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

Feynman integrals and periods in configuration spaces

We describe differential forms representing Feynman amplitudes in configuration spaces of Feynman graphs, and regularization and evaluation techniques, for suitable chains of integration, that give rise to periods of mixed Tate motives.Comment: 54 pages, LaTeX, 1 PDF figure; v2: correction and expansion of section

Notes on Feynman Integrals and Renormalization

I review various aspects of Feynman integrals, regularization and renormalization. Following Bloch, I focus on a linear algebraic approach to the Feynman rules, and I try to bring together several renormalization methods found in the literature from a unifying point of view, using resolutions of singularities. In the second part of the paper, I briefly sketch the work of Belkale, Brosnan resp. Bloch, Esnault and Kreimer on the motivic nature of Feynman integrals.Comment: 39

Essential norm of an integral-type operator from ω-Bloch spaces to μ-Zygmund spaces on the unit ball

In this paper, we give an estimate for the essential norm of an integral-type operator from $\omega$-Bloch spaces to $\mu$-Zygmund spaces on the unit ball