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