(Term 2, 2017/2018: Jan, 2018 -- April, 2018)

MATH
606D:201
Topics in Differential Equations.

"**Geometric
approaches to partial differential equations."**

Class:
TBA. **First organization meeting on Wednesday, Janunary 3, at
4:30pm in MathAnnex 1101.**

**Location:** TBA

**[Instructor] **Young-Heon
Kim

- yhkim "at" math "dot" ubc 'dot' ca
- Phone 604-822-6324
- Office MATH 216
- Office hours (subject
to change): TBA.

**[Tentative plan]**

- We will consider geometric methods for studying partial
differential equations, where one of the most important
challenges is to understand regularity/singularity of solutions.
First, we will cover convexity and related estimates, such as
Alexandrov estimates, in studying elliptic equations. This will
include the method of Caffarelli for studying the Monge-Ampere
equation. Some more recent advances will also be treated,
including quantitative stratification as developed by
Cheeger, Naber and collaborators, for estimating
singular/critical sets of elliptic equations, as well as the
geometric method of Logunov and Malinnikova for estimating nodal
sets of Laplace eigenfunctions.

**[Key references]**

- The Monge–Ampère Equation and Its Applications, by Alessio Figalli (Book, 2017, EMS)
- Quantitative Stratification and the Regularity of Harmonic Maps and Minimal Currents, by Jeff Cheeger, Aaron Naber. Communications on Pure and Applied Mathematics Vol 66, Issue 6 (2013), 965–990.
- Nodal sets of Laplace eigenfunctions: estimates of the
Hausdorff measure in dimension two and three. Aleksandr
Logunov and Eugenia Malinnikova.
https://arxiv.org/pdf/1605.02595.pdf

**[Prerequisites]**

- Solid background in measure theory and graduate level
partial differential equations. Background in differential
geometry will help.

**[Grading]**

- Class participation: 50%
- One assignment in the form of term paper (about 5 pages): 50%.