You are invited to a:

** Distinguished Lecture Series by**

**Prof. David Jerison (MIT)**

THE GEOMETRY OF LEVEL SETS

**1st lecture**: *Level surfaces of eigenfunctions and free boundaries*

Monday, January 20, 2020 at 15:30

**2nd lecture**: *Harnack inequalities for minimal surfaces and free boundaries*

Wednesday, January 22, 2020 at **16:30**

**3rd lecture**: *Global solutions and rigidity*

Thursday, January 23, 2020 at 15:30

**Abstracts:**

**Lecture 1: **in lecture 1 we will begin by describing the Hot Spots Conjecture of J. Rauch.

This question is an essential test of our understanding of the shapes of level sets of the simplest eigenfunctions. We will then relate our question to a variety of others about level sets and other kinds of interfaces – free boundaries, minimal surfaces, isoperimetric surfaces – as well as the KLS Hyperplane Conjecture in high dimensional convex analysis.

**Lecture 2: **in lecture 2 we will discuss how methods from geometric measure theory and elliptic regularity theory (developed for minimal surfaces) apply to level sets. We will focus on a version of the Harnack inequality that tells us how level surfaces influence each other. This gets us part way towards understanding hot spots.

**Lecture 3:** in lecture 3 we will explain how complex analysis and differential geometry, in particular as developed for minimal surface theory, can be used to characterize global solutions and prove rigidity and regularity results for free boundaries. This gives further insights into the missing ingredients that will be needed to understand level sets of eigenfunctions.

All lectures will be in Amado 232.

Light refreshments will be given in the faculty lounge on the 8th floor.