Weil's Conjecture on Tamagawa Numbers

The proof of the function field case by Gaitsgory and Lurie

The Weil conjecture on Tamagawa numbers is a statement about semisimple simply-connected linear algebraic groups that originated in a reformulation of the Siegel mass formula. The ideas in this reformultion go back to Tamagawa and Weil. The number field case was established by Langlands-Lai-Kottwitz in several stages between 1966 and 1988, while the function field case was proved by Gaitsgory and Lurie in 2014. It was first announced in 2011 at [4].
This semester we are organising a reading seminar with the goal to understand (part of) the recent paper by Dennis Gaitsgory and Jacob Lurie [1].
We will meet in bio 1.1.34. The first meeting is on Monday 20th April 2015 at 10ct, and every Monday after that starting at the same time. This may be subject to change which will be posted here.
The first three sessions are devoted to gain an overview of the problem and its origins. A good reference for this are the lecture notes taken by Aaron Mazel-Gee of a lecture series by Jacob Lurie at the Young Topologists Meeting 2014 [3]. An overview can be gained in [2]. It might also be helpful to look at [5].
After that the program is not set in stone and can be adjusted depending on interest. Suggestions are welcome as always. As the amount of material is overwhelming we will not be able to read the whole paper, let alone understand the details. Thus we will focus on specific parts of it. Some suggestions:

• Nonabelian Poincaré Duality.
• Chiral Homology.
• Grothendieck-Lefschetz Trace formula for stacks.

Date	Speaker	Topic
20th April 2015	Niko Naumann	Weil's conjecture for number fields (Section 1.1-2.1 of [3], see also Section 1.1 of [1]).
27th April 2015	Veronika Ertl	Weil's conjecture for function fields (Section 2.2-3.3 of [3], see also Section 1.2 of [1]).
4th May 2015	Veronika Ertl	See above.
11th May 2015	Niko Naumann	Overview of the proof (Section 4.1- 5.4 of [3], see also Section 1.4 and 1.3 of [1]).

Topological chiral homology

The remainder of the seminar will focus on chiral homology, which plays a big role in the proof of the Tamagawa number conjecture. This part of the semianr is selfcontained, it is not necessary that one has attended the first part. After an introductory part into the language of infinity categories, our goal is to read Section 5.5 of Lurie's book Higher Algebra. We will try to cover approximately one subsection per week, but decide on this as we go.

Date	Speaker	Topic
18th May 2015	Kim Nguyen	Introduction: Infinity categories and symmetric monoidal infinity categories.
26th May 2015	Kim Nguyen	Introduction: Infinity operads and little cubes. Note the change of the day. And new time: 14-16 ct.
1st June 2015	John Lind	Towards the definition of chiral homology: Parts of Sections 5.1 and 5.4 of [6].
8th June 2015	No seminar.	Summer school on K-homology.
15th June 2015	John Lind	Definition as left Kan extension and via integration on the Ran space, some properties: Parts of Section 5.5 of [6].
22nd June 2015	Georgios Raptis	Relation to TQFT.
29th June 2015	Georgios Raptis	Factorization homology and nonabelian Poincare duality.
6th July 2015	No seminar.	--

References

[1] D. Gaitsgory, J. Lurie: Weil's conjecture for function fields, http://www.math.harvard.edu/~lurie/papers/tamagawa.pdf, (2014).

[2] A. Kahn: Weil conjecture on Tamagawa numbers , entry in the nlabwiki, http://ncatlab.org/nlab/show/Weil%20conjecture%20on%20Tamagawa%20numbers, (2014).

[3] J. Lurie: Tamagawa numbers via nonabelian Poincaré duality, 5 lectures at Young Topologists Meeting 2014, notes taken by Aaron Mazel-Gee, https://math.berkeley.edu/~aaron/livetex/lurie-tamagawa-poincare.pdf, (2014).

[4] J. Lurie: Tamagawa Numbers via Nonabelian Poincaré Duality, talk at FRG Chern-Simons workshop, (Jan. 15-17, 2011).

[5] J. Lurie: Tamagawa Numbers via Nonabelian Poincaré Duality (282y), lecture notes http://www.math.harvard.edu/~lurie/282y.html, (2014).

[6] J.Lurie: Higher algebra, http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf, (2014).

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