# Geometric Analysis and Topology Seminar

#### A New Vertical-Versus-Horizontal Isoperimetric Inequality on the Heisenberg Group, with Applications to Metric Geometry and Approximation Algorithms

**Speaker:**
Assaf Naor, Princeton

**Location:**
Warren Weaver Hall 1314

**Date:**
Thursday, November 10, 2016, 10 a.m.

**Synopsis:**

In this talk we will show that for every measurable subset of the Heisenberg group of dimension at least 5, an appropriately defined notion of its "vertical perimeter" is at most a constant multiple of its horizontal (Heisenberg) perimeter. We will explain how this new isoperimetric-type inequality solves open questions in analysis (an endpoint estimate for a certain singular integral on \(W^{1,1}\)), metric geometry (sharp nonembeddability into \(L_1\)) and approximation algorithms (asymptotic evaluation of the performance of the Goemans-Linial algorithm for the Sparsest Cut problem). Joint work with Robert Young.