# Trimester Seminar

## Venue: HIM, Poppelsdorfer Allee 45, Lecture Hall

## Thursday, April 19^{th}, 14:00

## Differential equations in characteristic 0 and p

### Marius van der Put (Groningen University)

Abstract

Linear differential equations over the field of complex numbers are largely governed by differential Galois theory, monodromy and Stokes matrices. The good notion of linear differential equation over a field of positive characteristic is called "iterative differential module'' or ''stratified bundle''. Differential Galois theory exists in this case. The Tannaka group of the stratified bundles on a variety is the algebraic fundamental group in characteristic p. The main problem is producing stratified bundles and the inverse problem for differential Galois group. We will compare characteristic 0 and p. Further methods for the construction of stratified bundles on curves are presented.

## Thursday, April 19^{th}, 11:00

## Cohomology of arithmetic groups, periods and special values II

### Günter Harder (MPIM)

## Wednesday, April 18^{th}, 14:30 (Venue: MPIM)

**First order differential equations **

### Marius van der Put (Groningen University)

**Abstract**

We consider a first order differential equation of the form f(y′;y)=0 with f∈K[S;T] and K a differential field either complex or of positive characteristic. We investigate several properties of f, namely the 'Painlevé property' (PP), solvability and stratification. A modern proof of the classication of first order equations with PP is presented for all characteristics. A version of the Grothendieck-Katz conjecture for first order equations is proposed and proven for special cases. Finally the relation with Malgrange's Galois groupoids and model theory is discussed.

## Tuesday, April 17^{th}, 14:30

## Modular forms in Pari/GP

### Henri Cohen (Université de Bordeaux)

Abstract

The aim of this talk is to describe the new modular forms package available in Pari/GP which has a number of features not available in other packages, in particular expansion at all cusps and computation of arbitrary Petersson products.

## Tuesday, April 17^{th}, 11:00

## Cohomology of arithmetic groups, periods and special values I

### Günter Harder (MPIM)

## Monday, April 16^{th}, 16:30

## Serre-Tate theory

### Piotr Achinger (Instytut Matematyczny PAN)

## Friday, April 13^{th}, 14:00

## Prismatic site (after Bhatt-Scholze)

### Vadim Vologodsky (National Research University)

## Friday, April 13^{th}, 11:00

## Canonical candidates

### Vadim Vologodsky (National Research University)

## Thursday, April 12^{th}, 14:30

## P-adic multiple zeta values and p-adic pro-unipotent harmonic actions II

### David Jarossay (University of Geneva)

Abstract

We will compute the p-adic analogues of multiple zeta values in a way which keeps track of the motivic Galois action on the pro-unipotent fundamental groupoid of the projective line minus three points. This will be expressed by means of new objects which we will call pro-unipotent harmonic actions.

## Wednesday, April 11^{th}, 14:30 (Venue: MPIM)

**P-adic multiple zeta values and p-adic pro-unipotent harmonic actions I**

### David Jarossay (University of Geneve)

**Abstract**

Multiple zeta values are periods of the pro-unipotent fundamental groupoid of the projective line minus three points. We will explain a way to compute their p-adic analogues, which keeps track of the motivic Galois action, and which has an application to the finite multiple zeta values recently studied by Kaneko and Zagier. The computation will be expressed by means of new objects which we will call p-adic pro-unipotent harmonic actions.

## Friday, April 6^{th}, 14:30

## Interpreting Lauricella hypergeometric system as a Dunkl system

### Dali Shen (Instituto de Matemática Pura e Aplicada)

Abstract

In the 80's of last century, Deligne and Mostow studied the monodromy problem of Lauricella hypergeometric functions and gave a rigorous treatment on the subject, which provides ball quotient structures on $\mathbb{P}^n$ minus a hyperplane configuration of type $A_{n+1}$. Then some 20 years later Couwenberg, Heckman and Looijenga developed it to a more general setting by means of the Dunkl connection, which deals with the geometric structures on projective arrangement complements. In this talk, I will briefly review the Lauricella system first and then explain how to fit it into the picture of Dunkl system.

## Thursday, April 5^{th}, 16:30

## Mahler measures and L-functions

### Wadim Zudilin (The University of Newcastle)

Abstract

The talk outlines a method for reducing the values of L-functions to periods, in particular, to Mahler measures of polynomials in two variables.

Based on joint work with Mat Rogers and related work by Anton Mellit and François Brunault.

## Thursday, April 5^{th}, 14:00

## Many (more) zeta values are irrational

### Wadim Zudilin (The University of Newcastle)

Abstract

We considerably improve the asymptotic lower bound on the number of irrational odd zeta values as originally given in the Ball–Rivoal theorem. The proof is based on the construction of several linear forms in odd zeta values with related coefficients.

This is joint work with Stéphane Fischler and Johannes Sprang, http://arxiv.org/abs/1803.08905,

and the sole contribution, http://arxiv.org/abs/1801.09895.

## Thursday, April 5^{th}, 11:00

**Computing Peterson products in half-integral weight (after Nelson and Collins)**

### Henri Cohen (Université de Bordeaux)

## Wednesday, April 4^{th}, 14:30

**The polylog quotient and the Goncharov quotient in computational Chabauty-Kim theory**

### Ishai Dan-Cohen (Ben Gurion University)

**Abstract**

Polylogarithms are those multiple polylogarithms which factor through a certain quotient of the de Rham fundamental group of the thrice punctured line known as the polylogarithmic quotient. In joint work with David Corwin, building on work that was partially joint with Stefan Wewers, we push the computational boundary of our explicit motivic version of Kim's method in the case of the thrice punctured line over an open subscheme of Spec ZZ. To do so, we develop a greatly refined version of an algorithm I constructed a few years ago, tailored specifically to this case, and we focus attention on the polylogarithmic quotient with a vengeance. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth-one part of the mixed Tate Galois group studied extensively by Goncharov. We also discover an interesting consequence of the symmetry-breaking nature of the polylog quotient which forces us to symmetrize our polylogarithmic version of Kim's conjecture. Finally, we apply our refined algorithm to verify Kim's conjecture in an interesting new case.

## Tuesday, March 27^{st}, 14:00 (**Venue: MPIM**)

**Cohomology of GL**_{N}(ℤ) for N>7, trivialilty of K_{8}(ℤ) and arithmetical applications

_{N}(ℤ) for N>7, trivialilty of K

_{8}(ℤ) and arithmetical applications

### Philippe Elbaz-Vincent (Université Grenoble Alpes/HCM)

## Friday, March 23rd, 16:30

### $L(X_0(15),3)$

Wadim Zudilin (The University of Newcastle)

(based on joint work with Armin Straub, http://arxiv.org/abs/1801.06002)

## Friday, March 23rd, 14:00

### Mixed Tate motives and zeta values

Clément Dupont (Université de Montpellier)

(handwritten lecture notes can be requested from the speaker)

## Wednesday, March 21^{st}, 14:30 (joint seminar of MPIM and HIM, slot of NT lunch seminar of MPIM, **Venue: MPIM**)

**The Artin-Hasse isomorphism of perfectoid open unit disks and a Fourier-type theory for continuous functions on Q**_{p}

_{p}

### Francesco Baldassarri (Universitá di Padova/MPIM, Bonn)

## Wednesday, March 21st, 10:30

**Quantization of the moduli space of vector bundles**

### Shehryar Sikander (Abdus Salam International Centre for Theoretical Physics)

## Tuesday, March 20th, 14:00 (Seminar on Algebra, Geometry and Physics of MPIM, **Venue: MPIM**)

**Hodge theory of Kloosterman connections**

### Javier Fresán (École Polytechnique, Palaiseau)

Abstract

Recently, Broadhurst studied the L-functions associated with symmetric powers of Kloosterman sums and conjectured a functional equation. We show how the irregular Hodge filtration allows one to “explain” the gamma factors at infinity in a similar way to Serre's recipe for usual motives (ongoing joint work with Claude Sabbah and Jeng-Daw Yu).

## Tuesday, March 20th, 11:00

### **What is … a hypergeometric motive? **

### Madhav Nori (The University of Chicago)

## Monday, March 19^{th}, 16:30

**Hypergeometric exploration of the geography of motives**

### David Roberts (University of Minnesota Morris)

## Monday, March 19^{th}, 13:00

**GdT: Weight 3 GSp(2)-paramodular non-lifts**

### Chair: Gonzalo Tornaria

## Friday, March 16^{th}, 16:30

**On Siegel modular forms**

### Gonzalo Tornaría (Universidad de la República)

## Thursday, March 15^{th}, 16:30

**Hypergeometric and q-hypergeometric solutions of the quantum differential equations of cotangent bundles of flag varieties**

### Alexander Varchenko (University of North Carolina at Chapel Hill)

## Thursday, March 15^{th}, 15:00

**De Rham epsilon factors**

### Michael Groechenig (Imperial College)

## Thursday, March 15^{th}, 13:00

**Irregular Hodge filtration**

### Javier Fresan (École Polytechnique)

## Tuesday, March 12^{th}-14th

**Conference on Arithmetic and Automorphic Forms on the occasion of Günter Harder's 80th birthday, March 12 - 14, 2018**

Venue: MPIM

Program

## Friday, March 9^{th}, 16:30

**Differential equations associated to normal functions, and the transcendental regulator for a K3 surface and its self-product**

### James Lewis (University of Alberta)

## Thursday, March 8^{th}, 16:30

**An attractive Attractor**

### Duco van Straten (JGU Mainz)

(joint work with P. Candelas and X. de la Ossa)

## Thursday, March 8^{th}, 15:00

**l-adic ramification theory**

### Haoyu Hu (MPIM Bonn)

## Thursday, March 8^{th}, 13:00

**On the Stokes phenomenon**

### Jean-Baptiste Teyssier (KU Leuven)

## Wednesday, March 7^{th}, 16:30 (NT lunch seminar of MPIM, **Venue: MPIM**)

**On the periodicity of geodesic continued fractions**

### Hohto Bekki (MPIM)

**Abstract**

In this talk, we present some generalizations of Lagrange's periodicity theorem in the classical theory of continued fractions. The main idea is to use a geometric interpretation of the classical theory in terms of closed geodesics on the modular curve. As a result, for an extension F/F' of number fields with rank one relative unit group, we construct a geodesic multi-dimensional continued fraction algorithm to "expand'' a basis of F over the rationals, and prove its periodicity. Furthermore, we show that the periods describe the relative unit group. By extending the above argument adelically, we also obtain a p-adelic continued fraction algorithm and its periodicity for imaginary quadratic irrationals.

## Wednesday, March 7th, 16:30

**Techniques to compute monodromies of quantum DEs**

### Vasily Golyshev (IITP RAS Moscow)

## Wednesday, March 7^{th}, 14:30 (joint seminar of MPIM and HIM, slot of NT lunch seminar of MPIM, **Venue: MPIM**)

**Hecke's integral formula and Kronecker's limit formula for an arbitrary extension of number fields**

### Hohto Bekki (MPIM)

**Abstract**

The classical Hecke's integral formula expresses the partial zeta function of real quadratic fields as an integral of the real analytic Eisenstein series along a certain closed geodesic on the modular curve. In this talk, we present a generalization of this formula in the case of an arbitrary extension E/F of number fields. As an application, we present the residue formula and Kronecker's limit formula for an extension E/F of number fields, which gives an integral expression of the residue and the constant term at s=1 of the "relative'' partial zeta function associated to E/F. This gives a simultaneous generalization of two different known results given by Hecke-Epstein and Yamamoto. This result grew out of the study on geodesic multi-dimensional continued fractions and their periodicity. I would like to explain this original motivation after the tea.

## Wednesday, March 7th, 14:00

**Short random walks towards Apèry - part II**

### Jan Stienstra (Utrecht University)

## Tuesday, March 6th, 14:00

**Short random walks towards Apèry - part I**

### Jan Stienstra (Utrecht University)

## Tuesday, March 6th, 11:00

**What is... a log structure?**

### Piotr Achinger (Instytut Matematyczny PAN)

Abstract

I will give a gentle introduction to log(arithmic) geometry. Log structures, introduced by Fontaine, Illusie, and Kato, are a simple but quite powerful tool in the study of compactifications and degenerations. A neat construction due to Kato and Nakayama attaches to a log structure over C a topological space (called the "Betti realization"). The cohomology of this space is by definition the "Betti" cohomology, and there are de Rham and l-adic variants as well. All of this is (now) classical, but I will try to summarize also some recent developments in the subject, focusing on the topology of degenerations. No prior exposure to log geometry will be required.

## Wednesday, February 21^{st}, 14:00

**What is... the connection between the motivic coaction and QFT? **

### Oliver Schnetz (FAU Erlangen-Nürnberg)

## February 19th-23rd

**Bethe Forum Scattering Amplitudes in Gauge Theory, Gravity and Beyond**

#### Program

#### Venue and Travel Information

Remark: Some of the speakers are also expected to give a talk aimed at a more mathematical audience at the HIM workshop the week after.

## Tuesday, February 20th, 14:30 (Seminar on Algebra, Geometry and Physics of MPIM, **Venue: MPIM**)

**Motivic Gamma functions and periods associated to the Frobenius method**

### Spencer Bloch (University of Chicago/MPIM)

Abstract

(Joint work with Masha Vlasenko) In their work on the Gamma conjecture, Golyshev and Zagier introduced certain inhomogeneous Frobenius solutions defined in a neighborhood of a MUM point of a Landau model. We show in the case of the Apéry family that the variations of these inhomogeneous solutions about a nearby conifold point are periods, and the resulting generating function is a motivic gamma function in the sense of Golyshev. More generally, such solutions yield a variation of C-Hodge structure on a punctured disk about the MUM point. This variation admits a Q-Hodge structure if the monodromy of the inhomogeneous solutions about the conifold point satisfies the Picard-Lefschetz theorem.

## Thursday, February 15th, 14:30

**The modular forms of the simplest quantum field theory**

### Marianne Leitner (Dublin Institute for Advanced Studies)

Abstract

Much of the (2,5) minimal model in conformal field theory is described by very classical mathematics: Schwarz' work on algebraic hypergeometric functions, Klein's work on the icosahedron, the Rogers-Ramanujan functions etc. Unexplored directions promise equally beautiful results.

## Thursday, February 15th, 11:00

**Quantum fields as derivatives**

### Werner Nahm (Dublin Institute for Advanced Studies)

## Wednesday, February 14^{th}, 14:30 (joint seminar of MPIM and HIM, slot of NT lunch seminar of MPIM, **Venue: MPIM**)

**Hecke operators, buildings and Hall algebras**

### Oliver Lorscheid (IMPA, Rio de Janeiro)

**Abstract**

In this talk, we introduce such a variant: the graph of a Hecke operator. We explain a structure theorem for elliptic function fields and its applications to automorphic forms. We investigate its connection to Ronan's theory of adelic buildings, which is work in progress with Robert Kremser. We line out how Burban and Schiffmann's result about the Hall algebra of an elliptic curve can be used to determine the graphs of Hecke operators for GL(n), which is a recent result by Roberto Alvarenga.

## Tuesday, February 13th, 11:00

**Mixed Tate motives, bi-arrangements, and irrationality proofs**

### Clement Dupont (Université de Montpellier)

## Monday, February 12th, 4:30

**On some problems of Additive Combinatorics**

### I.D. Shkredov (Steklov Mathematical Institute)

**Abstract**

Additive Combinatorics is a mathematical field between Number Theory and Combinatorics studying any combinatorics on a group G which can be expressed via the group operation. This area is closely connected with Harmonic Analysis and Ergodic Theory and has a lot of deep applications to Number Theory, Analysis, Group Theory, Computer Science and so on. We will make a short survey on the subject.

## Wednesday, February 7^{th}, 14:30 (joint seminar of MPIM and HIM, slot of NT lunch seminar of MPIM, **Venue: MPIM**)

**Arakelov geometry, another take on L-functions**

### Vincent Maillot (Institut de Mathématiques de Jussieu, Paris)

**Abstract**

I will report on old and also not so old results relating Arakelov geometry and the theory of arithmetic L-functions.

## Wednesday, February 6^{th}, 14:00 (Seminar on Algebra, Geometry and Physics, **Venue: MPIM**)

**Macdonald polynomials and counting parabolic bundles**

### Anton Mellit (University of Vienna)

**Abstract**

Schiffmann obtained a formula for the (weighted) number of vector bundles with nilpotent endomorphism over a curve over a finite field. This talk will be about counting parabolic bundles with nilpotent endomorphism. The result we obtain gives an interesting new interpretation of Macdonald polynomials. Our formula turns out to be similar to the conjecture of Hausel, Letellier and Rodriguez-Villegas, which gives the mixed Hodge polynomials of character varieties. This allows us to obtain a new confirmation of their conjecture: we prove its implication for the Poincare polynomials of character varieties.

## Thursday, February 1^{st}, 14:30

**Multiple modular motives II **

### Richard Hain (Duke University)

## Thursday, February 1^{st}, 11:00

**Multiple modular motives I **

### Francis Brown (University of Oxford)

## Wednesday, January 31^{th}, 14:30 (joint seminar of MPIM and HIM, slot of NT lunch seminar of MPIM, **Venue: MPIM**)

**Euler-Kronecker constants**

### Kumar Murty (University of Toronto)

**Abstract**

Ihara defined and began the systematic study of the Euler-Kronecker constant of a number field. In some cases, these constants arise in the study of periods of Abelian varieties. For abelian number fields, they can be explicitly connected to subtle problems about the distribution of primes. In this talk, we review some known results and describe some joint work with Mariam Mourtada.

## Wednesday, January 31^{th}, 11:00

**What is… an exponential period? **

### Javier Fresan (École Polytechnique)

## Tuesday, January 30^{th}, 16:30

**Computing multiple polylogarithms (after Akhilesh)**

### Henri Cohen (Université de Bordeaux)

## Tuesday, January 30^{th}, 15:30 - 16:00

**A conjectural generalization of Zagier's formula for zeta(2,..,2,3,2,...,2) **

### Nobuo Sato (Kyoto University)

## Tuesday, January 30^{th}, 14:45 - 15:15

**Iterated integrals and symmetrized multiple zeta values **

### Minoru Hirose (Kyushu University)

## Tuesday, January 30^{th}, 14:00 - 14:30

**Bowman-Bradley type relations for symmetrized multiple zeta values**

### Steven Charlton (MPIM)

## Tuesday, January 30^{th}, 11:00 - 12:00

**Totally odd multiple zeta values and period polynomials **

### Koji Tasaka (Max Planck Institute for Mathematics)

## Monday, January 29^{th}, 16:30

**What is... a main conjecture?**

### Matthias Flach (California Institute of Technology)

## Monday, January 29^{th}, 15:15 - 15:45

**Rational associator in small depths**

### Nils Matthes (Kyushu University)

## Monday, January 29^{th}, 14:00 - 15:00

**Rooted tree maps**

### Tatsushi Tanaka (Kyoto Sangyo University)

## Monday, January 29^{th}, 11:00 - 12:00

**Multiple harmonic q-series at roots of unity and finite & symmetrized multiple zeta values**

### Henrik Bachmann (MPIM)

## Thursday, January 25^{th}, 16:30

**Mahler's measure and L(E,3)**

### Fernando Rodriguez Villegas (The Abdus Salam Centre for Theoretical Physics)

## Thursday, January 25^{th}, 14:30

**Tessellations, Bloch groups, homology groups**

### Rob de Jeu (Vrije Universiteit Amsterdam)

Abstract

Let k be an imaginary quadratic number field with ring of integers R. We discuss how an ideal tessellation of hyperbolic 3-space on which GL2(R) acts gives rise to an explicit element b of infinite order in the second Bloch group for k, and hence to an element c in K3ind(k), which is cyclic of infinite order. The regulator of c equals -12 ζk'(-1), and the Lichtenbaum conjecture for k at -1 implies that a generator of K3ind(k) can be obtained by dividing c by the order of K2(R). This division could be carried out explicitly in several cases by dividing b in the second Bloch group. The most notable case is that of Q(√-303}, where K2(R) has order ~22. This is joint work with David Burns, Herbert Gangl, Alexander Rahm, and Dan Yasaki.

## Thursday, January 25^{th}, 11:00

(please note the change of time)

**What is... a motivic gamma function?**

### Masha Vlasenko (Institute of Mathematics of the Polish Academy of Sciences)

## Wednesday, January 24^{th}, 14:30 (joint seminar of MPIM and HIM, slot of NT lunch seminar of MPIM, **Venue: MPIM**)

**Rational points on Picard modular surfaces**

### Dinakar Ramakrishnan (California Institute of Technology)

Abstract

Picard modular surfaces X, which are smooth compactifications of the quotients Y of the complex ball by a discrete subgroups Γ of SU(2,1), have been studied from various points of view. They are often defined over an imaginary quadratic field M, and we are interested in the rational points of X over finite extensions k of M. In a joint work with M. Dimitrov we deduced some finiteness results for the points on the open surfaces Y of congruence type with a bit of level (using theorems of Faltings, Rogawski and Nadel), which we will first recall.A recent result we have is an analogue, for polarized abelian threefolds with multiplication by OM and without CM factors, an analogue of the classical theorem of Manin asserting that for p prime, there is a universal r=r(p,k) such that for any non-CM elliptic curve E, the pr-division subgroup of E has no k-rational line. If time permits, we will explain the final objective of our ongoing program.

## Tuesday, January 23^{th}, 14:30

**Mixed motivic sheaves and weights for them**

### Mikhail Bondarko (MPIM)

## Tuesday, January 23^{th}, 11:00

**What are... Galois symbols on ExE ? (E an elliptic curve)**

### Dinakar Ramakrishnan (California Institute of Technology)

## Friday, January 12^{th}, 14:30

**What is... an associator?**

### Leila Schneps (Institut de Mathématiques de Jussieu)

## Friday, January 12^{th}, 11:00

**What is... relative completion?**

### Richard Hain (Duke University)

## Thursday, January 11^{th}, 11:00

**What is… a motivic Galois group?**

### Yves André (Institut de Mathématiques de Jussieu)

## Tuesday, January 9^{th}, 18:00

**Special Laurent polynomials and Apery numbers via normal functions**

### Matthew Kerr (Washington University in St. Louis)

## Tuesday, January 9^{th}, 16:30

**Introduction to mirror symmetry: special Laurent polynomials**

### Vasily Golyshev (National Research University Higher School of Economics (HSE))

## Monday, January 8^{th}, 18:00

**Introduction to mirror symmetry: generic Laurent polynomials**

### Hiroshi Iritani (Kyoto University)

## Monday, January 8^{th}, 16:30

**Introduction to mirror symmetry: four geographies**

### Vasily Golyshev (National Research University Higher School of Economics (HSE))

## Monday, January, 8th, 15:00

**Introductory Words and Talks**

### Christoph Thiele (HIM), Don Zagier (MPIM) and Spencer Bloch (The University of Chicago/MPIM)

## Friday, January, 5^{th}, 14:00

**Modular arrangements**

### Andrey Levin (LMS NRU HSE)

Lecture notes I Lecture notes II

Abstract

I want to present a potential source of periods of geometrical origin which can be interesting for number theory. The square X2 of the modular curve X=H/SL2(Z) is naturally equipped with a collection of curves, the so-called modular correspondences. A finite set of these curves is called a modular arrangement. A pair of arrangements in generic position determines some period, corresponding to the cohomology group of the complement of the first arrangement modulo the second.