On Thursday 28th of May 2026, in VRII-157, at 13:20, Cintia Pacchiano Camacho (Universidad Nacional Autónoma de México) will give a lecture for the Mathematical Colloquium.
Title: Variational methods and metric measure spaces.
Abstract: Variational methods appeared as an answer to the problem of finding minima of functionals. It is about giving a necessary and sufficient condition for the existence of the minimum, as well as conditions that allow its calculation and algorithms that let us compute it. Variational calculus is intimately linked with the theory of partial differential equations since the conditions for the existence of a solution to the minimization problem normally depend on the fact that said solution satisfies a certain differential equation. My research revolves around extending classical variational calculus results to metric measure spaces, focusing on methods associated with the existence and regularity of nonlinear parabolic and elliptic partial differential equations (PDEs). Instead of the classical Euclidean setting, we work purely on a variational level in the setting of a doubling metric measure space supporting a Poincaré inequality. In this talk, I will present one of my main research themes:
• The Total Variation Flow (TVF).
During the past two decades, a theory of Sobolev functions and first-degree calculus has been developed in this abstract setting. A central motivation for developing such a theory has been the desire to unify the assumptions and methods employed in various specific spaces, such as weighted Euclidean spaces, Riemannian manifolds, Heisenberg groups, graphs, etc. Analysis on metric spaces is nowadays an active and independent field, bringing together researchers from different parts of the mathematical spectrum. It has applications to disciplines as diverse as geometric group theory, nonlinear PDEs, and even theoretical computer science. This can offer us a better understanding of the phenomena and also lead to new results, even in the classical Euclidean case.