1,449 research outputs found
Toric embedded resolutions of quasi-ordinary hypersurface singularities
We build two embedded resolution procedures of a quasi-ordinary singularity
of complex analytic hypersurface, by using toric morphisms which depend only on
the characteristic monomials associated to a quasi-ordinary projection of the
singularity. This result answers an open problem of Lipman in Equisingularity
and simultaneous resolution of singularities, Resolution of Singularities,
Progress in Mathematics No. 181, 2000, 485-503. In the first procedure the
singularity is embedded as hypersurface. In the second procedure, which is
inspired by a work of Goldin and Teissier for plane curves (see Resolving
singularities of plane analytic branches with one toric morphism,loc. cit.,
pages 315-340), we re-embed the singularity in an affine space of bigger
dimension in such a way that one toric morphism provides its embedded
resolution. We compare both procedures and we show that they coincide under
suitable hypothesis.Comment: To apear in Annales de l'Institut Fourier (Grenoble
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
Motives: an introductory survey for physicists
We survey certain accessible aspects of Grothendieck's theory of motives in
arithmetic algebraic geometry for mathematical physicists, focussing on areas
that have recently found applications in quantum field theory. An appendix (by
Matilde Marcolli) sketches further connections between motivic theory and
theoretical physics.Comment: LaTeX 35 pages, article by Abhijnan Rej with an appendix by
M.Marcolli. Version II/Final: cosmetic changes to bibliography, added a small
subsection on triangulated categories to section 6. Accepted for publication
in the MPIM-Bonn "Renormalization, combinatorics and physics" proceedings
volum
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