The Fourth Annual Spring Institute on Noncommutative Geometry and Operator Algebras

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, Universität Göttingen 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 squareintegrable, 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 wellknown identity $Z_1=Z_2$. Lucia Di Vizio, Université 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.