# Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory

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*Alan Carey*

A publication of the European Mathematical Society

This collection of expository articles grew out of the
workshop “Number Theory and Physics” held in March 2009 at
The Erwin Schrödinger International Institute for Mathematical
Physics, Vienna. The common theme of the articles is the influence of
ideas from noncommutative geometry (NCG) on subjects ranging from
number theory to Lie algebras, index theory, and mathematical
physics.

Matilde Marcolli's article gives a survey of relevant aspects of NCG in
number theory, building on an introduction to motives for beginners by Jorge
Plazas and Sujatha Ramdorai. A mildly unconventional view of index theory, from
the viewpoint of NCG, is described in the article by Alan Carey, John Phillips,
and Adam Rennie. As developed by Alain Connes and Dirk Kreimer, NCG also
provides insight into novel algebraic structures underlying many analytic
aspects of quantum field theory. Dominique Manchon's article on pre-Lie
algebras fits into this developing research area. This interplay of algebraic
and analytic techniques also appears in the articles by Christoph Bergbauer,
who introduces renormalization theory and Feynman diagram methods, and Sylvie
Paycha, who focuses on relations between renormalization and zeta function
techniques.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in number theory and physics.