Summer Northwestern Analysis Program

SNAP 2017

Northwestern University

Week 2 - Topics in PDE

July 31-August 4, 2017

All lectures to be held in Frances Searle Building Room 1-441. Click here for a campus map.

Please be aware that because of construction on campus some paths to the Frances Searle Building may be closed. Please give yourself an extra time to find the building on Monday morning.

Group Photo

	Monday, July 31	Tuesday, August 1			Wednesday, August 2			Thursday, August 3	Friday, August 4
8.45am - 10.15am	Roy-Fortin	Roy-Fortin		9.00am - 10.30am	Visan		9.00am - 10.30am	Visan	Roy-Fortin
10.45am - 12.15pm	Auffinger	Auffinger		11.00am - 12.30pm	Roy-Fortin		11.00am - 12.00pm	Problem Session Auffinger	Problem Session Roy-Fortin
2.00pm - 3.30pm	Visan	Visan			Free afternoon		2.00pm - 3.30pm	Auffinger	Auffinger
4.00pm - 5.00pm	Problem Session Roy-Fortin	Problem Session Visan			Free afternoon		4.00pm - 5.00pm	Problem Session Visan	Problem Session Auffinger

Abstracts:

Antonio Auffinger   (Northwestern). Title: Probabilistic methods in PDE

Abstract: This mini-course is devoted to basic connections between partial differential equations and probability theory. First, we will introduce Brownian motion (BM) and derive some of its main properties. We will study hitting and exit times of subsets of R^d. Then, we will learn how solutions of classic parabolic and elliptic PDEs can be expressed using expectations of functionals of BM. In the last lecture, we will go over some interacting particle systems where PDEs appear naturally as hydrodynamic limits.

Guillaume Roy-Fortin   (Northwestern). Title: Eigenfunctions and eigenvalues

Abstract: It is the middle of the summer, the temperature is very warm and you hear a nice drum beat as you quietly sit on the beach by the lake shore. Your mind starts to wander: I can hear that drum, but can't quite see it. How big is it? Does it have to be circular? Can there be more than one drum producing such a soothing sound? Haunted by these fascinating questions, you eagerly leave the beach (don't forget your flip-flops) and attend this mini-course about eigenvalues and eigenfunctions of the Laplace operator.

• In the first lecture, we will start with the model cases of vibrating membranes and discuss the famous question of Mark Kac "Can one hear the shape of a drum?", hopefully answering some of the aforementioned interrogations that made you leave the lakefront. Topics: Dirichlet/Neumann eigenfunctions on rectangle and disk, counting eigenvalues and the Gauss circle problem, isospectrality.
• In the second lecture, we will focus on eigenvalues, present some of their most basic properties and discuss how to count them. Topics: Variational formulae, domain monotonicity, isoperimetric inequalities, Weyl's law.
• The third lecture will transfer the focus to eigenfunctions on surfaces. We will use the spherical harmonics to highlight some of the extreme behavior of eigenfunctions, such as L^p norms and doubling index. Topics: spherical harmonics, Gaussian beams, Sogge's L^p bounds, Donnelly-Fefferman growth bound.
• In the final lecture, we will talk about the so called nodal set of eigenfunctions. When one considers an eigenfunction as the probability density of a quantum particle in a given energy state, the zero - or nodal - set can be thought of as the set where that particle is least likely to be found. A famous conjecture by Yau predicted the size of that set and we will discuss recent advances on that front. Topics: nodal density, Courant's nodal domain theorem, Yau's conjecture, the Logunov papers.

• Lecture notes of Yaiza Canzani on the Laplacian
• Trends and tricks in spectral theory by Boulton and Levitin
• Lower bounds on measure of nodal sets by Sogge-Zelditch
• Lower bounds on measure of nodal sets II by Sogge-Zelditch
• Roy-Fortin's paper on zero sets and average local growth
• A nice video of Chladni patterns
  
  

Monica Visan   (UCLA). Title: Introduction to the nonlinear Schrödinger equation

Abstract: We introduce the Schrödinger equation as an example of a dispersive equation. Focusing on one concrete model, we illustrate some of the tools and techniques used to prove the existence of solutions and describe their asymptotic behavior.