926 research outputs found
Feynman integrals and motives
This article gives an overview of recent results on the relation between
quantum field theory and motives, with an emphasis on two different approaches:
a "bottom-up" approach based on the algebraic geometry of varieties associated
to Feynman graphs, and a "top-down" approach based on the comparison of the
properties of associated categorical structures. This survey is mostly based on
joint work of the author with Paolo Aluffi, along the lines of the first
approach, and on previous work of the author with Alain Connes on the second
approach.Comment: 32 pages LaTeX, 3 figures, to appear in the Proceedings of the 5th
European Congress of Mathematic
3-manifold groups are virtually residually p
Given a prime , a group is called residually if the intersection of
its -power index normal subgroups is trivial. A group is called virtually
residually if it has a finite index subgroup which is residually . It is
well-known that finitely generated linear groups over fields of characteristic
zero are virtually residually for all but finitely many . In particular,
fundamental groups of hyperbolic 3-manifolds are virtually residually . It
is also well-known that fundamental groups of 3-manifolds are residually
finite. In this paper we prove a common generalization of these results: every
3-manifold group is virtually residually for all but finitely many .
This gives evidence for the conjecture (Thurston) that fundamental groups of
3-manifolds are linear groups
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