Abstract:
We give a new method for the reduction of tensor integrals to finite integral representations and UV divergent analytic expressions. This includes a new method for the handling of the gamma-algebra. TYPO IN EQUATION (5) CORRECTED, MACROS REORDERED.

Abstract:
We summarize recent results connecting multiloop Feynman diagram calculations to different parts of mathematics, with special attention given to the Hopf algebra structure of renormalization.

Abstract:
The recently discovered formalism underlying renormalization theory, the Hopf algebra of rooted trees, allows to generalize Chen's lemma. In its generalized form it describes the change of a scale in Green functions, and hence relates to the operator product expansion. Hand in hand with this generalization goes the generalization of the ordinary factorial $n!$ to the tree factorial $t^!$. Various identities on tree-factorials are derived which clarify the relation between Connes-Moscovici weights and Quantum Field Theory.

Abstract:
We discuss the structure of Dyson--Schwinger equations in quantum gravity and conclude in particular that all relevant skeletons are of first order in the loop number. There is an accompanying sub Hopf algebra on gravity amplitudes equivalent to identities between n-graviton scattering amplitudes which generalize the Slavnov Taylor identities. These identities map the infinite number of charges and finite numbers of skeletons in gravity to an infinite number of skeletons and a finite number of charges needing renormalization. Our analysis suggests that gravity, regarded as a probability conserving but perturbatively non-renormalizable theory, is renormalizable after all, thanks to the structure of its Dyson--Schwinger equations.

Abstract:
We investigate to what extent renormalization can be understood as an algebraic manipulation on concatenated one-loop integrals. We find that the resulting algebra indicates a useful connection to knot theory.

Abstract:
We sketch a systematic approach to multiloop calculations. We focus on the Z-factors of a renormalizable theory and show that they can be obtained by purely algebraic methods based on power-counting and the forest structure of the divergent graphs. (submitted contributions to the conference proceedings "Confronting the Infinite", Adelaide February 1994)

Abstract:
Work is reported on finite integral representations for 2-loop massive 2-, 3- and 4-point functions, using orthogonal and parallel space variables. It is shown that this can be utilized to cover particles with arbitrary spin (tensor integrals), and that UV divergences can be absorbed in an algebraic manner. This includes a classification of UV divergences by means of the topology of the graph, interpreted in terms of knots.

Abstract:
We discuss the consistency of a new \gf -scheme with renormalization. In particular we study the power-counting behaviour of multiloop graphs to prove its consistency. As a side effect we obtain a short proof of the Adler-Bardeen theorem. Further we show that this \gf-scheme does not modify the BRST identities at any loop orders in contrast to BM type schemes.

Abstract:
We report on some conceptual changes in our present understanding of Quantum Field Theory and muse about possible consequences for the understanding of $v>c$ signals.