# Workshop Programme

## for period 8 - 12 April 2013

### Grothendieck-Teichmüller Theory and Multiple Zeta Values

8 - 12 April 2013

Timetable

 Monday 8 April 09:00-09:25 Registration 09:25-09:30 Welcome from INI Director, Christie Marr 09:30-10:30 Alekseev, A (Université de Genève) Divergence and super-divergence cocycles on the Grothendieck-Teichmueller Lie algebra Sem 1 The Grothendieck-Teichmueller Lie algebra grt can be viewed as a Lie subalgebra of derivations of the free Lie algebra in two generators. We use this observation to define two cocycles: the divergence cocycle on grt and the super-divergence cocycle on its even part. The divergence cocycle serves to define the Kashiwara-Vergne Lie algebra which is conjecturally isomorphic to grt. The super-divergence cocycle plays a role in the Rouviere's theory of symmetric spaces, and it is conjectured to be an injective map on the even part of grt. 10:30-11:00 Morning Coffee 11:00-12:00 Broadhurst, DJ (Open University) Honorary MZVs and modular forms Sem 1 $\sum_{a>b>c>d>e>0}(-1)^{b+d}/(a^3b^6c^3d^6e^3)$ is an alternating sum with weight 21 and depth 5, yet has a conjectural expression as a Q-linear combination of multiple zeta values (MZVs) that includes a sum of depth 7. The MZV Data Mine [arXiv:0907.2557] contains many other examples of "honorary MZVs", i.e. alternating sums that are reducible to MZVs, sometimes at the expense of an increase of depth by an even integer. This talk concerns the conjecture that the enumeration of MZVs that are not reducible to MZVs of lesser depth, yet are reducible to alternating sums of lesser depth, is generated by an enumeration of modular forms. 12:15-13:15 Lunch at Wolfson Court 13:30-14:30 Fresse, B (Université Lille 1) Towards higher dimensional analogues of the torsor of Drinfeld's associators Sem 1 The purpose of this talk is to give a description of the set of homotopy classes of formality quasi-isomorphisms for a Sullivan model of the Little n-discs operads, where we consider any n>1. The Sullivan model of a topological operad combines a commutative dg-algebra structure, reflecting the rational homotopy of the spaces underlying the operad, and a cooperad structure, reflecting the composition structures of the operad. I will explain the definition of an obstruction spectral sequence for the formality of these Sullivan models of operads. I will give a description of the obstruction spectral sequence associated to the little discs operads, and I will explain the connection with the definition of the Drinfeld associators. 14:30-15:00 Afternoon Tea 15:00-16:00 Horozov, I (Washington University in St. Louis) Multiple Dedekind Zeta Functions Sem 1 In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider them as generalizations of Euler's multiple zeta values to arbitrary number fields. Over imaginary quadratic fields MDZV capture in particular multiple Eisenstein series [ZGK]. We give an analogue of multiple Eisenstein series over real quadratic field. And an alternative definition of values of multiple Eisenstein-Kronecker series [G]. Each of them as a special case of multiple Dedekind zeta values. MDZV are interpolated into functions that we call multiple Dedekind zeta functions (MDZF). We show that MDZF have integral representation and can be written as infinite sums, and have an analytic continuation. Finally, we prove that the multiple residue of a multiple Dedekind zeta function at (1,...,1) is a period. 16:30-17:30 Marin, I (Université d'Amiens) Associators and representations of braids Sem 1 >From their very construction 20 years ago as elements of a torsor for the Grothendieck-Teichmueller group, Drinfeld associators provide representations of the (usual) braid groups. The study of representation-theoretic consequences of this construction reverberate into questions on associators, and provide conjectural views on what to expect from generalizations of associators to other types of braid groups, where the symmetric group is replaced by more general reflection groups. This talk will review some of these questions as well as the state of the art in this area. 17:30-19:00 Wine Reception & Poster session
 Tuesday 9 April 09:30-10:30 Hoffman, ME (U.S. Naval Academy) Multiple zeta values and quasi-symmetric functions Sem 1 In the past 15 years many insights about multiple zeta values (MZVs) have come from the observation that they are homomorphic images of elements of the Hopf algebra QSym of quasi-symmetric functions. I will discuss some aspects of homomorphisms from QSym to the reals related to MZVs, including connections with topology and recent results on MZVs of even arguments. 10:30-11:00 Morning Coffee 11:00-12:00 Ohno, Y (Kinki University) On multiple zeta-star values Sem 1 Multiple zeta-star values are defined with non-strict inequalities in the summation range while the multiple zeta values are defined with strict inequalities. To understand the structure of multiple zeta algebra, we study relations among multiple zeta-star values. In this talk, I will present some families of relations among multiple zeta-star values. 12:15-13:15 Lunch at Wolfson Court 13:30-14:30 Ünver, S (Koç University) Cyclotomic p-adic multi-zeta values Sem 1 We will define the generalization of p-adic multi-zeta values to roots of unity and compute these values for depth less than or equal to 2. The method is to solve the fundamental differential equation satisfied by the crystalline frobenius using rigid analytic methods and to study the coefficients of its power series expansion. 14:30-15:00 Afternoon Tea 15:00-16:00 Joyner, S (Brandeis University) A shuffle product formula for generalized iterated integrals Sem 1 In generalized iterated integrals, one can integrate complex powers of certain holomorphic 1-forms on Riemann surfaces. In this talk, I will present a shuffle product formula on such integrals. Applications will include expressions of Dedekind zeta functions of abelian number fields as series of certain polyzeta functions, as well as identities involving the Riemann zeta function. 17:00-18:00 Zagier, D (Max-Planck-Institut fuer Mathematik, Bonn, and Collège de France, Paris) RVP Lecture: From modular forms to finite groups Sem 1 Modular forms occur in many different parts of number theory and other fields of mathematics. In this talk we will discuss two places where generalised modular forms appear in connection with the theory of finite groups. One is the occurrence of "quasimodular forms" in the representation theory of symmetric groups (and in related questions concerning coverings of Riemann surfaces), the other, the fairly recent discovery of an unsuspected connection between the theory of "mock modular forms" and the representation theory of one of the sporadic simple groups. 18:00-19:00 RVP Wine Reception
 Wednesday 10 April 09:00-10:00 Hain, R (Duke University) Higher Genus Polylogarithms Sem 1 Are there polylogarithms in higher genus? Classical polylogarithms are defined on P1-{0, 1,1}, which is the moduli spaceM0,4 of 4-pointed genus 0 curves. The elliptic polylogarithms of Beilinson and Levin are defined on M1,1, the moduli space of elliptic curves and on M1,2, the punctured universal elliptic curve over it. In this talk I will give a uniform definition of polylogarithms of all genera which specializes to these in genera 0 and 1. I will then explain that there are are countable many polylogarithms in genus 2 — though they appear to be less interesting than elliptic polylogarithms — and that, when g > 2, there are very few. The upside of this “rigidity” of higher genus moduli spaces is that one can construct a theory of characteristic classes of rational points of curves of genus g > 2. 10:00-10:30 Morning Coffee 10:30-11:30 Dupont, C (Institut de Mathématiques de Jussieu) The Hopf algebra of dissection polylogarithms Sem 1 Grothendieck's theory of motives has given birth to a conjectural Galois theory for periods. Replacing the periods with their motivic avatars, one gets an algebra of motivic periods that are acted upon by a motivic Galois group. Recently, the computation of this action for multiple zeta values has been studied and used by Deligne, Goncharov and Brown among others. In this talk we will introduce a family of periods indexed by some combinatorial objects called dissection diagrams, and compute the action of the motivic Galois group on their motivic avatars. This generalizes the case of (generic) iterated integrals on the punctured complex plane. We will show that the motivic action is given by a very simple combinatorial Hopf algebra. 11:45-12:45 Pollack, A (Princeton University) Periods of modular forms and relations in the fundamental Lie algebra of Universal Mixed Elliptic Motives Sem 1 Hain and Matsumoto have defined a category of so-called universal mixed elliptic motives, universal in the sense that such objects should be thought of as living over the moduli of all elliptic curves. They have shown that this category is neutral Tannakian. An interesting question then is understand explicitly the fundamental Lie algebra of this category. We make some progress in this direction, by proving a result about relations between a minimal set of generators for this Lie algebra. In particular, we find that periods of modular forms are closely connected to these relations. This work is closely related to older work of Schneps, and it also appears that there may be some connection to work of Gangl-Kaneko-Zagier. 12:45-13:30 Lunch at Wolfson Court
 Thursday 11 April 09:30-10:30 Brown, F (IHES) Anatomy of the motivic Lie algebra Sem 1 The motivic Lie algebra is contained in the Grothendieck-Teichmuller Lie algebra, and is isomorphic to the free graded Lie algebra with one generator in every odd degree >1. Using motivic MZV's one can define canonical generators for this algebra, but their arithmetic properties are very mysterious. In this talk, I will explain how elements of the motivic Lie algebra admit a kind of Taylor expansion with a rich internal structure. This is closely connected with the theory of modular forms, universal elliptic motives, and some other unexpected algebraic objects. 10:30-11:00 Morning Coffee 11:00-12:00 Stieberger, S (Max-Planck-Institut für Physik, München) Motivic Superstring Amplitudes Sem 1 We review the recent advances in open superstring N-point tree-level computations. One basic ingredient is a basis of (N-3)! generalized Gaussian hypergeometric functions (generalized Euler integrals) encoding all string effects through their dependence on the string tension $\alpha'$. The structure of the open superstring amplitudes is analyzed. We find a striking and elegant form, which allows to disentangle its $\alpha'$ power series expansion into several contributions accounting for different classes of multiple zeta values. This form is bolstered by the decomposition of motivic multiple zeta values, i.e. the latter encapsulate the $\alpha'$-expansion of the superstring amplitude. Moreover, a morphism induced by the coproduct maps the $\alpha'$-expansion onto a non-commutative Hopf algebra. This map represents a generalization of the symbol of a transcendental function. In terms of elements of this Hopf algebra the alpha'-expansion assumes a very simple and symmetric form, which carries all the relevant information. 12:15-13:15 Lunch at Wolfson Court 13:30-14:30 Schnetz, O (Friedrich-Alexander-Universität Erlangen-Nürnberg) Proof of the zig-zag conjecture Sem 1 In quantum field theory primitive Feynman graphs give--via the period map--rise to renormalization scheme independent contribution to the beta function. While the periods of many Feynman graphs are multiple zeta values there exists the distinguished family of zig-zag graphs whose periods were conjectured in 1995 by Broadhurst and Kreimer to be certain rational multiples of odd single zetas. In joint work with F. Brown it was possible in 2012 to prove the zig-zag conjecture using the theory of graphical functions, single valued multiple polylogarithms and a theorem by D. Zagier on multiple zeta values of the form zeta(2,...,2,3,2,...,2). 14:30-15:00 Afternoon Tea 15:00-16:00 Schlotterer, O (University of Cambridge) Polylogarithms, Multiple Zeta Values and Superstring Amplitudes Sem 1 Superstring Theory is one of the most promising candidates for a quantum theory of gravity. Its interactions involve all kinds of multiple zeta values, and I will present how they emerge at the leading order (so-called tree level) of open string scattering amplitudes. The expansion of the underlying generalized Gaussian hypergeometric functions in the string tension gives rise to iterated integrals within the [0,1] interval. I will discuss polylogarithm manipulations required for their systematic reduction to multiple zeta values. 19:30-22:00 Conference Dinner at Corpus Christi College
 Friday 12 April 09:30-10:30 Schneps, LC (Institut de Mathématiques de Jussieu) Elliptic Grothendieck-Teichmueller theory Sem 1 I'll talk about the definition by Benjamin Enriquez of an elliptic (genus one) version of the Grothendieck-Teichmueller Lie algebra grt. I will compare it to the work of Hain, Matsumoto and Pollack on the fundamental Lie algebra of the category of elliptic mixed Tate motives. I will define conjectural analogous elliptic double shuffle and Kashiwara-Vergne Lie algebras. Finally, I will discuss a structure conjecture that emerges from work of Pollack, and prove the first case of it. 10:30-11:00 Morning Coffee 11:00-12:00 Hoshi, Y (Kyoto University) Grothendieck-Teichmuller Groups in the Combinatorial Anabelian Geometry Sem 1 By a result of Harbater and Schneps, the Grothendieck-Teichmuller groups may be regarded as natural objects in the study of the combinatorial anabelian geometry. In this talk, we discuss some results on the Grothendieck-Teichmuller groups that relate to the phenomenon of the tripod synchronization. In particular, I explain the surjectivity of the tripod homomorphism and a non-surjectivity result on the combinatorial cuspidalization. 12:15-13:15 Lunch at Wolfson Court 13:30-14:30 Furusho, H (Nagoya University) On multiple L-functions and their p-adic analogues Sem 1 This is based on my current joint work with Y.Komori, K.Matsumoto and H.Tsumura. The first half of my talk is to review and discuss basic properties of complex multiple L-functions and those of their special values at integer points. Based on them, in the second half, I will explain our construction of p-adic multiple L-functions which generalizes that of Kubota-Leopoldt's p-adic L-function and then will show their several basic properties particularly on their special values at both positive and negative integer points. 14:30-15:00 Afternoon Tea