Student Seminar on Geometry of Fundamental Lemmas (Fall 2017)

Logistics

Time: 2:30 pm - 4:00 pm on Wednesdays
Location: 528 Mathematics
Lunch: Meet at 1:00 pm in the Mathematics Building Lobby

This semester Chao Li, Yihang Zhu and I are organizing a learning seminar on the geometry of fundamental lemmas. The seminar abstract is as follows:

The Fundamental Lemma of Langlands–Shelstad is a pivotal ingredient in the Langlands program. It has appeared indispensable to many major modern-day achievements in number theory, including Fermat's last theorem, the local Langlands correspondence, the zeta function of Shimura varieties, the construction of Galois representations, the Birch and Swinnerton-Dyer formula, just to name a few. It also has variants in the setting of relative trace formulas, with applications to the Gan–Gross–Prasad conjectures, etc. After decades of work of many people, the LSFL was proved by Ngô Bảo Châu around 2010. The proof was geometric in nature, a beautiful blend of algebraic geometry and topology, using Hitchin fibrations, Weil conjectures, perverse sheaves, etc.

The aim of our seminar is to study the geometric nature of this proof. To make it more accessible (even for non number-theorists), we shall first study Yun's proof of the Jacquet–Rallis Fundamental Lemma (JRFL). This should minimize the prerequisites from automorphic forms and trace formulas in order to understand Ngô's key geometric ideas. All are welcome to participate.

A program with more detailed information, written by Chao and Yihang, can be found here.

Please email me if you would like to join the mailing list for this seminar.

References

This list will be expanded soon.

Notes

Here are my live-TeXed notes for the seminar. Due to my own lack of understanding of the materials, I have inevitably introduced both mathematical and typographical errors in these notes. Please email me if you have any corrections or comments.

Schedule

The schedule is tentative and subject to change as the seminar moves along.

  Date  Speaker  Topic  References 
Lecture 1  September 13  Yihang Zhu  Introduction   
Lecture 2  September 20  Yihang Zhu  Introduction (continued)   
Qixiao Ma  Statement of the Jacquet–Rallis fundamental lemma  [Yun11, 2.1-2.2] 
Lecture 3  September 27  Qixiao Ma  Statement of the Jacquet–Rallis fundamental lemma (continued)  [Yun11, 2.3-2.5] 
Samuel Mundy  [Yun11, 2.5] 
Lecture 4  October 4  Samuel Mundy  Reformulation of the Jacquet–Rallis fundamental lemma  [Yun11, 2.6-2.7] 
Lecture 5  October 11  Shizhang Li  Hitchin moduli space for general linear groups, and spectral curves  [Yun11, 3.1-3.2] 
Lecture 6  October 18  Daniel Litt  Hitchin moduli space for unitary groups, and product formula  [Yun11, 3.3-3.4] 
No talk  October 25    Canceled   
Lecture 7  November 1  Daniel Litt  Smallness of invariant maps  [Yun11, 3.5] 
Chao Li  Formulation of the global matching theorem  [Yun11, 4] 
No talk  November 8    Canceled due to IAS Workshop   
Lecture 8  November 15  Chao Li  Global matching theorem  [Yun11, 4] 
Lecture 9  November 22  Yihang Zhu  Proof of the Jacquet–Rallis fundamental lemma  [Yun11, 5] 
Lecture 10  November 29       
Lecture 11  December 6       
No talk  December 13    Canceled due to EPFL Conference   
Lecture 12  December 20       

Abstracts


Last updated: November 16, 2017.

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