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Research Areas

The research areas that can be developed by our students are:

Applied mathematics

Our research covers the areas of numerical analysis, inverse problems and optimization. In numerical analysis we focus on partial differential equations that typically represent problems of mathematical physics and electromagnetism. We study numerical methods and their approximation properties, in addition to related numerical linear algebra problems. We currently cover finite element and boundary methods, time-stepping methods, and Petrov-Galerkin methods for singularly disturbed problems. A particular topic is the development of fast numerical algorithms. These include the analysis of their precision and efficiency, and pre-conditioning and compression strategies for large matrix systems.

In reverse problems the objective is an indirect reconstruction of uncertain parameters using measurements of an amount affected by said parameters. Common models include differential or integral equations, where the unknowns are parameters of the equation, the measurements correspond to partial knowledge of the solution, and the objective is to determine the unknown coefficients. In such scenarios, the study of inverse problems falls heavily on functional analysis techniques and their applications, including partial differential equations, integral equations and operator theory.

An optimization problem seeks to find the best solution (under certain criteria) over a given set of alternatives. We focus on the development of new algorithmic techniques and development of lower bounds for such problems. Depending on the type of problems, the techniques used have a continuous or discrete nature, including computational complexity, convex analysis, discrete mathematics and algorithm design. The area finds multiple applications, particularly for problems that come from computer science, operations research, economics and engineering.

Related academics: Matías Courdurier, Thomas Führer, Norbert Heuer, Federico Castillo, José Samper,  Mircea Petrache, Pablo Palacios.


Partial differential equations

The existence, uniqueness and regularity of the systems solutions of partial differential equations are analyzed. The formation of singularities and patterns, the evolution of certain geometric objects, and the concentration of energy in localized structures are also studied.

It is sometimes sought to model the behavior of a physical phenomenon by means of systems of equations, or to simplify existing models by means of rigorous approximations and asymptotic analyzes. The emphasis is on equations that require nonlinear analysis.
Geometry


We study the geometric properties of varieties from different points of view. The rigidity of the Riemannian varieties against certain geometric objects is studied, as well as the evolution of these varieties under different flows. Geometric and analytical aspects of univalent and locally univalent functions are also studied in domains of the complex plane, harmonic transformations of the plane and its relation with minimal surfaces, as well as generalizations to several complex variables. On the other hand, the geometry of the algebraic surfaces and their moduli spaces is studied, and the surfaces K3 in relation to derived categories. The main tools come from complex analysis, differential geometry and algebraic geometry and differential equations.
Algebraic and Arithmetic Geometry
 
Our group focuses on algebraic geometry, number theory, and the interactions between these important areas in the framework of arithmetic geometry. We study the geometry of algebraic surfaces especially of general type, with emphasis on its geography and the compactification of Kollár-Shepherd-Barron - Alexeev of its moduli spaces, developing new tools explicit in Mori theory. At the same time, we attack problems on surfaces on the existence of low gender curves together with their potential hyperbolicity, in the sense of the absence of entire curves, all motivated by themes of both geometric and arithmetical origin. We also investigate rational and algebraic points in algebraic varieties by means of Arakelov geometry, equidistribution methods, Galois representations, Diofanthine approximation, analytical number theory and automorphic forms. These tools allow us to develop new methods in the arithmetic of points in varieties of Shimura and elliptic curves, with applications to fundamental topics such as dynamics of Hecke operators, special points, ranges of elliptic curves and the conjectures of Vojta. In addition, our group develops applications of all of the above in connection with mathematical logic, in the context of the tenth Hilbert problem.

Our team maintains strong international networks and generates relevant activities in the area. These include seminars, conferences and a constant flow of guests from other research centers. In particular, we aspire to offer an environment of excellence to welcome young people who seek to specialize in these issues.
 


Odds

The main research topics in the area of ​​probability are stochastic partial differential equations, the KPZ equation, particle systems, random media, random marches in random media, and non-commutative probability.

Dynamic systems

Dynamic Systems is an important area of ​​mathematics. Its origin can be placed with the work of Poincaré celestial mechanics at the end of the 19th century. The main objective is to understand the qualitative behavior of deterministic systems. The techniques, questions are varied and range from the use of probabilistic methods, to purely topological techniques. It is a discipline closely related to physics. For example, the methods and ideas of statistical mechanics have influenced the area.

On the other hand, dynamic systems have found applications in the most diverse areas of mathematics, from number theory to partial differential equations. The applications are varied, from finance to sociology. Parts of the theory of dynamic systems such as fractals and chaos has received great media attention in recent times. In our Faculty there is a group with diverse interests, with strong international networks, with seminars, congresses and guests that make it a perfect place to specialize in this subject.


Mathematical Physics

Many models of modern Mathematical Physics require the transversal management of mathematical tools such as Non-Commutative Geometry, Differential Geometry, Theory of Representation, Theory of Operators, Spectral Theory, EDP, Probabilities, Special Functions. In the group of Mathematical Physics a wide range of subjects is studied. These topics range from the Theory of Dispersion (Scattering theory) to Statistical Mechanics through Resonance Theory, the study of Schrödinger Operators, magnetic Hamiltonians, Random Operators, Ergodic Systems, Non-Autonomous Quantum Systems, Open Quantum Systems, Systems Integrables, Topological Quantum Systems and several others.