The goal is to familarize ourselves with $\ell$-adic sheaves and perverse sheaves (complex or $\ell$-adic). Primary goals:
(a) understand Deligne’s proof of Weil conjecture/Riemann hypothesis for function fields, and
(b) understand cohomological interpretation of exponential sums.
Saturday 2pm ~ 4pm? at Gross somewhere
Overview
Classical derived category, triangulated category, truncation and $t$-structure
Infinity category and stable infinity category ???
Constructible sheaves
Complex perverse sheaves
Formalism of $\ell$-adic sheaves, after (iv)
$\ell$-adic perverse sheaves
add more
(i) Weil Conjectures, Perverse Sheaves and $\ell$-adic Fourier Transform, by Reinhardt Kielhl and Rainer Weissauer
(ii) Exponential Sums and $\ell$-adic Cohomology: a Survey, by Gerard Laumon
(iii) Perverse sheaves and applications to representation theory, by Pramod N. Achar.
Chapter 5 of this contains an quick introduction to $\ell$-adic (perverse) sheaves. Chapter 1 to 3 of this book seems to be a thorough account of complex perverse sheaves.
(iv) Weil’s Conjecture for Function Fields I, by Dennis Gaitsgory and Jacob Lurie.
This is not the Weil conjecture we are aiming to understand, but the Weil conjecture on the Tamagawa number. Nevertheless, the chapters 2, 3 contain a $\infty$-categorical way of constructing the category of $\ell$-adic sheaves, which seems to be neater than the classical way. A caveat: they in the book require the scheme to be quasi-projective, but most of the results should remain true in the algebraic scheme under some dimension assumption.
(v) The decomposition theorem, perverse sheaves and the topology of algebraic maps, by Mark Andrea A. De Cataldo and Luca Migliorini.
(vi) D-Modules, Perverse Sheaves, and Representation Theory, by Ryoshi Hotta, Kiyoshi Takeuchi and Toshiyuki Tanisaki
I want to toss out some applications of perverse sheaves that looks interesting to me. People might also feel motivating from these. Due to my lack of knowledge, I’m not able to list applications in algebraic geometry and Hodge theory.
Function-sheaf dictionary [(i), Theorem 12.1] provides a correspondence between functions and perverse sheaves over an algebraic scheme over a finite field. In the context of Braverman-Kazhdan program, there is an IC sheaf, namely a simple object in the category of perverse sheaves, that corresponds to the basic function in the positive characteristic case. See [Ngo20].
Weights [(i), $\S$ 1], more or less the eigenvalues of Frobinius, along with the Grothendieck’s trace formula allow us to interpret a complete exponential sum / a Kloosterman sum as an alternating sum of dimensions of $\ell$-adic cohomologies, giving the best estimate. The main tools here are Deligne-Fourier transform and Artin-Schreier sheaves. See [(ii)] and [(i), $\S$ 5].
Geometric Langlands and geometric representation theory. These I don’t know well, but they are function field version of the classical Langlands over number fields. The common language used in that realm is surely perverse sheaves. For me I would like to know what is the content of geometric Satake, or derived Satake.
Fundamental lemma. As the same suggested, it is one of the fundamental lemma in matching of local orbital integrals. By the work of Waldspurper, the problem is reduced to the local positive characteristic case. It is finally proved by Ngo, which earns him a Field medal. For the history and the tools used, see [Ngo10] and [(v), $\S$ 4.6].
Riemann-Hilbert correspondence gives an equivalence bewteen perverse sheaves and regular holonomic $D$-module, invented by Saito and Kashiwara. Please refer to [(vi)] because I don’t know well on this.