## Xiaorun Wu(吴肖润)

Department of Mathematics
Columbia University
Room 509, MC 4406
New York, NY 10027
Email: xiaorunw at math dot columbia dot edu

• I am a second-year graduate student at Columbia University, expected to graduate in May 2027. My advisor is Professor Eric Urban. I obtained my undergraduate degree in May 2022 at Princeton University, under the supervision of Professor Shou-Wu Zhang and Professor Christopher Skinner.
• My research interest is in number theory & arithmetic geometry. In particular, problems related to Iwasawa theory, automorphic forms and representations, Langlands program, and special values of L-functions, and applications in the context of Diophantine geometry: for example, BSD conjectures, and the existence of rational points in certain varieties.

• ## Education

Ph.D. in Mathematics   Columbia University   (Expected 2027).

A.B. in Mathematics   Princeton University   (2018 - 2022).

## Research Publications

My papers can also be found on arXiv here. (The papers may not reflect current research interests).

### Accepted for publication

Higher moments for lattice point discrepancy of convex domains and annuli
2021. Journal of Number Theory, accepted.
arXiv Journal Talk by myself Abstract
Given a domain $$\Omega \subseteq \mathbb R^2$$, let $$\mathcal{D}(\Omega,x,R)$$ be the number of lattice points from $$\mathbb Z^2$$ in $$R\Omega -x$$, for $$R \ge 1$$ and $$x\in \mathbb T^2$$, minus the area of $$R\Omega$$: $\mathcal{D}(\Omega,x,R) = \# \{ (j,k) \in \mathbb{Z}^2 :(j-x_1,k-x_2) \in R\Omega \} - R^2|\Omega|.$ We call $$\int_{\mathbb{T}^2}|\mathcal{D}(\Omega,x,R)|^pdx$$ the $$p$$-th moment of the discrepancy function $$\mathcal{D}$$. In 2014, Huxley showed that for convex domains with sufficiently smooth boundary, the fourth moment of $$\mathcal{D}$$ is bounded by $$\mathcal{O}(R^2\log R)$$, and in 2019, Colzani, Gariboldi, and Gigante extended this result to higher dimensions.

In this paper, our contribution is twofold: first, we present a simple direct proof of Huxley's 2014 result; second, we establish new estimates for the $$p$$-th moments of lattice point discrepancy of annuli of radius $$R$$, and any fixed thickness $$0< t<1$$ for $$p\ge 2$$.

## Honors and Awards

• 2022 GSAS Fellowship, Columbia University
• 2022 MAA Outstanding Presentation Award, Joint Mathematics Conference
• 2021 MAA Outstanding Paper Award, Joint Mathematics Conference
• 2018/2019 Putnam Competition Top 7% Contestants
• 2015/2016 AIME Distinction, USAMO Qualifier
• 2015 Gold Medal, Singapore Mathematical Olympiad (SMO).

• ## Seminars Organized

• 2023 Fall Diophantine Geometry Seminar
• 2023 Spring Automorphic Forms & Representations Seminar
• 2023 Spring $$p-$$adic Hodge Theory Seminar
• 2022 Spring Princeton Univergraduate Student Seminar

• ## Talks

• Invited Speaker, AMS Contributed Paper Session for Graduate Student & Research Scholars on Algebraic and Arithmetic Geometry and Commutative Algebra, 2022, Seattle, WA [Program Link]
• Poster Presenter, Joint Mathematics Conference, AMS-PME Student Poster Session, 2022, Seattle, WA [Program Link]
• Poster Presenter, Mathematics Symposium, UIS, 2021, Chicago, IL [Poster]
• Invited Speaker, Young Mathematician Conference, 2021, Columbus, OH [Slides]
• Invited Speaker, MAA MathFest Student Paper Session, Virtual, 2021 [Abstract]
• Invited Speaker, Brown University Symposium for Undergraduates in Mathematical Sciences (SUMS) , 2021, Virtual [Talk]
• Invited Speaker, Joint Mathematics Conference, 2021, Virtual [Poster] [Talk]
• Poster Presenter, Mathematics Symposium, UIS, 2020, Chicago, IL [Poster]
• Invited Speaker, Young Mathematician Conference 2020, Columbus, OH [Slides]
• Poster Presenter, Joint Mathematics Conference, Student Poster Session, 2020, Denver, CO [Poster]
• Poster Presenter, Princeton SRC Conference, 2019, Princeton, NJ [Poster]

• MATH GU4042 Intro to Modern Algebra II, Fall 2023, Assistant Instructor.
• MATH S3028 Partial Differential Equations, Summer II 2023, Assistant Instructor.
• MATH S1012 Calculus II, Summer I 2023, Assistant Instructor.

• See my contribution to Columbia Directed Reading Program .

I am a member of Sigma Xi honor society, as well as a member of National Association of Mathematicians .

Last updated on Sep 16, 2023