SPRING CENTRAL SECTIONAL SAMPLER space 푅푛, one needs a notion of volume of a set and a Heat Flow and Sets notion of perimeter of a set. The rigorous and deep un- derstanding of these two notions has motivated many of of Finite Perimeter the spectacular developments in geometric measure the- ory throughout the 20th and beginning of the 21st cen- turies. Our modern understanding of the notion of vol- Fabrice Baudoin ume is largely based on the seminal works (1901–1902) by H. Lebesgue. Nowadays, one understands volume in A basic motivating question is: what is a good mathematical the category of measure spaces. The notion of perime- structure on a space that will allow us to define an intuitively ter is more elusive and can be understood from different reasonable and useful notion of perimeter for “good” sets? The viewpoints, yielding different generalizations. The Italian question is, of course, a little vague but thinking about it school in geometric measure theory played a major role in from different viewpoints is certainly fruitful and yields an advancing the theory. interesting mathematical adventure. The first idea for how to define the perimeter ofaBorel We will argue that Dirichlet spaces provide a convenient set 퐸 ⊂ 푅푛 is due to R. Caccioppoli (1904–1959) and is framework in which we can define sets of finite perimeter based on a generalization of the notion of rectifiability for and prove theorems generalizing classical results from Eu- a curve. Essentially, the perimeter of a set is the limit of clidean space, like the classical isoperimetric inequality, in the surface areas of polyhedra approximating 퐸. This is an elegant way. a perfectly reasonable take on what quantity the perime- The isoperimetric inequality in the plane. Dido was, ac- ter of a set should represent. However, this definition is cording to ancient Greek and Roman sources, the founder somehow rigid. It makes strong use of the Euclidean struc- and first Queen of Carthage (in modern-day Tunisia). The ture of the space, using the notion of a polyhedron and its legend says that when Dido arrived in 814 BCE on the coast area. Also, this definition is difficult to reconcile with the of Tunisia, she asked for a piece of land. Her request would variational interpretation of perimeter that is given by the be satisfied provided that the land could be encompassed calculus of variations (the Gauss-Green formula). A some- by an oxhide. This land became Carthage and Dido be- how dual, more flexible and easier to work with, definition came the queen. But: what is the shape of the land chosen of the perimeter of a set is also due to R. Caccioppoli. It by Queen Dido? Mathematically, we can first simplify and coincides with the previous one and is given by rephrase this problem as: in the Euclidean plane, what is the curve which encloses the maximum area 퐴 for a given 푃(퐸) = sup ∫ 퐝퐢퐯(휙)푑푥, perimeter 푃? 퐸 This is the isoperimetric problem in the plane. If we where the supremum is taken over the set of smooth and 푛 restrict ourselves to smooth closed curves and understand compactly supported vector fields in 푅 such that ‖휙‖∞ ≤ the perimeter of a set as the length of its boundary, then the 1. We note that this definition only requires a differen- problem can be solved using basic calculus: the method tial structure and a Riemannian metric to define the diver- of Lagrange multipliers. As is intuitively clear, the curve gence of vector fields; it can therefore be extended toany which encloses the maximum area 퐴 for a given perimeter Riemannian manifold. 푃 is the circle and one has the isoperimetric inequality More recently, following works by E. De Giorgi (1928– 1 1996) and M. Ledoux, a characterization of sets with finite 퐴 ≤ 푃2. 푛 4휋 perimeter in 푅 that uses very little structure was discov- 퐸 푅푛 퐸 So the city limits of Carthage formed a (half) circle. ered. If is a Borel set in , then has finite perimeter if We can still visit the ruins of Carthage today, and a small and only if museum there explains the isoperimetric story of Queen √휋 lim inf ∫ ∫ |1퐸(푥) − 1퐸(푦)|푝푡(푥, 푦)푑푥 푑푦 < +∞, Dido. 푡→0 2√푡 푅푛 푅푛 Modern footprints. As mathematicians, we like to formu- ‖푦−푥‖2 1 − late problems at different levels of generality and in a form where 푝푡(푥, 푦) = 푒 4푡 is the Euclidean heat ker- that can be generalized to different settings. To formulate (4휋푡)푛/2 nel. Moreover, in that case the isoperimetric problem in the 푛-dimensional Euclidean √휋 Fabrice Baudoin is a professor of mathematics at the University of Connecticut. 푃(퐸) = lim ∫ ∫ |1퐸(푥) − 1퐸(푦)|푝푡(푥, 푦)푑푥 푑푦. 푡→0 푛 푛 His email address is [email protected]. 2√푡 푅 푅 For permission to reprint this article, please contact: This characterization of sets of finite perimeter, which is [email protected]. due to M. Ledoux for balls and in 2007 to M. Miranda Jr., DOI: https://doi.org/10.1090/noti2257 D. Pallara, F. Paronetto, and M. Preunkert in the general

582 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 4 case, only requires a heat semigroup and is therefore ap- pealing since heat semigroups might be defined in the large category of Dirichlet spaces. Thus, at least formally, sets of finite perimeter may be perfectly well defined inany Dirichlet space and we do not need a gradient or a distance. This is the point of view we will take and will explain the associated theory. A key point is understanding the nor- 1 malizing factor which, in Euclidean space, reflects its √푡 ”smooth” structure, but which has to be modified to un- derstand sets of finite perimeter in nonsmooth spaces like fractals.

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