Summer Northwestern Analysis Program

SNAP 2017

Northwestern University

Week 3 - Advanced Topics in PDE

August 7-11, 2017

All lectures to be held in Technological Institute Room L361. Click here for a campus map.

Registration and coffee starts at 8.30am on Monday, August 7, in Technological Institute Room L361.

Group Photo

Monday, August 7 Tuesday, August 8 Wednesday, August 9 Thursday, August 10 Friday, August 11
9.00am - 10.30am Naber-Valtorta Wunsch Naber-Valtorta Naber-Valtorta Naber-Valtorta
11.00am - 12.30pm Wunsch Zelditch Zelditch Zelditch Zelditch
2.30pm - 4.00pm Naber-Valtorta Naber-Valtorta Free afternoon Wunsch Wunsch
4.30pm - 5.30pm Problem Session Naber-Valtorta Problem Session Wunsch Free afternoon Problem Session Zelditch Problem Session Naber-Valtorta


Monday August 7, 5.45pm there will be Pizza in the Mathematics Department Common Room, 2nd floor Lunt Hall



Aaron Naber   (Northwestern) and Daniele Valtorta   (Zürich). Title: Singularities of harmonic maps

Abstract: In this mini-course, we will discuss new results about the structure of singular sets of harmonic maps.

In particular, we will show that the singular set of a minimizing harmonic map u between two Riemannian manifolds M and N is rectifiable with codimension 3 in the domain, with uniform volume bounds depending only on its energy.

This result is based on a refined analysis of the monotonicity formula for harmonic maps and a Reifenberg-type theorem for the singular set. The technique used to prove this result is new, and it can be adapted to various different problems in geometric analysis.

During the course, we will give a brief introduction on harmonic maps and the quantitative stratification technique before focusing on the proof of the main result.

References for this result can be found on arXiv at arXiv:1504.02043, arXiv:1611.03008, arXiv:1612.08052.

TA: Zahra Sinaei (Northwestern)

Lecture Notes

Jared Wunsch   (Northwestern). Title: Microlocal analysis and evolution equations

Abstract: We will discuss the applications of microlocal analysis to evolution equations, with related applications in spectral asymptotics. We will begin by some warmup discussion of estimates for the wave and Schrödinger equations in Euclidean space, and then discuss what changes we might need to make to prove the same sorts of estimates in curved spaces. This will motivate the introduction of the calculus of pseudodifferential operators, which will be presented essentially axiomatically: the point of the minicourse will be to show that the tools of microlocal analysis are good for, and to encourage the student to read one of the many excellent and careful treatments of the technical details at a later date. We will then return to discussion of wave and Schrödinger equations, this time with more general geometric assumptions, and see how we can employ microlocal methods in developing energy estimates, including the propagation of singularities for the wave equation and, more generally, for operators of real principal type. Our work on the wave equation will lead to some classic results in spectral asymptotics.

TA: Perry Kleinhenz (Northwestern)

Lectures Notes
Problem Set 1

Steve Zelditch   (Northwestern). Title: Local and global analysis of eigenfunctions

Abstract: This mini-course is devoted to the behavior of eigenfunctions of the Laplacian of a compact Riemannian manifold as the eigenvalue λ tends to infinity. Eigenfunctions represent stationary states of quantum systems. The limit as λ tends to infinity is the classical limit in which high frequency waves get connected to geodesics. The main theme of the course is to relate eigenfunctions and geodesic flow.

Two important invariants of eigenfunctions are their nodal (zero) sets and their microlocal defect measures. The last year has seen significant new results in both areas, and I plan to cover parts of them in my mini-course. One is the recent proof by A. Logunov of the lower bound conjecture of S.T. Yau on hypersurface measure of nodal sets, along with a polynomial upper bound. Another is the use of ergodicity of the geodesic flow to obtain lower bounds on numbers of nodal domains on negatively curved surfaces.

No prior knowledge of eigenfunctions is assumed. Foundational results such as the Weyl law, Egorov's theorem, quantum ergodicity and the local structure of nodal sets will be presented in early lectures.

TA: Robert Chang (Northwestern)

Lecture Notes 1
Lecture Notes 2
Lecture Notes 3
Lecture Notes 4
Problem Set 1