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.

It often occurs that Taylor coefficients of Feynman amplitudes with rational parameters are not only "periods", but actually "multiple zeta values". In order to determine, at least heuristically, whether it is so in concrete instances, the philosophy of motives suggests an arithmetic method: counting points of related algebraic varieties modulo sufficiently many primes p and checking that the number of points varies polynomialy in p.
On the other hand, Kapranov has introduced an object, the "motivic zeta function", the role of which is precisely to "interpolate" between zeta functions of reductions modulo different primes p. We propose a survey of the developments of this idea.

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 the work of Broadhurst and Kreimer in the 90's, it has been observed numerically that massless scalar Feynman integrals evaluate to multiple zeta values in all known examples. On the other hand, the work of Belkale and Brosnan leads one to expect that the motive of a graph hypersurface is of general type. The big question is to try to reconcile these apparently contradictory results, and determine the precise arithmetic nature of perturbative quantum field theories.

In this talk, I will explain a new method for the exact calculation of Feynman integrals, and from it, deduce a sufficient criterion for a Feynman graph to evaluate to multiple zeta values. I will then prove this criterion for some infinite families of graphs. This explains, in particular, the examples known to the physicists.

We define a real mixed hodge structure on the motivic rational homotopy type of a complex variety by writing a Feynman integral.

The first three lectures will be introductory. They aim to acquaint physicists, and mathematicians with no background in quantum field theory, with the Hopf algebraic structure of renormalizable quantum field theory.

A very elementary introduction into the Hopf algebras of graphs and trees with an emphasis how to relate graphs to trees. We will include a short reminder of the role of overlapping divergences.

We study how to obtain sub Hopf algebras and their role in quantum field theory. In particular, we study the structure of a proof of renormalizability by local counterterms from the Hochschild cohomology of Hopf algebras.

Lecture 3: Dyson-Schwinger equations

We augment perturbation theory by equations of motions, obtained from the Hochschild cohomology studied in the previous lecture. We review these equations for various renormalizable field theories, and discuss
simplifications.

Lectures 4 and 5: We will discuss how to use the techniques reported on in the first three lectures to make progress with the understanding of quantum field theory. The lectures will be related to the talks by Bloch and Yeats, but will also comment on the structure of non-renormalizable field theories like quantum gravity.

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.

Lecture I: Algebraic cycle complexes

After a brief discussion of algebraic cycles, Weil cohomology and Grothendieck's categories of pure motives, we make our first approach to categories of mixed motives via cycle complexes.

Beilinson and Lichtenbaum have conjectured the existence of motivic complexes and motivic cohomology, that would give an algebraic version of the singular co-chain complex, and relate to algebraic K-theory the way that singular cohomology relates to topological K-theory. The first candidate for these complexes was supplied by Bloch's cycle complexes and his higher Chow groups. Later, Suslin defined a complex that gives an algebraic version of the complex of singular chains. We will discuss these constructions and their basic properties.

Beilinson reinterpreted his conjectures on motivic complexes by seeing them as arising from a category of mixed motives, a ``motivic" version of the category of sheaves on a topological space. This category has still not been constructed. However, categories playing the role of the derived category of Beilinson's motivic sheaf category have been constructed. We will describe Voevodsky's constructions of these ``derived categories", and their relation to Suslin homology and Bloch's higher Chow groups

Although many conjectures about categories of motives remain open, much is known about the part of the motivic category generated by extensions of Tate motives. This subcategory has a close relation to algebraic K-theory, motivic cohomology, as well as polylogarithms, zeta values and multiple zeta values, motivic versions of fundamental group and the moduli spaces of pointed rational curves.

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.

The representation theory of a (central extension) of a smooth loop group LG, for a compact semisimple Lie group G, extends to the fractionally differentiable loops (in Sobolev sense) Lq G of differentiability degree q > 1/2. However, for 0 < q < 1/2 the situation is different: even the central extension becomes ill-defined. A renormalization is needed which leads to an abelian extension by a normal subgroup. In this respect the group LqG has similarities with groups of smooth maps C^{\infty} (M,G), with M a compact manifold.

We describe a general framework in which we use KMS states for circle actions on a C*-algebra to construct Kasparov modules and semifinite spectral triples. By using a residue construction analogous to that used in the semifinite local index formula we associate to these triples a twisted cyclic cocycle on a dense subalgebra of A. This cocycle pairs with the equivariant KK-theory of the mapping cone algebra for the inclusion of the fixed point algebra of the circle action. The pairing is expressed in terms of spectral flow between a pair of unbounded self adjoint operators that are Fredholm in the semifinite sense.

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.

Mirror Symmetry for Elliptic Curves: 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.


The Modularity of Calabi-Yau Threefolds over Q: We will discuss the modularity of Galois representations associated to Calabi-Yau threefolds over Q. My talk will consists of two parts.

In Part I, we show that any rigid Calabi-Yau threefold over Q is modular, i.e. its two-dimensional Galois representation comes from a modular form of weight 4 on some some congruence subgroup of PSL(2,Z). The construction of rigid Calabi-Yau threefolds over Q is a challenging problem. We will construct a number of examples of rigid Calabi-Yau threefolds over Q and determine the associated modular forms of weight 4.

In Part II, we will focus on Calabi-Yau threefolds over Q which are non-rigid. When a Calabi-Yau threefold is non-rigid, the dimension of the Galois representation gets rather large, and the modularity question poses a serious challenge. We will construct explicit examples of non-rigid Calabi-Yau threefolds fibered over P1 by non-constant semi-stable K3 surfaces and reaching the Arakelov-Yau upper bound. For these examples, we prove that the "interesting" part of their L-series do come from modular forms. Also we will consider more general K3-fibered Calabi-Yau threefolds over Q and look into the modularity of motives arising from them.