My current research focuses on the algebro-geometric theory of conformal blocks of vertex operator algebras, building on the foundational work of Beilinson-Drinfeld and Frenkel-BenZvi, and further developed by Damiolini-Gibney-Krashen-Tarasca. I am collaborating with Jianqi Liu and Yiyi Zhu to extend this theory to orbifold settings. Additionally, I am working with Angela Gibney, Daniel Krashen, and Jianqi Liu to develop a tensor product theory for modules of vertex operator algebras, using an algebro-geometric approach informed by the aforementioned works and findings from previous projects.
I am also interested in Bruhat-Tits buildings and $p$-adic representations, particularly in problems related to simplicial distance and concave functions. The goal is to develop a combinatorial geometry on Bruhat-Tits buildings based on the concept of simplicial distance, and to apply this geometry to the study of $p$-adic representations. In this context, concave functions play a crucial role in connecting the combinatorial and group-theoretic aspects of Bruhat-Tits buildings.
My interest also extends to tensor triangulated geometry, particularly in the (co)stratification of tt-categories arising from important algebraic objects. The (co)stratification of representations of finite group schemes plays a key role in the development of tensor triangulated geometry and serves as a benchmark for many important theories. This project aims to extend the work of Barthel-Benson-Iyengar-Krause-Pevtsova to finite Hopf algebroids, with the ultimate goal of studying the stratification of Adams Hopf algebroids, a class of important objects in homotopy theory.
Other interests include algebraic analysis, $p$-adic geometry, higher category theory, homotopical algebras, transcendental number theory, etc.
Xu Gao, Jianqi Liu, and Yiyi Zhu, Twisted restricted conformal blocks of vertex operator algebras II: twisted restricted conformal blocks on totally ramified orbicurves, arXiv:2403.00545.
In this paper, we introduce a notion of twisted restricted conformal blocks on totally ramified orbicurves and establish an isomorphism between the space of twisted restricted conformal blocks and the space of twisted conformal blocks. The relationships among twisted (restricted) conformal blocks, $g$-twisted (restricted) correlation functions, and twisted intertwining operators are explored. Furthermore, by introducing a geometric generalization of Zhu’s algebra and its modules, we obtain a description of the space of coinvariants by modules over associative algebras and show it is finite-dimensional under some conditions.
In particular, a more conceptual proof of the $g$-twisted fusion rules theorem in vertex operator algebra theory is provided.
Xu Gao, Jianqi Liu, and Yiyi Zhu, Twisted restricted conformal blocks of vertex operator algebras I: $g$-twisted correlation functions and fusion rules, arXiv:2312.16278.
In this paper, we introduce a notion of $g$-twisted restricted conformal block on the three-pointed twisted projective line $\mathfrak{x}\colon\overline{C}\to\mathbb{P}^1$ associated with an untwisted module $M^1$ and the bottom levels of two $g$-twisted modules $M^2$ and $M^3$ over a vertex operator algebra $V$. We show that the space of twisted restricted conformal blocks is isomorphic to the space of $g$-twisted (restricted) correlation functions defined by the same datum and to the space of intertwining operators among these twisted modules. As an application, we derive a twisted version of the Fusion Rules Theorem.
Xu Gao and Ang Li, “The stable Picard group of finite Adams Hopf algebroids with an application to the $\mathbb{R}$-motivic Steenrod subalgebra $\mathcal{A}^{\mathbb{R}}(1)$”, Journal of Pure and Applied Algebra, Volume 228, Issue 11, 2024.
In this paper, we investigate the rigidity of the stable comodule category of a specific class of Hopf algebroids known as finite Adams, shedding light on its Picard group.
Then, we establish a reduction process through base changes, enabling us to effectively compute the Picard group of the $\mathbb{R}$-motivic mod $2$ Steenrod subalgebra $\mathcal{A}^{\mathbb{R}}(1)$.
Our computation shows that $\operatorname{Pic}(\mathcal{A}^{\mathbb{R}}(1))$ is isomorphic to $\mathbb{Z}^4$, where two ranks come from the motivic grading, one from the algebraic loop functor, and the last is generated by the $\mathbb{R}$-motivic joker $J$.
Xu Gao, “Simplicial distance in Bruhat-Tits buildings of split classical type”, UCSC Ph.D. Dissertation.
This study delves into \emph{simplicial distances} on Bruhat-Tits buildings of split classical types (namely, types of $A_n$, $B_n$, $C_n$, $D_n$, and any combination thereof).
Simplicial distance serves for measuring the proximity between vertices within the simplicial structure of a building.
The purpose of this research is three-fold: (i) to present a concrete characterization of the simplicial distance; (ii) to better understand simplicial balls; and (iii) to derive a formula for the simplicial volume and explore its asymptotic growth.
To accomplish these goals, vertices in Bruhat-Tits buildings are carefully analyzed under three frameworks: root systems, norms, and lattices.
By leveraging concave functions, we interpret simplicial balls as fixed-point sets of Moy-Prasad subgroups and deduce a formula for the simplicial volume.
Additionally, the theory of $q$-exponential polynomials is developed to facilitate the asymptotic study.
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.
Under construction
need updatesneed updatesneed updates