at 2:30 p.m. Santiago time

Abstract:  Our experiencing and, thus, our practices are co-enabled by entwined fields of knowledge, fields of power and forms of subjectivity. Through the articulation of different research pieces, and one protagonist, in this SIPo session we will tackle the following question: how free are we? Our protagonist is a student enrolled in Chile’s Secondary Technical VocationalEducation and Training (S-TVET) system. All research pieces used to drive then narrative of the presentation are the result of a postdoctoral research endeavor. These include research articles published in indexed journals (WoS and Scopus), cut-motion videos and a peer-reviewed book published by the Organization of Ibero-American States for Education, Science and Culture (OEI, for its acronym in Spanish). It is hoped that the space provided by SIPo can enable a conversation between researchers from different areas of knowledge where we can ponder on the underlying rationalities and logics behind our experiencing and practices.

 

Venue: John Von Neumann Seminar Room, CMM, Beauchef 851, North Tower, 7th Floor

Speaker: Alejandra Muñoz Arancibia
Center for Mathematical Modeling, U. de Chile
Date: Monday, June 03, 2024 at 2:30 p.m. Santiago time

Abstract:  The Automatic Learning for the Rapid Classification of Events (ALeRCE) is a Chilean-led astronomical alert broker, and one of the seven brokers selected to receive the full alert stream from the Rubin Observatory. Since 2019 ALeRCE has been ingesting the public stream from the Zwicky Transient Facility, which contains hundreds of thousands of alerts per night, providing classifications and visualization for the astronomical objects detected.

In this talk I describe the ALeRCE project and its different science cases, showing how it has enabled the follow-up and study of young supernovae and other transient events. I highlight some of our discoveries, plus unforeseen benefits that we have found throughout our search. The advent of new surveys that target the dynamic sky, like the ones conducted now by the Asteroid Terrestrial-impact Last Alert System and soon by Rubin, opens a promising future for continuing the collaborative exploration of the changing universe.

 

Venue: John Von Neumann Seminar Room, CMM, Beauchef 851, North Tower, 7th Floor

Speaker: Haritha Cheriyath
Center for Mathematical Modeling, U. de Chile
Date: Monday, May 06, 2024 at 2:30 p.m. Santiago time

Abstract:  Symbolic dynamics, initially used as a tool for analyzing general dynamical systems, has later caught more attention due to its independent applications in other areas including information theory and coding. From a given directed graph, we construct a symbolic space consisting of infinite paths on it. We are interested in studying its complexity by counting paths of fixed lengths. The topological entropy of this system is given by the growth rate of this complexity function.
When some of the edges are removed from the graph, the entropy of the new perturbed system drops. What if instead of edges, we make some paths to be forbidden? How much does the entropy drop when the lengths of these forbidden paths increase? These questions can be re-framed as enumeration of finite strings where a given collection of strings are forbidden. We explore more on this enumeration problem and if time permits, we discuss its applications in seemingly unrelated scenarios that include game theory and pattern matching algorithm.

 

Venue: Sala John Von Neumann, 7th floor, Beauchef 851

Speaker: Emeric Bouin
Université Paris-Dauphine
Date: Tuesday, April 23, 2024 at 2:30 p.m. Santiago time

Abstract: In this talk, I will present some facets of the best known equation in reaction-diffusion, which is called the Fisher KPP equation. I will explain where it comes from and what kind of results are interesting for mathematicians. Nobody needs knowledge in PDEs, the only thing to know is what is a Laplacian in R^d, or partial derivatives of order 2.

 

Venue: Sala John Von Neumann, 7th floor, Beauchef 851