I will discuss joint work with Frederic Rochon involving an application of pseudodifferential operators and renormalized traces to the index theory of manifolds with fibered hyperbolic cusps. We start with a family of Dirac-type operators and if these are Fredholm, or can be perturbed by smoothing operators to be Fredholm, we find a formula for the Chern character of their index bundle. Upon specializing to Riemann surfaces with cusps or to parabolic bundles over Riemann surfaces, we can identify the cusp contributions to the curvature of the determinant line bundle in formulas of Takhtajan-Zograf with eta forms - as well as extend these formulas to the full Chern character.

The problem of renormalization of Feynman amplitudes leads to limiting mixed Hodge structures. I will describe how to think about the underlying monodromy, which is linked to the combinatorics of flags of 1PI subgraphs of the graph. The fantasy is that underlying the monodromy is a limiting mixed motive whose periods are multiple zeta values.

Single scale quantities in Quantum Field Theories as anomalous dimensions, Wilson coefficients or hard-scattering subsystem cross sections have a particularly simple structures if considered in Mellin space. Up to 3-loop orders they can be represented in terms of polynomials of finite nested (colored) harmonic sums $S_{a_1,...,a_k}(N)$, with $N$ a positive integer, and $a_i$ integer. These sums obey algebraic relations since they are elements of a quasi-shuffle algebra. The analytic continuation of these functions from $N$ integer to $N$ complex is required to solve the associated renormalization group equations and to analyze experimental data. We construct the complex analysis of the finite nested harmonic sums up to weight {\sf w=6} in explicit form. Rational argument relations and differentiation w.r.t. $N$ leads to new relations between the finite harmonic sums, the structural relations. It turns out that only a few numbers of harmonic sums or associated Mellin transforms have to be considered to construct the above physical quantities. Beyond the level of {\sf w=6} other finite sums may contribute, which obey similar algebraic and structural relations.

Toeplitz operators are a generalization of pseudodifferential operators, whose symbolic calculus live on a symplectic cone (not necessarily a cotangent bundle). Although index theory cannot give rise to a nice topological formula for these, "asymptotic equivariant" index does; it is related to M.F. Atiyah's equivariant index for relatively elliptic pseudodifferential operators. (Wodzicki's residual trace appears in relevant cases as a special instance of this.)

Since A. Grothendieck proposed the first definition of a "motif", in the early 60's, this notion has undergone several interesting developments and enrichments. Very recently, some applications of a new theory of noncommutative motives to number-theory and quantum field theory have been found with the support of techniques supplied by Noncommutative Geometry and Operator Algebras.

The talk will outline very recent developments in the theory of "endomotives".

This series of lectures will give a leisurely introduction to pseudodifferential operators. I will discuss in particular the famous regularized traces (a la Wodzicki and Kontsevich-Vishik). Then I will report on my recent joint work with Henri Moscovici and Markus Pflaum on regularized traces in the parameter dependent pseudodifferential calculus. While 'ordinary' traces give rise to K-theory invariants I will advertise that the mentioned traces give rise to \emph{relative} K-theory invariants. There is an intriguing formula which expresses the Atiyah-Patodi-Singer eta-invariant in terms of a Chern-character like formula involving a regularized trace.

After introducing connected filtered Hopf algebras, I'll explain the renormalization mechanism of Connes-Kreimer in this context. Locality, renormalization group and beta function will also be introduced in the context of connected graded Hopf algebras.

A canonical sum, respectively a canonical integral whose values live in meromorphic maps, are defined on holomorphic germs of symbols. They generalize to multiple sums of germs of symbols with conical constraints, respectively multiple integrals of germs of symbols with linear constraints. We can build this way
i) multiple zeta functions seen as multiple sums of symbols with conical constraints and
ii) Feynman type integrals seen as multiple integrals of symbols with linear constraints.
We discuss different renormalization procedures which yield renormalized values that factorize over independent sets of constraints, a condition reminiscent of locality in quantum field theory. In particular, the corresponding renormalized multiple zeta values obey stuffle relations.

Let X be a closed manifold. This talk will be about the topology of the space of invertible pseudodifferential operators of order zero acting on smooth functions on X. We will show that its homotopy groups can be expressed in terms of the K-theory of the cosphere bundle of X. The proof will use the long exact sequence of homotopy groups associated to the Serre fibration defined by the principal symbol. This also leads to interesting results concerning the subgroup of invertible compact perturbations of the identity.

In this talk we present a summation theory for indefinite nested sums and products based on difference fields, and we demonstrate how the underlying algorithms (telescoping, creative telescoping, recurrence solving) can support the evaluation of Feynman integrals from Perturbative Quantum Field Theory. Besides this, we focus on the following spin off: In our summation theory the sums and products are represented in the so called $\Pi\Sigma$-fields by transcendental field extensions. Motivated by this fact, we construct a difference ring monomorphism for such fields (or subrings) into the ring of sequences. In particular, this construction transfers the transcendence properties from a given difference field into the sequence domain. Combining this construction with summation theorems, we can show, e.g., that the harmonic numbers $\{H^{(i)}_n|i\geq1\}$ with $H^{(i)}_n:=\sum_{k=1}^n\frac{1}{k^i}$ are algebraically independent over $Q(n)$.

A determinant functional on an algebra is a combination of more fundamental structures which are of some independent interest. For instance, index theory is a special case of these structures (rather than vice versa). On the other hand, determinant structures appear to extend easily to categories, and other exotic enviroments. The most natural formulation arises using methods of cyclic cohomology and non commutative geometry, leading to a number of natural conjectures. Indeed, at this point the theory is at an early stage of development and somewhat enigmatic.

I will report on work in progress on renormalization Hopf algebras of gauge theories. Their coaction on the coupling constants is described in the context of Batalin-Vilkovisky (BV) algebras. The so-called master equation satisfied by the action in the BV-algebra implies the existence of certain Hopf ideals in the renormalization Hopf algebra.

Continuing the ideas introduced by Kreimer at this conference I will discuss what we can say and what we can conjecture about the simplified differential equations coming from Dyson-Schwinger equations in a variety of examples with different characteristics, including toy examples as well as QED and QCD.

We look into the formula, due to Douglas and Dijkgraaf, on the generating function F_g(q) of the number of simply ramified covers of genus g > 0 over a fixed elliptic curve with marked points. Their result is that F_g(q) is a quasimodular form of weight 6g-6 on the full modular group PSL_2(Z). There are two ways of computing F_g(q): the fermionic count and the bosonic count. The fermionic counting is a mathematical treatment, and we will give a mathematical proof to the formula. On the other hand, the bosonic counting rests on physical arguments, which involves path integrals on trivalent Feynman diagrams. We will compute F_g(q) for small genera with bosonic count.