Keywords:
Geometric Analysis, Inverse Problems, Design Optimization, Dirichlet to Neumann (DtN) Map, Steklov Eigenvalues, Spectral Theory, Semiclassical Analysis, Quantum Ergodicity
Synopsis (16 page pdf)
Description
I have surveyed literature on spectral geometry of the Laplacian, including works of Lichneroqicz-Obata, Gromov, Li-Yau, Hoffman-Ostenhoff-Nadirashvili. Specifically, I am focusing on the nodal geometry of Laplace eigenfunctions in the compact setting, including works of Mangoubi (inner radius estimates), Colding-Minicozzi and Georgiev-Mukherjee. I am highly interested in local elliptic techniques, as exemplified in the recent work of Logunov-Malinnikova on the size of nodal sets and also in understanding how such frequency function based local elliptic techniques carry over to the Steklov setting. I am currently reading some works of Bellova-Lin, Zhu, Georgiev-Roy-Fortin, in that direction. Also, currently beginning to survey global techniques of semiclassical analysis, starting from the works of Sogge-Zelditch and related problems of eigenfunction mass concentration and quantum ergodicity. For instance, the billiard flow problems as in Marzuola, Hassell-Hillairet-Marzuola, Cekic-Georgiev-Mukherjee.
Nodal Geometry
Heat Equation, Wave Equation and Geometric problems
- Harmonic Maps in Analysis of Nonlinear Geometric PDEs based on Prof. Aaron Naber’s lectures at IITB.
Minimal Surfaces
- Studied relationship of Steklov Eigenvalues and Minimal Surfaces, extremal eigenvalue problems in connection to minimal surfaces. Notes based on Prof. Richard Schoen’s talk, includes overview of his groundbreaking work in collaboration with Prof. Ailana Fraser.