AMUSE

Date: February 10, 2020

Time: 6:00PM - 7:00PM

Location: BLOC 220

Speaker: Patricia Alonso Ruiz, Texas A&M University

Title: Measuring a sponge: how to formulate the isoperimetric problem in fractals

Abstract: The yearly budget of a ranch owner allows him to purchase one mile of fence to delimit a piece of land for the cattle to graze. What shape will provide the largest possible space for the cattle? This question is known as the isoperimetric problem, which consists in finding among all sets with the same perimeter the one that maximizes its area. The problem can be posed in any dimension, and in the usual Euclidean space its solution is known to be the circle or, more generally, a ball. But what if our ambient space is rather porous, like a sponge or a lung, something "fractal"? To formulate the isoperimetric problem we need good notions of area and perimeter, but the standard Euclidean ones become useless here. So, how can we measure the area and the perimeter of a piece of sponge? In this talk we will present the Hausdorff measure and outline a newly developed concept of perimeter as the natural candidates to make sensible measurements in fractal sets.