Complex Geometry
We study the geometry of group actions on Riemann surfaces, their Jacobians and Abelian varieties. We also study algebraic surfaces and their moduli spaces. The main tools come from complex analysis, algebraic geometry and group representations.
Researchers: Gonzalo Riera, Rubí E. Rodriguez, Giancarlo Urzúa
Dynamical Systems
Researchers: Godofredo Iommi, Jan Kiwi, Mario Ponce, Juan Rivera-Letelier
The Fundamentals of Set Theory
We study theories with classes that address the various problems created by removing the axiom of regularity and/or replacing it with weaker and more natural versions. These problems are always associated with the various versions of the chosen axiom.
Researcher: Gloria Schwarze
Geometric Function Theory
We study geometric and analytic aspects of univalent and locally univalent functions in domains contained in the complex plane. Of particular interest is the relation between the Schwarzian derivative, univalence, distortion, and the description of simply connected domains. Techniques of analysis are combined with complex differential equations and differential geometry. Recently we have extended our research to harmonic transformations of the plane and their relation to minimal surfaces, as well as generalizations to several complex variables.
Researcher: Martin Chuaqui
Inverse Problems
In inverse problems the goal is an indirect reconstruction of unknown parameters, using the measurements of a quantity affected by those parameters.
A common approach is to model the phenomenon as governed by a differential or integral equation. The unknown parameters become parameters in the equation and the measurements correspond to the restricted knowledge of the solutions of the equation. The inverse problem, in such a case, consists in recovering the unknown parameters of the equation from the restricted knowledge of its solutions.
In the study of these problems, the techniques used in partial differential equations, integral equations and operator theory are particularly useful.
Researcher: Matias Courdurier
Logic
We study abstract algebraic logic and the types of algebras that arise from the process of algebrization of deductive systems. We also study non-classical logic, in particular paraconsistent logic and multivalued logic.
Researchers: Renato Lewin, Irene Mikenberg
Mathematical Physics
Some models require tools from Functional Analysis, Operator Theory in function spaces and, in general methods from Partial Differential Equations. In this sense, the area of mathematical physics includes Spectral Analysis of Operators in Hilbert spaces and the study of the properties of the dynamics generated by operators in different contexts. For example, we study Scattering Theory, Spectral theory, Schrodinger Operators, Magnetic Hamiltonians, Non-Autonomous Dynamics, Unitary Operators, etc..
We also study the asymptotic behavior and qualitative properties of solutions of various equations of evolution and spectral problems related to harmonic analysis on Lie groups and algebras.
Researchers: Maria Angelica Astaburuaga, Olivier Bourget, Victor Cortes, Claudio Fernandez, Alberto Montero, Gueorgui Raykov, Rafael Tiedra, Jan Felipe Van Diejen.
Numerical Analysis
We study numerical methods to solve partial differential equations and equations of boundary integral operators, in particular the finite element method and the boundary element method. We focus on problems in non-smooth domains with singular solutions and the development of higher order methods like p and h-p versions. We analyze a priori error estimates and a posteriori error estimates and work on problems related to numerical linear algebra as domain decompositions. Examples of specific applications include the modeling of elastic materials and electromagnetic waves.
Researcher: Norbert Heuer.
Partial Differential Equations
The existence, uniqueness and regularity of solutions of systems of partial differential equations are analyzed. We also study the formation of singularities and patterns, the evolution of certain geometric objects, and the concentration of energy in localized structures. We sometimes try to model the behavior of physical phenomena via systems of equations, and simplify existing models via rigorous analysis and asymptotic approximations. Emphasis is placed on equations that require nonlinear analysis.
Researchers: Carmen Cortazar, Manuel Elgueta, Marta Garcia-Huidobro, Duvan Henao, Alberto Montero, Monica Musso, Mariel Saez.
Probability
Researchers: Alejandro Ramirez, Rolando Rebolledo.
