| Name: | Max Lind |
|---|---|
| Email: | ml6723@princeton.edu |
| Home Institution: | Princeton University |
| Project: | Topics in Arithmetic, Geometry, and Dynamics |
| Mentor: | Professor Alex Kontorovich |
I met with Professor Kontorovich each day this week and settled on my problem: to compute eigenvalues of the Laplacian on compact hyperbolic surfaces. Another goal is to learn material related to the Langlands Program.
I read articles about the Jacquet-Langlands correspondence, learned about Hejhal's algorithm for computing eigenvalues of the Laplacian on noncompact hyperbolic surfaces with a cusp, and tried unsuccessfully to modify Hejhal's algorithm so that it work in the compact setting.
This week, I tried to use Mathematica to compute eigenvalues of the Laplacian on compact arithmetic hyperbolic surfaces. I hadn't used Mathematica before and spent the week learning it and writing my program.
This week, I worked to compute eigenvalues of the Laplacian on compact arithmetic hyperbolic surfaces in Mathematica using the Jacquet-Langlands correspondence.
This week, I read expository articles about the Langlands program. These articles explained the Langlands program from the perspective of class field theory.
This week, I brushed up on the basic theory of lie algebras from Helgason's book "Differential Geometry, Lie Groups, and Symmetric Spaces" in order to read Bump's book "Automorphic Forms and Representations."
I read parts of "The Genesis of the Langlands Program" by Mueller and Shahidi, as well as "An Elementary Introduction to the Langlands Program" by Gelbart and "Riemann's Zeta Function and Beyond" by Gelbart and Miller.
I worked on my final presentation, which is about how the Langlands program might generalize Artin reciprocity.
I wrote my final paper on L-functions related to the Langlands program.