SYLLABUS

Geometric Inequalities (Math 8803) Fall 2024, Georgia Tech

Lecture
M W 3:30-4:45 Skiles 311

Instructor

Professor Mohammad Ghomi

Office: Skiles 203
Office hours: M W 2-3
Email: ghomi@math.gatech.edu

Course Description

The focus of this class will be the isoperimetric inequality in spaces of nonpositive curvature. The classical isoperimetric inequality states that in Euclidean space spheres bound any given volume with the least surface area. Generalizing this phenomenon to spaces of nonpositive curvature, or Cartan-Hadamard manifolds, has been one of the outstanding open problems in differential geometry, which has stimulated much work in Calculus of variations, PDEs, and even mathematical physics. We will survey the related literature and discuss many other related open problems.

We will begin by reviewing the various proofs of the isoperimetric inequality in Euclidean space, including connections with the theory of convex bodies, the Sobolev inequality, Faber-Krahn inequality, and then move to Riemannian manifolds. We will give a quick introduction to notion of curvature, so previous knowledge of Remannian Geometry is not necessary. Among other relevant topics we will explore CAT(0) spaces, various notions of convexity, and curvature flows.

Prerequisites

A previous course in differential geometry is recommended but is not strictly necessary.

Textbooks

Instructors notes, and various papers and texts which will be posted on the course canvass site.

Grading

The grade is based on class participation (the students are expected to attend all lectures), and a presentation on a topic related to the class discussions.