(Term 2, 2017/2018: Jan, 2018 -- April,
Topics in Differential Equations.
approaches to partial differential equations."
TBA. First organization meeting on Wednesday, Janunary 3, at
4:30pm in MathAnnex 1101.
- yhkim "at" math "dot" ubc 'dot' ca
- Phone 604-822-6324
- Office MATH 216
- Office hours (subject
to change): TBA.
- 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.
- 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
- Nodal sets of Laplace eigenfunctions: estimates of the
Hausdorff measure in dimension two and three. Aleksandr
Logunov and Eugenia Malinnikova.
- Solid background in measure theory and graduate level
partial differential equations. Background in differential
geometry will help.
- Class participation: 50%
- One assignment in the form of term paper (about 5 pages): 50%.