Undergraduate Research Project, Fall 2018-Spring 2019

as part of the Exchange Program at the Department of Mathematics, and the Discovering America Program at MSU

The undergrad research group won the Best Talk Award in the 16th Annual Student Mathematical Conference at MSU in May 2019. Here is the preprint the research group completed in the summer of 2019 on the Landscape Theory:

Project Title: Landscape theory for tight-binding Hamiltonians


Prof. Ilya Kachkovskiy

Assistant Professor

Department of Mathematics
Michigan State University
Email: ikachkov@msu.edu

Dr. Shiwen Zhang

Visiting Assistant Professor

Department of Mathematics 
Michigan State University
Email: zhangshiwen@math.msu.edu

Current Enrolled Undergraduate Students at MSU:

Buhl, John; Cinzori, Isaac; Ginnett, Isabella; Landry, Mark; Li, Yikang; Liu, Cincy

Current Weekly Meeting Hours:

Mon 12:30pm-1:30pm; Thu 5:10pm-6:00pm

Project Area:

Analysis, Spectral Theory, Mathematical Physics

Project Description:

Anderson localization is one of the central phenomena studied in modern mathematical physics, especially in dimensions 2 and 3, starting from Nobel-prize winning discovery by P. W. Anderson. Recently, a new approach for the 1D discrete model was proposed by M. Lyra, S. Mayboroda, and M. Filoche, which shows interesting relations with the Dirichlet problem on the lattice and also allows to significantly reduce complexity of some numerics related to the problem. The main goal of the project is to extend this this approach to higher (and most interesting physically) dimensions. The project will involve advanced reading, possible new results in finite and infinite-dimensional spectral theory, understanding physics behind some problems in linear algebra, and novel numerical experiments. Original research results are expected as an outcome for successful students.

Prerequisites:   Linear algebra
Recommended Background: Basic operator theory, some numerical linear algebra. Knowledge of spectral theory will be helpful but can also be included as advanced reading.

Notes on landscape theory

Project Contents on Overleaf (read-only).

Tentative Weekly Schedule:

Other Information and Additional resources:

Linear Algebra Done Wrong, by S. Treil. A free online book with a clean presentation, available here.

Academic Calendar;

MSU Mathematics Undergraduate Programs; MSU Mathematics Graduate Programs.