Hopf-in-Lux

Title: The Geometry of Renormalization

Abstract: Connes and Marcolli showed that the $\beta$ function of a quantum field can be uniquely written in terms of a flat connection of a renormalization bundle, defined by the Hopf algebra of Feynman diagrams associated to that theory. In this talk, I review the construction of this connection and show how the renormalization bundle can be used to define a $\beta$ function first locally, and then globally over a curved space-time background manifold. Finally, I construct a conformal renormalization bundle that accounts for conformal changes of metric on the background manifold, rather than the scalar changes considered in prior work.

Title: Graded Hopf algebras, X-infinity structures and deformation quantization

We shall present the graded `cofree' Hopf algebras needed to understand A-infinity and L-infinity structures, viz the tensor (co)algebra and the graded symmetric (co)algebra. We present a pedestrian approach to A- and L-infinity structures as graded coderivations of square zero and show how to transport them by homotopical perturbation theory on the corresponding coalgebras over the (co)homology spaces thereby presenting Gerstenhaber, Nijenhuis-Richardson and Schouten brackets. In this context we shall sketch the Hochschild-Kostant-Rosenberg Theorem for the Hochschild (co)homology of symmetric algebras. Formality will be introduced as an L-infinity morphism between graded Lie-structures. At the end we shall show Kontsevich's result that formality implies the integrability of associative deformations up to order two. As examples we shall take some universal envelopping algebras of Lie algebras.

Title: Hopf algebra structures and Parking functions

Abstract: We exhibit a family of well-known Hopf algebras which can be endowed with some finer bialgebra structure namely a tridendriform bialgebra structure. One example is the Hopf algebra of Parking functions which are very useful combinatorial objects. In their article Novelli and Thibon conjecture that these Parking functions are free with respect to the tridendriform algebra structure. We prove the conjecture and show that this bialgebra can be reconstructed from its primitives through a structure theorem. Moreover this primitive part is free as a Gerstenhaber-Voronov algebra.

LECTURE:

Title: Modern aspects of perturbative renormalization

Abstract: In this series of lectures we describe the process of renormalization in perturbative quantum field theory. In the first lecture we will introduce the basic concepts by using toy models. Based on the lectures at this school on quantum field theory (by A. Frabetti) and Hopf algebras (by A. Makhlouf) we will exemplify these concepts in concrete calculations in quantum field theories relevant to physics. Along these lines Connes-Kreimer's Hopf algebra of renormalization will be introduced. At the end we will outline a Lie theoretic approach to perturbative renormalization using the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras. These results rely on properties of Hopf algebras encapsulated in the notion of associated descent algebras.

TALK:

Title: On the pre-Lie Magnus and Fer expansions

Abstract: W. Magnus' classical paper on the exponential solution of differential equations for a linear operator received much attention since its appearance in 1954, both in applied mathematics as well as physics. More recently, Magnus' results were explored from a more algebraic-combinatorial perspective using the language of operads and combinatorial Hopf algebras. In this talk we review recent work on the classical Magnus and Fer expansions, unveiling a new structure by using the language of dendriform and pre-Lie algebras as well as Hopf algebras of rooted trees. Let us emphasize that a dendriform algebra may be seen at the same time as an associative as well as a (pre-)Lie algebra. Main examples of dendriform algebras are provided by the shuffle and quasi-shuffle algebras as well as Rota-Baxter operators, e.g. the Riemann integral map or Jackson's q-integral. We introduce the notion of linear equations in dendriform algebras, which include matrix initial value problems as a particular example, and investigate its solution theory. In the end we outline a new approach to Butcher's B-series in the context of pre-Lie algebras. Due to the intimate link between dendriform algebras and Rota-Baxter operators, motivations as well as applications naturally appear. They rang from numerical methods, to control theory to the process of renormalisation in perturbative quantum field theory. (This is joint work with D. Manchon and D. Calaque)

LECTURE

Title: Basic quantum field theory for mathematicians

Abstract: In these lectures I present some basic notions of quantum field theory with the aim of being comprehensible by mathematicians. The first lectures review classical and quantum interacting fields and their perturbative expansion as series over Feynman diagrams.

TALK

Title: Pro algebraic groups and renormalization

Abstract: I present an interpretation of the Connes-Kreimer renormalization Hopf algebra in terms of proalgebraic groups for the perturbative series previously described.

LECTURE:

Title: Incidence Hopf algebras

Abstract: The theory of incidence Hopf algebras provides a simple, unified framework for constructing Hopf algebras based on families of combinatorial structures, yielding a rich interplay between algebra and combinatorics. The aim of these lectures is to give both an introduction to and an overview of the subject, with emphasis on examples that reveal the combinatorial underpinnings of well-known Hopf algebras that occur in a variety of fields.

TALK:

Title: Noncrossing partition Hopf algebra

Abstract: We introduce and study the incidence Hopf algebra H of the family of noncrossing partition lattices of finite linearly ordered sets. After presenting some basic combinatorial results on noncrossing partitions, we give two formulas for the antipode S of H. The first of these is given as an alternating sum indexed by certain dissections of polygons and is cancellation-free. The second gives the value of S on the canonical set of generators in terms of a second set of generators, related to the first by Lagrange inversion; this formula is not only cancellation-free but is in fact sign-free. We demonstrate the relationship between S and reversion of formal power series and give a sign-free version of the Lagrange inversion formula. Finally, we use an edge labeling of noncrossing partition lattices, defined by Richard Stanley, to construct an isomorphism between H and the Hopf algebra of symmetric functions and an embedding of H in the Hopf algebra of parking functions. These maps have sign-free expressions in terms of the canonical bases of these Hopf algebras. (This joint work with Hillary Einziger.)