Research interests
My current research mainly focus on Bruhat-Tits buildings and $p$-adic representations.
Specifically, I am interested in studying the simplicial distance in Bruhat-Tits buildings,
which are related to many topics such as the Moy-Prasad subgroups of a reductive group and almost self-dual lattices in a $p$-adic symplectic/orthogonal space. The larger goal is to understand the fixed-point sets in Bruhat-Tits buildings and their applications to $p$-adic representations.
For more details see my Research Statement
Other interests include $p$-adic geometry, representation theory, mathematical physics, algebraic analysis, tensor triangular geometry, higher category theory, homotopical algebras, transcendental number theory, etc.
Publications and preprints
- Xu Gao and Ang Li, “The stable Picard group of finite Adams Hopf algebroids with an application to the $\mathbb{R}$-motivic Steenrod subalgebra $\mathcal{A}(1)^{\mathbb{R}}$”, arXiv:2306.12527.
- Xu Gao, “Simplicial volumes in Bruhat-Tits buildings of split classical type”, arXiv:2210.03328.
- Xu Gao, Ming Liu, Chengming Bai and Naihuan Jing, “Rota-Baxter Operators on Witt and Virasoro Algebras”, Journal of Geometry and Physics, vol.108, 2016, pp.1-20.
- Xu Gao, “Extensions and Non-abelian Cohomology of Pre-Lie Algebras”, Master degree thesis, 2015, Nankai University.
Expository writing
Under construction
- Review on Bruhat-Tits buildings, 2021.
- Henselian local rings, 2020.
- An English translation version of Chapter 16 of EGA, 2018.
- my reading notes on Deligne’s Equations Differentielles a Points Singuliers Reguliers, 2018.
need updates
- Note on Spectral Sequences, 2018.
- Reading notes on Complex $D$-modules, 2018.
need updates
- Homological algebra note, 2018.
need updates
- Note on Dirichlet’s Theorem of primes in arithmetic progressions, 2016.
- Solutions to Neukirch’s “Algebraic Number Theory”, 2015.
- Solutions for exercises in Humphreys’ GTM 9, 2013.