### Abstract - Dirk Kreimer

Feynman graphs: from operads to renormalization group to number theory

Renormalization has been settled as a self-consistent approach to the
treatment of short-distance singularities in the perturbative
expansion of quantum field theories thanks to the work of
Bogoliubov, Parasuik, Hepp, Zimmermann, and followers.
Nevertheless, its intricate Hopf algebra combinatorics went unrecognized for
a long time. In this talk we want to describe recent results devoted to the
Hopf algebra structure of quantum field theory (QFT).

The talk will review the results obtained so far in a fairly informative
style, emphasizing the underlying ideas and concepts. Furthermore, challenges for further investigation will be emphasized, as it more and more becomes clear that this Hopf algebra structure provides a fine tool for a better
understanding of a correct mathematical formulation of QFT as well
as for applications of concepts from operad theory to number theory.
A review can be found in http://arXiv.org/ps/hep-th/0010059.

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Last modified: March 9, 2001

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