Math apps

Langlands program tools

• Arthur’s Market
• Green formula apps
• Bruhat decomposition - Finds the Bruhat Decomposition of a symbolic matrix in $\mathrm{GL}_n$. Powered by PyScript and SymPy.
• Green Polynomials - allows one to compute Green polynomials for representations of $\mathrm{GL}_n\left( \mathbb{F}_q \right)$. It uses Theorem 2.6 from the article The Green polynomials via vertex operators by Naihuan Jing and Ning Liu.
• Nilpotentor - allows one to explore characters that correspond to nilpotent orbits for general linear groups as defined by David Ginzburg in Section 2 of the paper Certain conjectures relating unipotent orbits to automorphic representations . See also Section 5.1 of the paper On Fourier Coefficients of Automorphic Forms of GL(n) by Dihua Jiang and Baiying Liu.
• Root exchange demonstration - Demonstrates the root exchange algorithm done in Section 3.2 of the paper The generalized doubling method: $(k,c)$ models by Yuanqing Cai, Solomon Friedberg, Dmitry Gourevitch and Eyal Kaplan.

Computational laboratories in number theory

In Winter 2024 I taught MATH 476 at the University of Michigan. We followed Hugh Montgomery’s wonderful book Computational Laboratories in Number Theory. I implemeted many (variants) of these programs in HTML/JavaScript. They are now available on http://numbertheorylabs.com/apps/. I would like to thank Hugh for designing these labs, and for sharing the code of his original DOS programs with me.