# News

Juan David Samboní was under the supervision of Dr. Rommel Bustinza Pariona, academic of the Department of Mathematical Engineering

**Juan David Samboní Chicangana** is a PhD candidate in Mathematical Sciences at the Universidad del Cauca, located in the city of Popayán, Colombia. In the first instance, he studied Mathematics, a plan that is similar to the degree that is done in Chile, and “after that,” details Juan David, “the opportunity arose with who is currently my tutor, to do master's studies in mathematical sciences, and then the doctorate, all at the Universidad del Cauca.”

“I was looking for opportunities to study abroad, I applied to programs in England, especially looking for scholarships, because studying is very expensive and much more so over there, and I got some support, but it was not enough and I applied for the bicentennial doctoral excellence scholarships provided by the government and I benefited,” he explains.

Regarding his interests and lines of research, Juan David highlights “mainly the analysis of partial differential equations. More than anything, the continuous analysis, which is not the discretization part, but rather, in abstract mathematical domains without discretization, we do the analysis on those spaces of what we call continuous solution, which is in fact the one we seek to approximate later, once we know that it exists and is unique, by means of numerical methods. I also have an affinity for mathematical analysis and functional analysis, all tools that then lend themselves to the analysis of differential equations”.

His PhD tutor is Ramiro Miguel Acevedo, who obtained the highest academic degree at the Mathematical Engineering Research Center, CI²Ma, at UdeC in 2008. “He was a student of Professor Rodolfo Rodriguez and did his studies in computational electromagnetism which is, in fact, the line in which we try to focus what I do in my studies. Basically he, since he studied here and has had, in fact, several regular visits here at CI²Ma. A requirement of the scholarship with which I am financing my doctoral studies in Colombia, is to carry out a research stay abroad for at least three months. Thus the idea arose to come to Concepción, under the guidance of Professor Rommel Bustinza, during the summer months. Incidentally, to also take advantage of the opportunity to attend and participate in WONAPDE 2024”.

Dr. Rommel Bustinza Pariona stated that “Juan David is one of those people who can transmit knowledge in a simple and pleasant way, even when the mathematical tools involved are quite technical. In the period we were working with him, he demonstrated an exemplary level of independence and collaboration. Discussions were always fruitful, even when we did not reach the objective we had originally planned”.

Meanwhile, the director of CI²Ma, Dr. Rodolfo Araya Durán, emphasized that “the arrival of foreign students is always a source of joy for our center, even more so when the thesis director is a former student of our doctoral program. We are confident that Juan David's work, together with Dr. Bustinza, has been very fruitful”.

### The most enriching experience

Regarding the scientific objectives of his stay, Juan David explains that “when studying computational electromagnetism, a structure known as degenerate is presented, when one of the main variables of our model suffers some perturbation or affectation by some operator, and this is very common in computational electromagnetism due to the presence of two different types of domains; that type of problem has an abstract structure and we try to identify the type of structure it has, what terms appear, what operators, in what form, and we propose a formulation and try to demonstrate existence and uniqueness, and then look at application models such as those presented in electromagnetism that fit that theory and be able to demonstrate the existence and uniqueness for that type of problem”.

“The theory that we developed at the beginning was thought in my thesis was designed for Hilbert spaces, in which you have many properties and you can do a lot of mathematical analysis and it gives you many tools to obtain your objectives. But then, what happens when we go to let's say 'more realistic' applications, for example, fluids, sometimes there are some nonlinearities for which Hilbert spaces are no longer useful to describe them, so we have to get out of there and see how things could behave if we work in Banach spaces, we lose many properties, but the idea was that we already know that this theory works, to apply it to this other type of problems, that was basically the objective we set for our stay here, to develop it in an abstract way and see if it was possible to develop them, to see if, at least, we can obtain an example of application of fluids and we obtained results, we were able to verify that it is possible under certain hypotheses”.

Leveraging his stay at UdeC, Juan David participated in WONAPDE 2024, giving the talk 'Existence and Uniqueness of Solution for a Family of Nonlinear Degenerate Mixed Parabolic Equations and Its Applications to Eddy Current Model' in the Computational Electromagnetics session coordinated by Prof. Rodolfo Rodríguez.

“I had the opportunity to talk with him. It is the first time I am in an international and English-speaking event, so it was a challenge, it put some pressure on me to be presenting in front of so many personalities, there were people whose articles I had studied, read their books, whose work I have been reviewing since my master's degree in this field of computational electromagnetism, it was very impressive to be able to share with them at a more human level. This has been the most enriching experience I have had so far", Juan David emphasized .