Schedule

Lectures: WSC-N-U-3.05.
Coffee: WSC-S-U-3.01.
Discussions and registration: WSC-O-3.46.

(*) Free afternoon: We will provide guided tours for the Folkwang Museum (modern and contemporary art).

Mini courses

Javier Fresán: E-functions and Geometry

E-functions were introduced by Siegel in 1929 with the motivation of generalising the Hermite–Lindemann–Weierstrass theorem on the transcendence properties of exponentials of algebraic numbers. They are power series with algebraic coefficients that satisfy a linear differential equation and a certain growth conditions of an arithmetic nature. The Laplace transforms of E-functions, obtained by multiplying their n-th coefficients by n!, are called G-functions. According to the Bombieri–Dwork conjecture, they all come from geometry, for example in the sense that every G-function is annihilated by a Picard–Fuchs differential operator. This suggests that the ring of special values of G-functions agrees with the ring of periods.

In this series of lectures, I will explain how exponential period functions provide a geometric source of E-functions that conjecturally accounts for all of them. More precisely, we will prove that every exponential period function is a linear combination of E-functions with quasi-unipotent monodromy, with coefficients in the field generated by periods, values of the gamma function at rational arguments, and the Euler constant. The proof relies on fine properties of exponential motives, such as the analogue, in this setting, of the theorem of the fixed part. The lectures are based on joint work with Peter Jossen.

Peter Jossen: Exponential Motives and the Period Conjecture

Lecture 1: On perverse sheaves on the affine line

No pain no gain. In this lecture, I will introduce and show some properties of the category Perv_0, consisting of perverse sheaves on the affine line with vanishing cohomology. This might sound a bit intimidating, however, as I will show, these perverse sheaves and the morphisms between them can be described in a very elementary and explicit manner. As it turns out, Perv_0 comes with a tensor product (additive convolution) turning it into a neutral tannakian category. I will then show a couple of simple propositions related to tannakian Galois groups, for instance characterising objects with finite/abelian Galois groups. Finally, I will discuss possible enrichments of Perv_0, for instance to Hodge modules, which results in the category of Exponential Hodge structures introduced by Kontsevich and Soibelman.

Lecture 2: Exponential motives

The great motivation. Given an algebraic variety X over a subfield of the complex numbers, and a regular function f on X, I will introduce several kinds of cohomology groups H^n(X,f), in particular one kind that lives in Perv_0. Motivated by that, we construct the category of exponential motives, Nori style. It is a neutral, Q-linear tannakian category. Done that, I will explain some of the basic properties of the category of exponential motives, and how it relates to classical Nori motives. If time permits, I will also talk about the conjectural relationship with the triangulated category of motives on the affine line, Voevodsky style.

Lecture 3: The period pairing

Good things come in pairings. In this lecture we will have a closer look at the period pairing for exponential motives and what goes into the proof of the fact that it is non-degenerate. In its essence, this proof goes back to work of Marco Hien and Céline Roucairol (2008). To explain it, we will require some real algebraic geometry (the real oriented blowup) and some analysis (Poincaré lemmas).

Lecture 4: Examples and consequences of the period conjecture.

It all comes together now. We can now formulate the period conjecture for exponential motives, generalising the classical period conjecture. Done that, I will spend whatever time remains on examples, calculating explicitly some motivic fundamental groups and period matrices for selected motives. We will see how several folklore conjectures, such as for example the algebraic independence of pi and e, or the transcendence of Euler's constant, are consequences of the exponential period conjecture.

Claude Sabbah: Introduction to exponential mixed Hodge structures

Lecture 1. Exponential mixed Hodge structures (without Hodge filtration) after Kontsevich-Soibelman.
Lecture 2. The irregular Hodge filtration
Lecture 3. Hodge and irregular Hodge filtrations
Lecture 4. Finite monodromic exponential mixed Hodge structures and ulterior motives

You can find here lecture notes for this mini course.

References:

• Kontsevich-Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, 2011, Section 4.
• Fresán-Sabbah-Yu, Hodge theory of Kloosterman connections, 2018, Sections A3-A7,
• Sabbah-Yu, Hodge properties of Airy moments, 2023, Sections 2-5.

Swann Tubach: D-modules and Mixed Hodge Modules on the affine line

Lecture 1. D-modules on the affine line.

We introduce D-modules on the affine line, provide examples and prove its basic properties. We discuss the notions of regularity and holonomicity and observe their interaction with the Fourier transform.

Lecture 2. Mixed Hodge Modules on the affine line.

In this second lecture we provide Hodge theoretic enhancements to the first lecture. After some reminders on Hodge structures, we study variations of such on a curve and finish with mixed Hodge modules.

Lecture 3. Perverse Nori motives on the affine line.

The third lecture aims to give a motivic flavour to the first two. It introduces perverse Nori motives and gives a list of its basic properties. The end of the lecture will be devoted to a proof that a suitable subcategory of perverse Nori motives on the affine line is equivalent to the category exponential Nori motives defined in P. Jossen’s lecture.