Trimester Seminar

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.

Abstract

First order differential equations

Marius van der Put (Groningen University)

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

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.

Thursday, April 12th, 14:30

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 11th, 14:30 (Venue: MPIM)

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.

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.

Recorded Talk

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.

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.

Recorded Talk

Recorded Talk

Wednesday, April 4th, 14:30

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.

Recorded Talk

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)

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

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).

Venue: MPIM

Program

Abstracts

Thursday, March 8th, 16:30

Duco van Straten (JGU Mainz)

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

Wednesday, March 7th, 16:30 (NT lunch seminar of MPIM, Venue: MPIM)

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, 14:30 (joint seminar of MPIM and HIM, slot of NT lunch seminar of MPIM, Venue: MPIM)

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.

Tuesday, March 6th, 11:00

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.

Lecture notes

February 19th-23rd

Bethe Forum Scattering Amplitudes in Gauge Theory, Gravity and Beyond

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)

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

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.

Lecture notes

Recorded Talk

Wednesday, February 14th, 14:30 (joint seminar of MPIM and HIM, slot of NT lunch seminar of MPIM, Venue: MPIM)

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.

Lecture notes

Monday, February 12th, 4:30

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 7th, 14:30 (joint seminar of MPIM and HIM, slot of NT lunch seminar of MPIM, Venue: MPIM)

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 6th, 14:00 (Seminar on Algebra, Geometry and Physics, Venue: MPIM)

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.

Recorded Talk

Recorded Talk

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

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.

Recorded Talk

Recorded Talk

Recorded talk

Recorded talk

Recorded talk

Lecture notes

Thursday, January 25th, 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.

Recorded Talk

Thursday, January 25th, 11:00

(please note the change of time)

Lecture notes

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

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.

Recorded talk

Slides

Friday, January, 5th, 14:00

Andrey Levin (LMS NRU HSE)

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.