5/2/06 ========= Abstracts ========= Prakash Belkale, University of North Carolina at Chapel Hill Matroids, motives and conjecture of Kontsevich Abstract: (joint work with Patrick Brosnan) Let G be a finite connected graph. The Kirchhoff polynomial of G is a certain homogeneous polynomial whose degree is equal to the first betti number of G. These polynomials appear in the study of electrical circuits and in the evaluation of Feynman amplitudes. Motivated by work of D. Kreimer and D. J. Broadhurst associating multiple zeta values to certain Feynman integrals, Kontsevich conjectured that the number of zeros of a Kirchhoff polynomial over the field with q elements is always a polynomial function of q. We show that this conjecture is false by relating the schemes defined by Kirchhoff polynomials to the representation spaces of matroids. Moreover, using Mnev's universality theorem, we show that these schemes essentially generate all arithmetic of schemes of finite type over the integers. =========================================================================== Alain Connes, College de France, IHES, Vanderbilt University 1) From Renormalization as Birkhoof decomposition to the Riemann-Hilbert correspondence. 2) The cosmic Galois group, the group of diffeographisms and diffeomorphisms of coupling constants, the role of motivic Galois theory. 3) A candidate mental picture of space-time, after renormalization and taking the standard model seriously. =========================================================================== Katia Consani, Johns Hopkins University An introduction to the arithmetic of motives Abstract: In these talks I will review the notion of the L-series of a motive, focusing in particular on the description of the local Euler factors and their properties. =========================================================================== Kurusch Ebrahimi-Fard (IHES, Bures-sur-Yvette) Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion Abstract: We describe in this talk a unification of several factorizations arisen from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations. This was obtained from a particular Baker-Campbell-Hausdorff recursion formula. We will exemplify this in the context of the renormalization problem in pQFT. Using a simple representation of the combinatorics of renormalization in terms of triangular matrices, we thereby recover in the presence of a Rota-Baxter operator the matrix representation of the Birkhoff decomposition of Connes and Kreimer. =========================================================================== Bertfried Fauser, Max Planck Institute for Mathematics, Leipzig Title: Hopf algebras in Arithmetic and Renormalization Abstract: We consider addition and multiplication of nonnegative integers and dualize these notions. 'Virtual numbers' lead to the Grothendieck ring of arithmetic in a Hopf algebraic version analogous to recent developments in quantum field theory. It turns out, that the binary operations in question are not homomorphisms but only multiplicative maps in the sense of number theory. Only the multiplicative version leads to an interesting modeling of number theoretic functions via Hopf algebra techniques, e.g. the antipode has as generating function the shifted M\"obius function $\mu(s-1)$. We show, that there is strong evidence that renormalization of quantum fields has the same origin as multiplicativity in number theory, allowing a mutual benefit of these two seemingly different fields. Talk based on: BF, P.D. Jarvis, The Dirichlet Hopf algebra of arithmetics, Journal of Knot Theory and its Ramifications, accepted, math-ph/0511079 BF, Renormalization : A number theoretical model, math-ph/0601053 =========================================================================== Frederic Fauvet, Universite de Strasbourg 1 Alien calculus in Quantum Field Theory Abstract: I will report on several results in the field of Ecalle's theory of resurgent functions and alien differential calculus, that are relevant for perturbative series in QFT. In particular, we will see how, in this context, trees and polylogarithms naturally fit in. I shall also discuss direct relations with the study of the transverse structure of some dynamical systems at irregular singular points. =========================================================================== Alessandra Frabetti, Univeriste de Lyon 1 QED Hopf algebras and groups of combinatorial series Abstract: I will present Brouder's tree-expanded perturbative solution of Dyson-Schwinger equations for QED, the Hopf algebras describing QED renormalization in this context, and finally their dual groups of formal series of combinatorial type. =========================================================================== Li Guo, Rutgers University Some Constructions and Applications of Rota-Baxter Algebras Abstract: We will give an introduction to Rota-Baxter algebras which started with a probability study of Spitzer in 1950s and found interesting applications in the work of Connes and Kreimer on renormalization of QFT. We will discuss their basic properties, the constructions of the free objects and applications to multiple zeta values by renormalization method. Some other applications will be given in Kurusch Ebrahimi-Fard's talk. =========================================================================== Minhyong Kim, University of Arizona Non-abelian cohomology varieties in Diophantine geometry Abstract: We will give an overview of the ideas surrounding cohomology varieties for unipotent fundamental groups, and how they might be used to study Diophantine problems. =========================================================================== Marcelo Laca, University of Victoria Equilibrium and symmetries of Hecke C*-algebras Abstract: I will discuss the interplay between the KMS equilibrium condition and the symmetries of some C*-algebraic dynamical systems arising from number theory. =========================================================================== Matilde Marcolli, Max-Planck-Institute Bonn Noncommutative Geometry and Motives Abstract: I will cover in these lectures the recent joint work with Connes and Consani (math.QA/0512138), where we combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on l-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost-Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. =========================================================================== Ralf Meyer, Universitaet Goettingen Homological algebra for Schwartz algebras of reductive p-adic groups Abstract: Let \(G\) be a reductive group over a non-Archimedean local field. Then the canonical functor from the derived category of smooth tempered representations of \(G\) to the derived category of all smooth representations of \(G\) is fully faithful. Here we consider representations on bornological vector spaces. As a consequence, if \(G\) is semi-simple, \(V\) and \(W\) are tempered irreducible representations of \(G\), and \(V\) or \(W\) is square-integrable, then \(\mathrm{Ext}_G^n(V,W)\cong0\) for all \(n\ge1\). We use this to prove in full generality a formula for the formal dimension of square-integrable representations due to Schneider and Stuhler. =========================================================================== Henri Moscovici, Ohio State University Noncommutative complex geometry on the moduli space of Q-lattices of rank 2 Abstract: I will talk about joint work (in progress) with A. Connes, in which we develop the hypoelliptic theory of the noncommutative complex space of rank two Q-lattices modulo commensurability. =========================================================================== Niranjan Ramachandran, University of Maryland Introduction to motives Abstract: This will be a gentle introduction to Grothendieck's vision of motives. We begin with Riemann surfaces (algebraic curves) and their Jacobians before going to the considerably sophisticated case of algebraic varieties. Also planned is an interlude on mixed Hodge structures, time permitting. =========================================================================== Bahram Rangipour, Ohio State University Bicrossed product Hopf algebras and their Hopf cyclic cohomology Abstract: This is joint work with Henri Moscovici. We develop an intrinsic method to compute the Hopf cyclic cohomology of bicrossed product Hopf algebras. Our motivation is the fact that bicrossed product Hopf algebras provide a good source of noncommutative and noncocommutative Hopf algebras. As an application we compute the Hopf cyclic cohomology of the family of Connes-Kreimer-Moscovici Hopf algebras and their extensions by direct methods. =========================================================================== Walter van Suijlekom, Max-Planck-Institute Bonn The Hopf algebra of Feynman graphs in QED Abstract: We report on the Hopf algebraic description of renormalization theory of quantum electrodynamics. The Ward-Takahashi identities are implemented as linear relations on the (commutative) Hopf algebra of Feynman graphs of QED. Compatibility of these relations with the Hopf algebra structure is the mathematical formulation of the physical fact that WT-identities are compatible with renormalization. As a result, the counterterms and the renormalized Feynman amplitudes automatically satisfy the WT- identities, which leads in particular to the well-known identity $Z_1=Z_2$. =========================================================================== Lucia Di Vizio, Universite Pierre et Marie Curie (Paris) P-adic q-difference equations Abstract: I will introduce p-adic q-difference equations in the case |q|=1. I'll survey joint work with Yves Andre and some recents results by Andrea Pulita. ===========================================================================