Several natural phenomena and models in science and technology may have their behavior modified by the application of external commands. For instance, an engineer can control a mechanical system using external forces, an economist can act on a financial equilibrium through fees, a chemist can modify a process by regulating the temperature, a medical doctor can control or cure diseases through drugs, etc. The mathematical theory that aims to understand how external actuators can be applied to physical systems so to change their behavior is called Control Theory. Within this line of research, we address control problems that can be formulated as follows: In an evolution system governed by an equation (differential or in partial derivatives), suppose that we can act on the system through a control (the right side of the equation, some boundary condition, ...). We want to know if it is possible to construct a control in such a way that the solution corresponding to a given initial state begins and that in a given time (control time) the solution is in a desired final state (exactly or approximately).

We are also interested in the stabilization of systems through feedback mechanisms, in which the control is regulated according to the responses obtained through its interaction with the system, and it is one of the topics of great importance in Control Theory, mainly for its applications in problems such as noise attenuation and the development of flexible structures.

In general, the study of the controllability of a physical system is quite delicate, especially in infinite-dimensional spaces (models in PDE), and even for the linear case. The differential equation that governs the evolution may be, for instance, of parabolic type (heat equation, Stokes system), of hyperbolic type (wave equation, Maxwell's equations), of dispersive type (Schrödinger's equation, Korteweg-de Vries equation), which induces intrinsic properties to the solutions, hence there is not a general method that does not depend on the type of equation. Within this line of research our main objective is the construction of efficient and economic control mechanisms for problems of different nature that arise in different fields of science and technology.

In this line of research, one studies the existence, non-existence and multiplicity of solutions of some classes of Elliptic Partial Differential Equations, defined in Euclidean domains, using analytical-functional methods such as: variational methods and topological methods. Qualitative properties of solutions of these equations are also addressed, such as: regularity, symmetry and minimum energy properties, asymptotic behavior, blow-up, among others. In order to accomplish these goals, it is necessary to study certain function spaces (Lebesgue, Sobolev, Orlicz and Besov spaces), as well as their properties and the development of new function spaces is sometimes imperative. Certain classes of equations can be studied using the variational method, which is done by searching for critical points of certain functionals that are defined in spaces of infinite dimension, together with the aid of mini-max theory and Morse theory. Many other equations do not have a variational structure and, therefore, other techniques have been used, such as the Brouwer and Leray-Schauder Degree Theory, Fixed Point Theorems and the Bifurcation Theory. Qualitative properties have been studied through the principles of maxima, the so-called Alexandrov-Serrin method (moving plane method) and its variants. Harnack-type inequalities and the De Giorgi-Nash-Moser theory are also used. Most of the researched problems are motivated by applications in other scientific areas, mainly in Physics, Astronomy, Climatology, Biology, Chemistry, Economics, among others.

This line of research is devoted to the study of qualitative properties associated with Evolution Partial Differential Equations. The models to be considered appear naturally in the description of various phenomena, such as the movement of quantum particles, heat propagation, movement of fluids, wave propagation, etc. More precisely, we seek to understand when the equations describing a given phenomenon have solutions, and also how these solutions behave as time evolves, that is to say, we will analyze properties such as well-posedness, asymptotic behavior, characterization and stability under disturbances of global attractors or pullback attractors, among others.

This line of research includes the study of geometric and analytic properties for solutions of Partial Differential Equations, having applications in the regularity theory for non-linear diffusive processes and free boundary problems arising from geometry, thermodynamics, fluid dynamics, material sciences, financial engineering, among others.

Functional Analysis is a branch of Mathematical Analysis, which investigates infinite-dimensional vector spaces endowed with a topology, as well as continuous linear operators defined on these spaces. The roots of Functional Analysis lie in the investigation of function spaces related to the solutions of differential equations. Nowadays, the techniques from Functional Analysis are applied not only to Mathematical Analysis, but also to other fields such as Quantum Information Theory, Probability and Combinatorics.

Dynamical systems are mathematical models for many problems in Physics, Biology, Economics, Engineering and so on. These dynamical systems are normally associated with differential equations that can be ordinary differential equations, partial differential equations, functional differential equations, partial-functional differential equations and discrete systems. Mathematical models are obtained using empirical laws, measurements, observations, etc. Here, our main concerns are associated with understanding the past and predicting the future with information from the present; in this sense, we give attention to the asymptotic sets of states; namely, the stable and unstable varieties, and the local, global, pullback and forwards attractors, results of existence, characterization and regularity of these sets are studies. In addition, it is known that often some of the influences that the system suffers are neglected during modeling (for simplicity of analysis) and the parameters of the approximate model can be determined with some error. Thus, practical models are only approximations of an ideal model and mistakes are inevitable. With this in mind, it is of fundamental importance that the models enjoy a certain stability with respect to all possible disturbances. Therefore, we also pay attention to results of continuity of asymptotic sets of states in the sense of the Hausdorff metric.

In Differential Geometry there are two major branch that sometimes interact with each other, namely: intrinsic and extrinsic geometry. In the first, we investigate how a variety equipped with a metric is characterized by restrictions on Curvature and its features like the Bonnet-Myers and Gauss-Bonnet theorems. Extrinsic geometry, on the other hand, is concerned with the geometric properties of the Submanifolds in an ambient variety that may be Riemannian, Lorentzian, similar or more general. These submanifolds can be obtained as critical points of some associated functionals such as length (geodesics) and area (the Plateau problem). We can investigate results of existence or characterization of submanifolds in certain geometries such as Hilbert and Bernstein's theorems.

Geometric Analysis is a mathematical area at the interface between Differential Geometry, Analysis, Partial Differential Equations and Mathematical Physics. It includes both the use of geometric methods in the study of Partial Differential Equations and the application of the theory of Partial Differential Equations to Geometry. Since its emergence in the 20th century, it has had a prolific capacity to solve problems not only in its generating lines, but in several other areas of which we can mention Algebraic Geometry, Topology and General Relativity. Among the most relevant topics in this line, we can mention the studies on Ricci Flow, Isoperimetric Problems, Stability, Allen-Cahn Equations, Min-max and prescribed curvature problems.

Singularity Theory is a broad field with vague boundaries. It is about the geometry and topology of spaces and maps defined by either polynomials or analytic equations that are not regular. The theory uses techniques from many areas of math, such as the algebraic geometry, knot theory, optics, robotic and computer vision. The applications of this theory made it quite successful in the last three decades. One of the main goals is to study the topology and classification of singular varieties and analytic maps, describing numerical invariants associated to these objects.

This research line includes hydrodynamic behavior of particle systems, stochastic processes with local interaction, asymptotic theory, and inference for stochastic processes.

The goal of this research area is the analysis of qualitative properties of Wiener functionals (Gaussian, semimartingales, etc), and also, the analysis of stochastic differential equations. The topics of interests are the following: (a) Existence and smoothness of probability distributions of functionals of stochastic processes and their relations with partial differential hypoelliptic operators; (b) Representations of stochastic processes by differential and integral operators on spaces of stochastic processes; (c) Representations of PDEs arising in fluid dynamics (Navier-Stokes, Burgers, etc) by forward-backward stochastic differential equations; (d) Study of evolution equations driven by Gaussian noises by means of the semigroup theory in Banach spaces, and, most commonly, in Hilbert spaces; (e) Study of stochastic differential equations, in particular, forward-backward, and their connection with parabolic PDEs; (f) Equations driven by a fractional Brownian motion, rough paths; (g) Non-Markovian path-dependent stochastic control and connections with viscosity solutions.