*Ten projects that will receive Fondecyt Regular 2024 funds from ANID in the areas of Mathematics and transdisciplinary research are led by researchers from the Center for Mathematical Modeling (CMM), in addition to one Fondecyt Postdoctoral project.*

## Fondecyt Regular

Of the 693 projects approved for this year in the Fondecyt Regular 2024 Project Program, ten CMM researchers were awarded: Rafael Correa, Sebastián Donoso, Joaquín Fontbona, Axel Osses, Matías Pavez, Daniel Remenik and Avelio Sepúlveda at the Universidad de Chile, Jocelyn Dunstan at the Universidad Católica, Pedro Pérez at the Universidad de O’Higgins (now at Universidad de Chile), and Manuel Solano at the Universidad de Concepción.

Most of these initiatives are part of the 121 grants awarded to the Universidad de Chile, which positioned itself as the institution with the highest number of projects benefited at the national level, with 29 of them originating in the Faculty of Physical and Mathematical Sciences.

“To promote scientific-technological based research in the various areas of knowledge, through the financing of individual research projects of excellence oriented to the production of knowledge”, is the objective declared by ANID for this contest, whose results were recently published.

It should be noted that most of the CMM projects were selected in the area of evaluation of Mathematics, except for the initiative of Professor Dunstan (UC), corresponding to Inter-transdisciplinary Research.

## Fondecyt Postdoctorado

Meanwhile, in the Fondecyt Postdoctoral Program 2024, the postdoctoral researcher David Villacís, who develops his work under the supervision of CMM associate researchers Pedro Pérez (UChile) and Emilio Vilches (UOH), was awarded a grant.

According to ANID, the objective of this competition is to stimulate the productivity and future scientific leadership of people recently initiated in research and who have a PhD academic degree, by carrying out research projects with a view to their labor insertion in the academic or other field and their interaction and collaboration with consolidated research groups.

## Center for Mathematical Modeling

CMM is currently the most active scientific research institution in mathematical modeling in Latin America. It is a center of excellence of ANID, located in the Faculty of Physical and Mathematical Sciences of the University of Chile and today integrates eight other associated universities. Its mission is to create mathematics to respond to problems of other sciences, industry and public policies. It seeks to develop science with the highest standards, excellence and rigor in areas such as data science, climate and biodiversity, education, resource management, digital mining and digital health.

## CMM projects benefited by Fondecyt Regular

### Rafael Correa (CMM / U. de Chile)

*Variational Analysis of Generalized Supremum Function with Applications*

*Variational Analysis of Generalized Supremum Function with Applications*

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### Sebastián Donoso (CMM / U. de Chile)

**Nonexpansiveness in topological dynamics and applications**

The concept of non-expansivity for actions of integers is a transversal property of all dynamical systems and has proven to be very useful in solving other problems in areas, in principle, not linked to topological dynamics. We recently introduced the context of geometric group theory to understand the notion of non-expansivity, which allowed us to introduce this concept for a vast class of groups. The ANID-funded project seeks to study various aspects of the notion of non-expansivity for finitely generated group actions, and to explore new applications topological dynamics, group geometry and combinatorics.

“The support granted by ANID will be very useful to develop the research program proposed by the project, since it will allow me to finance research stays, visits of researchers, and master’s thesis students who wish to work on these topics”, says Prof. Donoso.

### Jocelyn Dunstan (CMM / UC)

**Privacy-preserving methods for clinical natural language processing in Spanish**

Nowadays, large linguistic models (LLMs) are revolutionizing the way we interact with machines. For example, a growing community of researchers is interested in new ways in which LLMs, such as ChatGPT, could solve tasks involving unstructured text. These models require huge amounts of text to train and use hundreds or billions of parameters. Although pre-trained linguistic models work well with less data, the large number of parameters could lead to unwanted memorization of personal identification numbers, names, or addresses, making them susceptible to privacy attacks, such as inferring whether someone belongs to a dataset. Medical applications are a promising field in which to apply pre-trained LLM, as it deals with large amounts of free text from electronic medical records, such as diagnoses, prescriptions or inpatient notes. However, privacy preservation in medicine is a cornerstone, as exposing sensitive patient information violates human rights.

The goal of this project is to study, create and evaluate privacy-preserving methods to encourage the ethical use of clinical text data in LLM applications, formally ensuring the protection of sensitive patient data. This goal is crucial, as unstructured text can improve predictive tasks and enhance the exploitation of epidemiological information. In addition, this project will be one of the first to focus on the Spanish language.

### Joaquín Fontbona (CMM / U. de Chile)

**Propagation of chaos in particle systems in mean-field interaction in mathematical physics and mathematical biology**

We will study systems of particles or individuals with stochastic behavior and in mean-field interaction among them, associated to three types of models: one in mathematical ecology, another in neural networks in neurosciences, and a third in kinetic theory of gases. In each of these three models, we seek to establish mathematical results that prove that these systems of particles or individuals, when the number of these tends to infinity, approximate certain deterministic nonlinear dynamics, represented by certain nonlinear evolution equations that describe the macroscopic behavior of populations of neurons, animals in an ecological environment, or gas particles, and we seek to understand how macroscopic patterns emerge in these equations, from the microscopic behavior of individuals. The project will also have as young co-investigators Héctor Olivero, from the Universidad de Valparaíso, and Roberto Cortez, from the Universidad Andrés Bello.

### Axel Osses (CMM / U. de Chile)

**Inverse problems in wave and photon propagation arising in medical imaging and microscopy applications**

The goal of this project is to develop high-level mathematical analysis of inverse problems related to wave propagation and photon transport with two applications that have explosive and interdisciplinary interest in biology, biomedicine and physics: first, elasticity imaging, including elastography, cardiac fibers and geophysical surveying and, second, microscopy, including super-resolution microscopy and optical capture. These applications have promoted a number of new investigations and methods in the mathematical analysis of inverse problems. This includes developing mathematical models for direct and inverse problems; performing theoretical uniqueness and stability analyses; designing numerical reconstruction algorithms; and validating them with noisy or real synthetic data from physical experiments. Co-investigators are Matías Courdurier (Faculty of Mathematics UC) and Víctor Castañeda (Center for Medical Informatics, Faculty of Medicine, University of Chile).

### Matías Pavez-Signé (CMM-CNRS / U. de Chile)

**Randomness and expansion in Extremal Combinatorics**

“This project has many different edges, from extremal graph theory to problems in limit theory of discrete structures. On the one hand, I will investigate classical questions in extremal graph/hypergraph theory and Ramsey theory, using mainly probabilistic techniques. There will be a particular focus on solving questions about Ramsey numbers of trees”, details Matías Pavez. “On the other hand, I also want to study notions of «expansion» in hypergraphs with the goal of creating new techniques that are capable of solving problems in random graphs. Finally, I will study classical probabilistic problems on finite words or graphs, where, however, the random model comes from a «boundary» object (graphon, permutation, etc.). In this subject, different areas of mathematics are mixed, such as analysis and probabilities, with a purely combinatorial motivation”, he adds.

### Pedro Pérez (CMM / U. de Chile)

**On Geometric and Variational Properties of Probust Chance Constrained Optimization Problems**

Modern problems suggest the incorporation of a duality between a probabilistic and robust model, for example, to control a stochastic process over a time interval with high probability. These models are called probabilistic/robust random constrained optimization problems. The objective of this project is to investigate the geometric and variational properties of the probabilistic optimization problem with probabilistic/robust chance constraints and, at the same time, to devise efficient algorithms for its solution. The project is divided into three specific objectives:

- Geometric properties: to analyze and develop a framework that ensures the geometric properties of probability functions of the probabilistic/robust type and of the constraint sets generated by such functions, including properties such as convexity or generalized concavity.
- Variational properties: Develop a methodology to obtain first and second order variational information, including practical formulas for gradients and Hessians.
- Algorithms: Propose and implement efficient algorithms for solving optimization problems with chance constraints related to probabilistic/robust optimization models.

### Daniel Remenik (CMM / U. de Chile)

**Integrable fluctuations in the KPZ universality class**

“It is a continuation of the line of research that I have been carrying out for several years, in a class of random models that come from physics, and that include, for example, models of random growth of interfaces such as in the growth of a colony of bacteria, or that of the combustion front of a piece of paper, among many others,” says Prof. Daniel Remenik. “This class of models, which is known as the KPZ class, has a «universal» behavior, which means that, although the models may be very different from each other, their statistical behavior at the macroscopic level is the same. The goal of the project is to expand the class of models for which this behavior can be characterized, and to advance the understanding of this behavior and its connection to other phenomena and other areas of mathematics.”

### Avelio Sepúlveda (CMM / U. de Chile)

**Geometry of spin systems**

The goal of this project is to contribute to the understanding of the geometry of spin systems in dimension 2. Three main areas will be addressed: Gaussian free field (GFF) level sets, spin systems with continuous symmetries (\(O(N)\) model) and spin systems with discrete symmetries. In the first area, we will explore the fundamental properties of their level sets and seek to translate this understanding to other spin systems. For the second topic and third topic, particular phenomena will be investigated, such as the topological phase transition in the XY model, as well as the absence of topological transition in \(N\geq3\) cases and how this translates for the GFF to integer values and the discrete Coulomb gas. In addition, the application of these concepts to spin models in random flat maps will be explored, with the goal of making a significant contribution to the understanding of geometry in spin systems and its connection to the GFF, thus addressing open questions in statistical physics.

### Manuel Solano (CMM / U. de Concepción)

**The Transfer Path Method for non-coincident meshes. Applications to interface problems and non-body-fitted grids.**

Numerical methods for partial differential equations (PDEs) are usually based on a polyhedral discretization of the domain, originating a variational crime due the approximation of the geometry that restricts the accuracy of the numerical scheme. In the literature we can find two different approaches to overcome this limitation: fitted methods and unfitted methods. In the first one the discretization of the domain is adapted to the domain, which causes difficulties from the implementation point of view, especially when dealing with complicated structures or evolving domains. On the other hand, unfitted methods are based on a background mesh where the domain of interest is immersed, and the main challenge is to incorporate the boundary conditions on the computational boundary which not necessarily coincides with the true boundary. During the last decade unfitted methods have intensively developed, especially the CutFEM method, the Shifted Boundary Method, and the Transfer Path Method. The main scope of the project is to develop the Transfer Path Method in two contexts: non- coincident triangulations and evolving domains with non-body-fitted meshes.

This is the fourth consecutive Fondecyt for prof. Manuel Solano, who has been benefited with a Fondecyt de Iniciación (2013-2014), and three Fondecyt Regular projects (2016-2019, 2020-2023 and 2024-2027). “In all of them I have been Principal Investigator and sole investigator”, Prof. Solano points out.

CMM Communications