Professor Yau Mathematical Sciences Center
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## Research Interest

Number theory and automorphic representations. More specifically, I am interested in the interaction of the following topics as predicted in the Langlands Program:

- Arithmetic of Shimura varieties, Rapoport-Zink spaces, affine Deligne-Lusztig varieties, etc.
- Automorphic representations, Arthur-Selberg trace formulas and endoscopy
- Representation theory of real and p-adic groups.

## Recent teaching

- Spring 2024.
- Topics in the Langlands Program.
- Joint Lecture Series in Modern Number Theory (the Mordell Conjecture)

- Fall 2023.
- The Hodge-Tate period map on the perfectoid Shimura variety
- History of Class Field Theory (in Chinese) 类域论的历史 1 2 3 (You may need to refresh the video website twice.)

- Spring 2023. Representations of p-adic groups. Notes
- Spring 2022. Topic course on Shimura varieties. Notes
- Summer 2021. Mini course at BICMR: Counting points on Shimura varieties. (website), Notes
- Spring 2021, Math 621: Algebraic Number Theory II. Notes

## Papers & Preprints

*Some of the PDFs linked below are more updated than the arxiv versions.

- The stable trace formula for Shimura varieties of abelian type

(with Mark Kisin and Sug Woo Shin)

[pdf] [arxiv]

Revision coming! To appear in*Ann. Math. Stud.* - The stabilization of the Frobenius-Hecke traces on the intersection cohomology of orthogonal Shimura varieties.

[pdf] [arxiv]*Astérisque*To appear. - Stabilizers of irreducible components of affine Deligne-Lusztig varieties

(with Xuhua He and Rong Zhou)

[pdf] [arxiv]*J. Eur. Math. Soc.*To appear. - Twisted orbital integrals and irreducible components of affine Deligne-Lusztig varieties

(with Rong Zhou)

[pdf]*Camb. J. Math.*, 8 (2020), No. 1, 149–241. doi:10.4310/CJM.2020.v8.n1.a3 - Fine Deligne-Lusztig varieties and Arithmetic Fundamental Lemmas

(with Xuhua He and Chao Li)

[pdf]*Forum Math. Sigma*, 7 (2019), E15. doi:10.1017/fms.2019.45 (Open Access) - Arithmetic intersection on GSpin Rapoport-Zink spaces

(with Chao Li)

[pdf]

*Compos. Math.*, 154 (2018), No. 7, 1407-1440. doi:10.1112/S0010437X18007108 - Remarks on the arithmetic fundamental lemma

(with Chao Li)

[pdf]

*Algebra Number Theory*, 11 (2017), No. 10, 2425-2445. doi:10.2140/ant.2017.11.2425 - Report for the Oberwolfach workshop Automorphic Forms, Geometry and Arithmetic
(Aug 2021)

[pdf] - Proof of the arithmetic fundamental lemma in the minuscule case

(with Chao Li)

[pdf]

An appendix to Yifeng Liu,*Fourier-Jacobi cycles and arithmetic relative trace formula**Camb. J. Math.*, 8 (2020), No. 1, 149–241. DOI: 10.4310/CJM.2021.v9.n1.a1 - Introduction to the Langlands-Kottwitz method.

[pdf]

A chapter in the Paris Book Project*Shimura Varieties, London Mathematical Society Lecture Note Series: 457*

(edited by T. Haines and M. Harris, 2020). - Report for the Oberwolfach workshop Harmonic Analysis and the Trace Formula
(May 2017)

[pdf]

##### Preprints

##### Published

##### Other Published Writings

My name in Chinese characters: 朱艺航 (simplified); 朱藝航 (traditional).

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