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:

- J. Buhl, I. Cinzori, I. Ginnett, M. Landry, Y. Li, X. Liu,

Landscape Theory for Schrödinger Operators with General Hopping Terms on a Finite Lattice. arXiv:1907.00808.

**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

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

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

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.

- Week 1-4: Introduction to Quantum Mechanics and Spectral Theory; Adance reading on
*Linear Algebra Done Wrong*, by S. Treil, and ``baby spectral theory'' in*Methods of Modern Mathematical Physics-Volume 1*, by M. Reed-B. Simon.- Jan 14, Mon, lecture by I. Kachkovskiy:
*Introduction to qunatum mechanics*. Review problems set I for linear algebra. HW1 - Jan 24, Thu, problem solving sessions on eigenvalue problems
- Jan 28, Mon, disscusion on Harmonic Oscillator, simple examples of differential/difference equations, and Schrödinger/Jacobi matrix
- Feb 1st, 4:30 – 6:30, C-304 WH. Undergraduate Math/Stat Research Project Introductions Exchange and MSU Student Research Teams. Presenters: Cinzori, Isaac; ; Li, Yikang and Liu, Cincy. Talk Slides.

- Jan 14, Mon, lecture by I. Kachkovskiy:
- Week 5-8: Advance reading on,
*Landscape theory on discrete lattices*, by M. Lyra, S. Mayboroda, and M. Filoche.- Feb 4th, 12:35-1:25. Conjectures and open problems of landscape theory for Schrödinger operators on a finite lattice.
- Feb 7th, Thu, general audience talk by S. Zhang: From eigenvaleus to spectrum, part I. Talk Slides.
- Feb 11th, Mon, problem solving session.
- Feb 18th, Mon, problem solving session.
- Feb 21st, Thu, convex/concave functions and Maximum/Minimum Principle. Notes on Maximum Principle.

- Week 9-16: Research Projects
- Extension of landscape theory to general hopping operators
- Landscape theory on higher dimensional lattices
- Numerical experiments on landscape theory
- Landscape theory on infinite lattices, with application to half/full line Anderson model, and to periodic/quasi-periodic tight-binding Hamiltonians