Research Trimester on Multiple Zeta Values, Multiple Polylogarithms and Quantum Field Theory1 ICMAT, September 15 - December 19, 2014 Workshop on Multiple Zeta Values, Modular Forms and Elliptic Motives II ICMAT, December 1-5, 2014 Speakers: 1. C. Bogner (HU Berlin, Germany) 2. D. Broadhurst (The Open University, UK) 3. F. Brunault (ENS Lyon, France) 4. B. Enriquez (CNRS, Strasbourg, France) 5. G. Felder (ETH Z¨ urich, Switzerland) 6. R. Friedrich (MPIM Bonn, Germany) 7. Y. Iijima (RIMS, Kyoto, Japan) 8. M. Kaneko (Kyushu Univ., Japan) 9. M. Kim (Univ. of Oxford, UK) 10. U. K¨ uhn (Univ. of Hamburg, Germany) 11. M. Levine (Univ. of Duisburg-Essen, Germany) 12. Y. Manin (MPIM Bonn, Germany) 13. H. Nakamura (Osaka Univ., Japan) 14. A. Salerno (Bates College, USA) 15. L. Schneps (CNRS, Paris, France) 16. S. Stieberger (MPI Physik, M¨ unchen, Germany) 17. T. Terasoma (Univ. of Tokyo, Japan) ´ Bures-Sur-Yvette, France) 18. P. Vanhove (IHES, 19. K. Vogtmann (Cornell Univ., USA and Univ. of Warwick, UK) 20. S. Wewers (Univ. of Ulm, Germany) 1 www.icmat.es/RT/MZV2014 1 Schedule: Monday Tuesday Wednesday Thursday Friday Registration 10:30 K. Vogtmann 9:30 – 10:30 T. Terasoma 9:30 – 10:30 P. Vanhove 9:30 – 10:30 G. Felder 9:30 – 10:30 Coffee break 11:00 – 11:30 D. Broadhurst 11:30 – 12:30 Coffee break 10:30 – 11:00 F. Brunault 11:00 – 12:00 Coffee break 10:30 – 11:00 M. Levine 11:00 – 12:00 Coffee break 10:30 – 11:00 Y. Manin 11:00 – 12:00 Coffee break 10:30 – 11:00 S. Stieberger 11:00 – 12:00 C. Bogner 12:30 – 13:30 B. Enriquez 12:00 – 13:00 H. Nakamura 12:00 – 13:00 M. Kaneko 12:00 – 13:00 U. Kühn 12:00 – 13:00 L u n c h 13:30 – 15:30 R. Friedrich 15:30 – 16:30 L u n c h 13:00 – 15:30 A. Salerno 15:30 – 16:30 L u n c h 13:00 L u n c h 13:00 – 15:30 Y. Iijima 15:30 – 16:30 L u n c h 13:00 M. Kim 16:30 – 17:30 L. Schneps 16:30 – 17:30 S. Wewers 16:30 – 17:30 Titles + Abstracts: C. Bogner Sunrise integrals beyond multiple polylogarithms The vast majority of Feynman integrals considered in particle physics today can be expressed in terms of multiple polylogarithms, but for some relevant cases, this class of functions seems not to be sufficient. The massive two-loop sunrise integral is an important showcase for this problem. We discuss recent progress in the computation of this Feynman integral for the general case of arbitrary values of the particle masses. By examination of a Picard-Fuchs differential equation and an elliptic curve related to the Feynman graph, we arrive at two versions of our result: one in terms of integrals over complete elliptic integrals and the other in terms of a function resembling an elliptic dilogarithm. D. Broadhurst Multiple Deligne Values in Quantum Field Theory Multiple Deligne Values (MDVs) are iterated integrals on the interval x ∈ [0, 1] of the differential forms d log(x), −d log(1 − x) and −d log(1 − λx), where λ is a primitive sixth root of unity. MDVs of weight 11 enter the renormalization of the standard model of particle physics at 7 loops, via a counterterm for the self coupling of the 2 Higgs boson. I shall review a recent and intensive investigation of all of the118,097 MDVs with weights up to 11 and comment on the status of 6 conjectures on MDVs, one of which engages modular forms at even weights w = 12 and w > 14. F. Brunault Beilinson’s conjecture for Rankin–Selberg products of modular forms In this talk, we will investigate Beilinson’s conjecture for the Rankin– Selberg convolution of two modular forms. We construct explicit elements in the motivic cohomology of the product of two Kuga– Sato varieties, and show that their regulator is proportional to noncritical L-values of Rankin products of modular forms, as predicted by Beilinson’s conjecture. Further, we show that these elements extend to the boundary of the Kuga–Sato variety. This is joint work with Masataka Chida. B. Enriquez Flat connection on configuration spaces of surfaces We construct an explicit bundle with flat connection on the configuration space of n points of a complex curve. This enables one to recover the ‘formality’ isomorphism between the Lie algebra of the prounipotent completion of the pure braid group of n points on a surface and an explicitly presented Lie algebra (Bezrukavnikov). When the genus is one, one recovers a part of the elliptic analogue of the Knizhnik–Zamolodchikov–Bernard connection, which is at the origin of the theory of elliptic associators and MZVs. G. Felder Derived representation schemes and combinatorial identities Representation schemes parametrize representations of associative algebras on a given vector space. I will review a derived version of this theory, due to Berest, Khachatryan and Ramadoss, and present simple examples, such as the algebra of polynomials in two variables, featuring phenomena that are visible in computer experiments, and only partly understood mathematically. I will present some conjectures 3 that lead to new combinatorial identities partly proven and partly still conjectural. (Based on joint work with Y. Berest and A. Ramadoss and with Y. Berest, A. Patotski, A. Ramadoss and T. Willwacher.) R. Friedrich TBA TBA Y. Iijima A pro-l version of the congruence subgroup problem for mapping class groups of genus one Let l be a prime number. Then the natural outer action of the mapping class group on the surface group induces an outer action of the relative pro-l completion of the mapping class group on the pro-l completion of the surface group. In this talk, we discuss the faithfulness of this pro-l outer action in the case where genus is 1. Our main result is that the pro-2 case is faithful, but the pro-l case for l ≥ 11 is not faithful. In order to give a negative answer to the problem in the case where l ≥ 11, we also consider the issue of whether or not the image of the natural outer action of the absolute Galois group of a certain number field on the geometric pro-l fundamental group of a modular curve is a pro-l group. This is a joint work with Yuichiro Hoshi. M. Kaneko Finite and symmetric multiple zeta values Starting from the naive truncation and mod p of the usual multiple zeta values, we define ”finite multiple zeta values” as elements in an algebra over the rationals. We give some results and conjectures on these values as well as their real analogues which we call symmetric or ”finite real” multiple zeta values. Our main conjecture predicts that these two objects are beautifully connected with each other. This is a joint work with Don Zagier. 4 M. Kim Non-abelian reciprocity laws and iterated integrals We formulate iterative non-abelian reciprocity laws with coefficients in a hyperbolic curve and then give examples showing iterated integrals emerging in the process of making them explicit. U. K¨ uhn On the generators of a certain algebra of q-multiple zeta values The q-analogon of multiple zeta values given by the generating series of bi-multiple divisor sums, also refered to as bi-brackets, naturally contains the algebra of generating series of multiple divisor sums as well as the ring of quasi-modular forms. The bi-brackets satisfy a variation of the double shuffle relations. This allows us to study the number of generators via the linearised version of these relations. We obtain this way upper bounds for the dimension of this bi-filtered algebra for small length similar as for multiple zeta values. M. Levine TBA TBA Y. Manin Iterated integrals, Dedekind symbols, and Zeta polynomials In the first part of the talk, I will explain how the notion of a generalized Dedekind symbol can be extended to non commutative groups of values, and give examples of such symbols using iterated integrals of modular forms. In the second part, I will describe the somewhat mysterious construction of ”zeta polynomials” that can be thought of as ”local L-functions in characteristic one”. 5 H. Nakamura Monodromy of elliptic curves and Mordell transformations in Grothendieck-Teichmueller theory Classical Mordell transformation switches quartic models of elliptic curves to cubic forms. We discuss related Grothendieck–Teichm¨ uller theory and application. A. Salerno On the double shuffle Lie algebra structure: Ecalle’s approach The real multiple zeta values are known to form a Q-algebra; they satisfy a pair of well-known families of algebraic relations called the double shuffle relations. To study the algebraic properties of multiple zeta values, one can study the algebra of formal symbols subject only to the double shuffle relations. Quotienting this algebra by products, one obtains a vector space that Racinet proved carries the structure of a Lie coalgebra. The dual of this space is thus a Lie algebra, known as the double shuffle Lie algebra. Ecalle has developed a deep theory to explore combinatorial and algebraic properties of the formal multizeta values. In this talk, we will explain how Racinet’s theorem follows in a simple and natural way from Ecalle’s theory. This is joint work with Leila Schneps. L. Schneps TBA TBA S. Stieberger Periods and superstring amplitudes We present (some) connections and implications of superstring amplitudes from and to number theory. These relations include motivic multiple zeta values, single-valued multiple zeta values, Drinfeld, Deligne associators and Lie algebra structures related to Grothendiecks Galois theory. More concretely, we will show that tree-level superstring amplitudes provide a beautiful link between generalized hypergeometric functions and the decomposition of motivic multiple zeta 6 values. Furthermore, we establish relations between complex integrals on P 1 \{0, 1, ∞} as single-valued projection of iterated real integrals on RP 1 \{0, 1, ∞}. T. Terasoma Depth filtration and mixed elliptic motives On the Q-vector space generated by (motivic) multiple zeta values, there are two filtrations by weight and depth. The depth filtraion is defined by the numbers of dx/(1−x)’s in the iterated integral expression of the multiple zeta values. The conjecture by Broadhurst–Kreimer, it is strongly expected that this filtration is related to the space of elliptic modular forms. In this talk, we will try to explain the relation between the depth filtration and the representation of Tannaka’s fundamental group of mixed elliptic motives on the fundamental group of the Tate curve. P. Vanhove Feynman integrals, Elliptic poylogarithms and beyond We will discuss a class of Feynman integrals in two dimensions leading to a surprizingly rich mathematical structure. These Feynman integrals arise as period of mixed of Hodge structures, that we will describe in detail. At the lowest loop order the integrals are given by elliptic polylogarithms, but a new class of function is needed at higher loop order. K. Vogtmann Hairy graphs, modular forms and the cohomology of Out(Fn ) The rational cohomology of Out(Fn ) vanishes in high and low dimensions, but Euler characteristic calculations show that there must be many classes in the middle range. In this talk I will summarize what we know and then show how tools from topology, representation theory and modular forms can be used to construct a large number of cocycles. Those within range of computer calculations have been shown to give nontrivial cohomology classes, and it is conjectured that essentially all of them should. This is joint work with J. Conant, A. Hatcher and M. Kassabov. 7 S. Wewers TBA TBA 8

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