The modern discipline of geometry interacts virtually with every branch of science, from mathematics to physics and to the applied sciences.  For instance, in string theory and particle physics, geometric invariants have been used to classify properties of solutions to equations from field theories, while in computer graphics and simulation, the study of triangulations and their deformations plays a crucial role.  Moreover, within different branches of mathematics, geometry has also played an important role, being perhaps one of the most remarkable examples, the proof of the Poincaré conjecture using the Ricci flow.

The wide range of problems related to geometry require an ample set of tools to address them, which vary from analytic to purely algebraic. For instance, on one side an important theme in the area has been the development of sophisticated techniques from the theory of partial differential equations, which provided insight of curvature and topological properties of manifolds, while on the algebraic side, important breakthroughs have come from the study of the birational geometry and moduli spaces of algebraic varieties.

Our group develops research in geometry from various points of view (both analytic and algebraic). Some of the topics that we cover are the study of the geography and compactified moduli spaces of surfaces of general type, the geometric and analytic aspects of univalent and locally univalent functions in domains of the complex plane, harmonic transformations of the plane and their link to minimal surfaces, geometric flows and their applications, variational problems arising from geometry (such as minimal surfaces, harmonic maps and optimal point configurations) and  more generally, the interplay between geometric objects and the theory of partial differential equations.

RESEARCHERS