Xiang Li

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Welcome to my personal website! Here is the definition of me.

CV

You can find my detailed CV (curriculum vitae) here. Below is my education background.

1. Sep 2023 – Now
   PhD in Mathematics

   The University of Edinburgh, School of Mathematics (Hodge Institute)

2. Oct 2022 – Jun 2023
   MASt in Mathematics (Part III)

   University of Cambridge, Department of Pure Mathematics and Mathematical Statistics

3. Sep 2020 – May 2022
   BSc in Mathematics

   The University of Edinburgh, School of Mathematics

4. Sep 2018 – Jun 2020
   BSc in Mathematics

   South China University of Technology, School of Mathematics

Research

Recent Research Interests

With Martin Lüdtke, we establish foundations for the Chabauty–Kim method over number fields for the thrice-punctured line \(X = \mathbb{P}^1 \setminus \{0,1,\infty\}\). We investigate Kim's Conjecture for \(X\) over quadratic fields, verifying it in some cases and showing it fails in others. For imaginary quadratic fields \(K\) where \(S\) is empty or consists of a single Galois-unstable prime, we provide a proof of the \(S\)-Selmer section conjecture based on Chabauty–Kim methods.

Preprints

• \(S\)-integral Points on the Thrice-punctured Line over Cyclotomic Fields
  with Minhyong Kim, Martin Lüdtke · coming soon
• Polylogarithmic Chabauty–Kim loci over number fields
  with Martin Lüdtke · 2026 · arXiv:2608.20615

Expository Notes

• Tensor Category in Arithmetic, Geometry and Physics
  Dec 2023, GlaMS Group Project, with Julia Bierent, Gianni Gagliardo, Chun-Yu Bai
• The Modularity Theorem and the Modular Approach
  May 2023, Part III Essay
• Net of Conics and Cubic Curves
  Mar 2022, Undergraduate Final Year Project, with Yasmine Bligaard, Saffron Wang, Ruby David-Jekyll
• Algebra Theorem Proving in Lean
  Sep 2021, Undergraduate Summer Project

Talks

• Chabauty–Kim Methods over Number Fields: \(\mathbb{P}^1\backslash \left\{0,1,\infty\right\}\) 2 Jul 2026, 16:30 · MPIM Lecture Hall, Max Planck Institute for Mathematics

  Speed talk at ChaBONNty Conference. See the slides here.

• S-integral Points on the Thrice-punctured Line over Cyclotomic Fields 5 Sep 2025, 15:00–15:30 · C17 Pope Building, University of Nottingham

  Young Researchers in Algebraic Number Theory VII

• Rigid Geometry and Coleman Function 7 Feb 2025, 14:55–15:45 · Room 110, School of Mathematics and Statistics, University of Glasgow

  The GEARS seminar

• Introduction to Chabauty–Kim Methods on S-unit Equations 2 Aug 2024, 14:30–15:00 · TCC Room, Mathematical Institute, University of Oxford

  Young Researchers in Algebraic Number Theory VI

• Tannakian Category, Unipotent Completion, and de Rham Fundamental Group 1 Dec 2023, 10:45–11:15 · ICMS

  Examples Showcase at GlaMS

• Euler's Totient Theorem and the Prime Number Theorem 16 Feb 2023, 16:00–17:00 · Zoom / Huxley 410, Imperial College

  (with Ella Yu) — Seminar of London Learning Lean. Watch the recording here. See the slides here.

Travel

• Young Researchers in Algebraic Number Theory VIII 2 – 4 Sep 2026 · University of Bristol, UK
• ChaBONNty Conference 29 Jun – 3 Jul 2026 · Max Planck Institute for Mathematics, Germany
• Young Researchers in Algebraic Number Theory VII 3 – 5 Sep 2025 · University of Nottingham, UK
• The GEARS seminar 7 Feb 2025 · University of Glasgow, UK
• Arithmetic, Geometry, Space and Time: A workshop on the occasion of Minhyong Kim's 61st birthday 25 – 29 Nov 2024 · ICMS, Edinburgh, UK
• Young Researchers in Algebraic Number Theory VI 31 Jul – 2 Aug 2024 · University of Oxford, UK
• The Mordell conjecture 100 years later 8 – 12 Jul 2024 · MIT, US
• Gauge Fields in Arithmetic, Topology and Physics 15 – 19 Apr 2024 · ICMS, Edinburgh, UK
• Winter Workshop Chabauty-Kim 2024 14 – 16 Feb 2024 · Heidelberg University, Germany

Teaching

Past Tutoring

Codes