Speaker:  Christopher Cabezas
Center for Mathematical Modeling, U. de Chile
Date: Monday, April 21, 2025 at 2:30 p.m. Santiago time

Abstract:  

An important question in dynamical systems is the classification, i.e., to be able to distinguish two isomorphic dynamical systems. In this work, we focus on the family of multidimensional substitutive subshifts. Constant-shape substitutions are a multidimensional generalization of constant-length substitutions, where any letter is assigned a pattern with the same shape. We prove that in this class of substitutive subshifts, under the hypothesis of having the same structure, it is decidable whether there exists a factor map between two aperiodic minimal substitutive subshifts. The strategy followed in this work consists in giving a complete description of the factor maps between these substitutive subshifts. We will also discuss related results, such as a condition to ensure that the substitution defines a subshift, and some consequences on coalescence, automorphism group and number of symbolic factors. This is a joint work with Julien Leroy.

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

Speaker:  Mirabel Mendoza
Center for Mathematical Modeling, U. de Chile
Date: Monday, March 31, 2025 at 2:30 p.m. Santiago time

Abstract:  

We consider the Shortest Odd Path (SOP) problem, where given an undirected graph $G$, a weight function on its edges, and two vertices $s$ and $t$ in $G$, the aim is to find an $(s,t)$-path with odd length and, among all such paths, of minimum weight. For the case when the weight function is conservative, i.e., when every cycle has non-negative total weight, the complexity of the SOP problem had been open for 20 years, and was recently shown to be NP-hard. I’ll present a polynomial-time algorithm for the special case when the weight function is conservative and the set of negative-weight edges forms a single tree. The algorithm exploits the strong connection between SOP and the problem of finding two internally vertex-disjoint paths between two terminals in an undirected edge-weighted graph. This is a joint work with Alpár Jüttner, Csaba Király, Gyula Pap, Ildikó Schlotter, and Yutaro Yamaguchi.

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

Speaker:  María Fernanda Espinal
Center for Mathematical Modeling, U. de Chile
Date: Monday, March 17, 2025 at 2:30 p.m. Santiago time

Abstract:  

The Yamabe problem is a classical question in conformal geometry that seeks for existence of metrics with constant scalar curvature within a conformal class. The problem was posed by H. Yamabe in 1960 as a possible extension of the famous uniformization theorem, which states that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane or the Riemann sphere. After the conjecture was already confirmed by the work of R. Schoen, an alternative approach was proposed by R.Hamilton in 1989. He suggested to use a geometric flow, which is now known as the Yamabe flow. The k-Yamabe flow is a fully non-linear extension to the Yamabe flow that appears naturally in problems related to topological classification in higher dimensions. In this talk we describe the construction, classification and asymptotic behavior of radially symmetric gradient k-Yamabe solitons that are locally conformally flat [ES24], these are special solutions to this flow that play a central role in the theory. Our study extends the results obtained by P. Daskalopouolos and N. Sesum in [DS13] in the case n > 2k.

Joint work with Mariel Sáez.

References
[DS13] Panagiota Daskalopoulos and Natasa Sesum. The classification of locally conformally flat yamabe solitons. Advances in Mathematics, 240:346–369, 2013.
[ES24] Mar´ıa Fernanda Espinal and Mariel S´aez. On the existence and classification of k-yamabe gradient solitons. arXiv preprint arXiv:2410.06942, 2024.

 

Venue: Jacques L Lions Seminar Room, CMM, Beauchef 851, North Tower, 7th Floor

Speaker:  Sani Biswas
Center for Mathematical Modeling, U. de Chile
Date: Monday, December 16, 2024 at 2:30 p.m. Santiago time

Abstract:  

We present a general Milstein-type scheme for McKean-Vlasov stochastic differential equations (SDEs) driven by Brownian motion and Poisson random measure and the associated system of interacting particles where drift, diffusion and jump coefficients may grow super-linearly in the state variable and linearly in the measure component. The strong rate of -convergence of the proposed scheme is shown to be arbitrarily close to one under appropriate regularity assumptions on the coefficients. For the derivation of the Milstein scheme and to show its strong rate of convergence, we provide an It\^o formula for the interacting particle system connected with the McKean-Vlasov SDE driven by Brownian motion and Poisson random measure. Moreover, we use the notion of Lions derivative to examine our results. The two-fold challenges arising due to the presence of the empirical measure and super-linearity of the jump coefficient are resolved by identifying and exploiting an appropriate coercivity-type condition).

 

Venue: Jacques L Lions Seminar Room, CMM, Beauchef 851, North Tower, 7th Floor

Speaker:  Laura Jiménez
Center for Mathematical Modeling, U. de Chile
Date: Monday, December 02, 2024 at 2:30 p.m. Santiago time

Abstract:  

I will delve into three key topics of my research in quantitative ecology and how the outcomes contribute to understanding and preventing biodiversity loss. In each case, I will describe the ecological context, the data at hand, and the primary modeling tools used to address the problems of interest. First, I will talk about optimal survey design, which involves techniques to efficiently estimate population density by balancing sample size, spatial distribution, and survey effort. Next, I will explain how statistical calibration techniques are applied for error correction and data fusion from diverse sources to improve biomass or abundance estimates, which are then used to design management and conservation strategies for coral reef fish. Lastly, I will demonstrate the use of ecological niche models to describe where a species lives and predict its likely distribution under climate change and anthropogenic disturbances.

 

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

Speaker: David Torregrosa Belén
Center for Mathematical Modeling, U. de Chile
Date: Monday, November 18, 2024 at 2:30 p.m. Santiago time

Abstract:  

Many situations in convex optimization can be modeled as the problem of finding a zero of a monotone operator, which can be regarded as a generalization of the gradient of a differentiable convex function. In order to numerically address this monotone inclusion problem it is vital to be able to exploit the inherent structure of the monotone operator defining it. The algorithms in the family of the splitting methods are able to do this by iteratively solving simpler subtasks which are defined by separately using some parts of the original problem. In this talk, we will introduce some of the most relevant monotone inclusion problems and present their applications to optimization. We will describe the different difficulties arising in their treatment, which urge to consider specific splitting schemes suitable for each monotone operator.

 

Venue: Jacques L Lions Seminar Room, CMM, Beauchef 851, North Tower, 7th Floor

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