## Frontiers in Mathematics

**Date: ** November 29, 2023

**Time: ** 4:00PM - 5:00PM

**Location: ** Bloc 117

**Speaker: **Frontiers Speaker - Fabrice Baudoin, University of Connecticut

**Description: **Talk 1: Heat flow and sets of finite perimeter
It is well known that among all plane curves, the circle encloses the maximum area for a given perimeter. This question is called the isoperimetric problem in the plane. Mathematicians often generalize problems to encompass broader settings and different levels of abstraction. To effectively address the isoperimetric problem, we require precise definitions of area or volume and perimeter for a set. The rigorous development of these concepts has fueled remarkable advancements in geometric measure theory throughout the 20th and early 21st centuries. Building upon the work of H. Lebesgue (1901-1902), we now have a comprehensive understanding of set volume within the category of measure spaces. The concept of perimeter, however, is more elusive and can be approached from various perspectives. The foundational contributions of R. Caccioppoli (1927-1928) and E. De Giorgi (1950s) have laid the groundwork for our current understanding. This lecture will explore early approaches to the theory of sets of finite perimeter in abstract measure spaces. We will also delve into recent developments of the theory in the context of Dirichlet spaces, where it has been established that a coherent theory of perimeters is intricately linked to fundamental properties of solutions to the heat equation.