| Monday March 3 |
Tuesday March 4 |
Wednesday March 5 |
Thursday March 6 |
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| 9h00 -- 10h00 | Cheng | Nikulin | Gritsenko | Julia | |
| 10h00 -- 10h30 | Coffee break | ||||
| 10h30 -- 11h30 | Volpato | Scheithauer | Vanhove | Golyshev | |
| 11h45 -- 12h45 | Zagier | Murthy | Iohara | Clery | |
| 12h45 -- 15h00 | Lunch | ||||
| 15h00 -- 16h00 | Gaharamov | Skoruppa | Salvati Manni | Mertens | |
| 16h15 -- 17h15 | Manschot | Green | Pioline | discussions | |
| Diner | |||||
| Speaker | Title and Abstract | Files |
| Cheng Miranda | Umbral moonshine: Niemeier lattices and Mock Modular forms Monstrous Moonshine famously relates Hauptmoduls, a specific type of modular functions, and the representation theory of the sporadic group M, the Monster group. In this talk I will discuss a novel type of moonshine phenomenon, relating mock modular forms and interesting finite groups. I will explain the relation between this "umbral moonshine" and the 23 Niemeier lattices with non-trivial root systems (based on joint work with J. Duncan and J. Harvey). | |
| Clery Fabien | Siegel modular forms of genus 2 and level 2 After recalling one result of Igusa about the structure of the ring of scalar-valued Siegel modular forms of genus 2 on $\Gamma[2]$, we will explain how to get vector-valued Siegel modular forms on $\Gamma[2]$ by using two different constructions (Rankin--Cohen brackets and symmetric powers of gradients of odd theta constants.) Then we will describe one module of vector-valued modular forms by giving its generators and relations. This is a joint work with Gerard van der Geer and Samuel Grushevsky. | paper |
| Gahramanov Ilmar | Mathematical structures behind supersymmetric dualities The purpose of this talk is to give an overview on very interesting mathematical structures behind superconformal indices in various dimensions. The superconformal index is one of the efficient tools in the study of supersymmetric gauge theories which provides the most rigorous mathematical check of supersymmetric dualities. The superconformal index of a four-dimensional supersymmetric gauge theory is expressed in terms of elliptic hypergeometric integrals. Such integrals are the new class of special functions and is of interest both in mathematics and in physics. Though we will mostly concentrate on four-dimensional theories, we will outline some recent results on superconformal indices of two-, three- and four-dimensional supersymmetric gauge theories and relations between them. | |
| Golyshev Vasily | Congruence sheaves and congruence differential equations | |
| Green Michael B | String scattering amplitudes, Feynman diagrams and M-theory | |
| Gritsenko Valery | Theta-blocks: a hyperbolic approach | |
| Iohara Kenji | A
$Z^N$-graded generalization of the Witt algebra A generalization of the so-called generalized Witt algebra has been discovered by O. Mathieu and myself. Here, I explain how this algebra came out and show some examples of its representations. | |
| Julia Bernard | Exceptional fermions
After a brief review of key features of dualities in supergravity and string theories, we shall explain the relation between their Borcherds superalgebras of duaities and Kac-Moody algebras. We shall discuss a recent breakthrough relating the extended series of Lie groups of E type of rank between 1 and 8 that has been recognized in Del Pezzo middle cohomology and maximal supergravity with the list of Lie algebras in the magic square of Freudenthal and Tits. And if time permits more occurrences of division algebras in number theory and physics will be described or conjectured. | |
| Manschot Jan | Asymptotic formulas for coefficients of inverse theta functions We determine asymptotic formulas for the coefficients of a natural class of negative index and negative weight Jacobi forms. These coefficients can be viewed as a refinement of the numbers of partitions of $n$ into $k$ colors. Part of the motivation for this work is that they are equal to the Betti numbers of the Hilbert scheme of points on an algebraic surface $S$ and appear also as counts of Bogomolny-Prasad-Sommerfield (BPS) states in physics. | paper |
| Mertens Michael | Class Number Type Relations for Fourier Coefficients of Mock
Modular Forms The Hurwitz class numbers of binary quadratic forms satisfy many beautiful recurrence relations, the most famous of which are directly related to the Eichler-Selberg trace formula. In the first part of this talk, we prove a similar infinite series of class number relations, which were conjectured by Henri Cohen in 1975, using well-known results from the theory of mock modular forms. In the second part we show that in general Fourier coefficients of mock modular forms of weight 3/2 as well as those of mock theta functions satisfy similar recurrence relations. | |
| Murthy Sameer | Mathieu moonshine, mock modular forms and string theory I shall discuss a conjecture of Eguchi, Ooguri and Tachikawa from 2010 that relates the elliptic genus of K3 surfaces and representations of M24, the largest Mathieu group. The generating function of these representations is a mock theta function of weight one-half. After discussing some properties of this function, I shall present a particular appearance of this function in string theory that suggests a construction of a non-trivial infinite-dimensional M24-module. This is based on joint work with Jeff Harvey. | paper |
| Nikulin Viacheslav | Kahlerian $K_3$ surfaces and Niemeier lattices Kahlerian K3 surfaces and Niemeier lattices. We consider markings of Kahlerian K3 surfaces by Niemeier lattices and their applications. See our preprint arXiv:1109.2879 for some details. | |
| Pioline Boris | Rankin-Selberg methods for closed string amplitudes After integrating over supermoduli and vertex operator positions, scattering amplitudes in superstring theory reduce to integrals on a fundamental domain of the Poincaré upper half plane. A direct computation is in general unwieldy, but becomes feasible if the integrand can be expressed as a Poincarés series, i.e. a sum over images under a suitable subgroup of the modular group: if so, the integration domain can be extended to a simpler domain at the expense of keeping a single term in each orbit -- a technique known as the Rankin-Selberg method. I will apply this method to one-loop BPS-saturated amplitudes, where the integrand is the product of a Siegel-Narain lattice partition function times a weakly, almost holomorphic modular form. I will also discuss extensions to higher loop amplitudes. Work in collaboration with C. Angelantonj and I. Florakis. | |
| Salvati Manni Riccardo | On Superstring amplitude: an overview and some mathematical developments Starting from D'Hoker and Phong's program of finding multiloop superstring amplitudes by using factorization constraints, it has been reconsidered a conjecture, raised in the eighties, on the vanishing of the Schottky form on the moduli space of Riemann surfaces. We ( Grushevsky, _) disproved it. This fact leaded Codogni and Shepherd-Barron to disprove the existence of stable equations for moduli space of curves. We will discuss these results and possible applications. | |
| Scheithauer Nils | Automorphic products of singular weight We derive some new classification results for automorphic products of singular weight using the Riemann-Roch theorem and Serre duality. For example we show that the only holomorphic automorphic product of singular weight on an unimodular lattice is Borcherds' phi function, i.e. the theta lift of the inverse of the delta function. | |
| Skoruppa Nils |
How to construct explicitly Jacobi forms and vector valued modular
forms, and how to to turn elliptic modular forms into Jacob forms
In various theories one needs to construct explicitly spaces of vector valued elliptic modular forms. Examples for such theories are algebraic quantum field theory or the geometry of moduli spaces in algebraic geometry. A recent theorem shows that vector valued modular forms can always be realized as Jacobi forms. For the latter there are various efficient constructions available. In this talk we explain the mentioned theorem, various constructions for Jacobi forms, and we show how to assemble everything to obtain useful explicit formulas for the objects mentioned in the title. | |
| Vanhove Pierre | Feynman Integrals, Regulators and Elliptic Polylogarithms | |
| Volpato Roberto | Second Quantized Mathieu Moonshine
The Mathieu Moonshine is a conjectural relationship between a finite simple group, the Mathieu group M24, and the elliptic genus of K3. The evidence in favour of this conjecture, which is very reminiscent of the famous Monstrous Moonshine phenomenon in mathematics, is based on an impressive series of numerical coincidences. A good explanation for such coincidences, either from a physical or a mathematical point of view, is still missing. After reviewing the main features of the Mathieu Moonshine and of some generalizations, we describe its second quantized version, connecting the Mathieu group M24 to a set of Siegel modular forms. The latter admit a physical interpretation as generating functions for the multiplicities of 1/4 BPS states in certain four dimensional N=4 supergravity theories. The conjecture suggests the existence of some (unknown) mathematical structure organizing the spectrum of BPS states in theories with extended supersymmetry. | |
| Zagier Don | Higher spherical polynomials |