Dedekind sums in geometry , topology , and arithmetic October 11 – 16 , 2009 MEALS

S (in alphabetic order by speaker surname) Speaker: Abdelmejid Bayad (Université d’Evry Val d’Essonne) Title: Some facets of multiple Dedekind-Rademacher sums Abstract: We introduce two kind of multiple Dedekind-Rademacher sums, in terms of Bernoulli and Jacobi modular forms. We prove their reciprocity Laws, we establish the Hecke action on these sums and we obtain new Knopp–Petersson identies. We show how to deduce Dedekind’s, Rademacher’s, Sczech’s reciprocity formulas from our main results. Some applications in number theory (special values of some L-functions, Periods, etc.) will be discussed. Speaker: Pierre Charollois (Institut de Mathématiques de Jussieu (Paris 6)) Title: Integral Dedekind sums for GLn(Q) Abstract: This is a report on a joint work with Samit Dasgupta, based on the construction by R. Sczech of a rational valued cocycle for GLn(Q). Our refinement now provides an integral valued cocycle, that can be expressed by a simpler fomula in terms of Bernoulli numbers. Speaker: Samit Dasgupta (University of California Santa Cruz) Title: Dedekind Sums and Gross–Stark Units Abstract: In 2006 I stated a conjectural formula for Gross-Stark units over number fields. In this talk I will discuss the role played by Dedekind sums in this formula. I will also look towards the function field setting for inspiration, where the conjectural formula may be proven following work of Hayes and using the theory of Drinfeld modules. Speaker: Ricardo Diaz (University of Northern Colorado) Title: A Solid Angle Algorithm for Spherical Polytopes Abstract: An algorithmic procedure for computing the spherical measure of spherical polytopes is outlined, based upon downward induction on dimension, via the Divergence Theorem. Potential applications include the determination of canonical solid-angle weights that behave additively under decomposition of lattice polytopes. Speaker: Karl Dilcher (Dalhousie University) Title: Reciprocity relations for Bernoulli numbers Abstract: We start with the observation that several classical identities for Bernoulli numbers can be written as reciprocity relations, and then prove a new type of three-part reciprocity relation for Bernoulli numbers. As a consequence we obtain a quadratic recurrence for these numbers. This recurrence requires, surprisingly, the knowledge of only one third of the previous numbers. In a second part of this talk I will give a new elementary proof of the reciprocity law for the classical Dedekind sums, based on uniform distributions of integers in subintervals of the real line. (Joint work with T. Agoh and K. Girstmair). Speaker: Yoshinori Hamahata (Tokyo University) Title: Reciprocity laws of Dedekind sums in characteristic p Abstract: We introduce Dedekind sums for lattices defined over finite fields and show the reciprocity law for them. Also, we establish the similar thing over function fields.