332 research outputs found
Modular curves, C* algebras, and chaotic cosmology
We make some brief remarks on the relation of the mixmaster universe model of
chaotic cosmology to the geometry of modular curves and to noncommutative
geometry. In particular we consider a class of solutions with bounded number of
cycles in each Kasner era and describe their dynamical properties, which we
relate to the noncommutative geometry of a moduli space of such solutions,
given by a Cuntz-Krieger C*-algebra.Comment: 12 pages LaTeX, one eps figur
Noncommutative Geometry and Arithmetic
This is an overview of recent results aimed at developing a geometry of
noncommutative tori with real multiplication, with the purpose of providing a
parallel, for real quadratic fields, of the classical theory of elliptic curves
with complex multiplication for imaginary quadratic fields. This talk
concentrates on two main aspects: the relation of Stark numbers to the geometry
of noncommutative tori with real multiplication, and the shadows of modular
forms on the noncommutative boundary of modular curves, that is, the moduli
space of noncommutative tori. To appear in Proc. ICM 2010.Comment: 16 pages, LaTe
Seiberg-Witten Floer Homology and Heegaard Splittings
This paper presents the construction of the Seiberg-Witten-Floer homology of
three-manifolds with non-trivial rational homology, and some properties of the
invariant of three-manifolds obtained by computing the Euler characteristic.
This new version corrects a mistake and some misprints that appear in the
published version (Internat. J. Math., Vol.7, N.5 (1996) 671-696). Various
sections are rewritten in more detail.Comment: 46 Pages, LaTex, no figure
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
Principles and Parameters: a coding theory perspective
We propose an approach to Longobardi's parametric comparison method (PCM) via
the theory of error-correcting codes. One associates to a collection of
languages to be analyzed with the PCM a binary (or ternary) code with one code
words for each language in the family and each word consisting of the binary
values of the syntactic parameters of the language, with the ternary case
allowing for an additional parameter state that takes into account phenomena of
entailment of parameters. The code parameters of the resulting code can be
compared with some classical bounds in coding theory: the asymptotic bound, the
Gilbert-Varshamov bound, etc. The position of the code parameters with respect
to some of these bounds provides quantitative information on the variability of
syntactic parameters within and across historical-linguistic families. While
computations carried out for languages belonging to the same family yield codes
below the GV curve, comparisons across different historical families can give
examples of isolated codes lying above the asymptotic bound.Comment: 11 pages, LaTe
Motivic Information
We introduce notions of information/entropy and information loss associated
to exponentiable motivic measures. We show that they satisfy appropriate
analogs to the Khinchin-type properties that characterize information loss in
the context of measures on finite sets.Comment: 24 pages, LaTeX, 1 jpg figur
Motivic renormalization and singularities
We consider parametric Feynman integrals and their dimensional regularization
from the point of view of differential forms on hypersurface complements and
the approach to mixed Hodge structures via oscillatory integrals. We consider
restrictions to linear subspaces that slice the singular locus, to handle the
presence of non-isolated singularities. In order to account for all possible
choices of slicing, we encode this extra datum as an enrichment of the Hopf
algebra of Feynman graphs. We introduce a new regularization method for
parametric Feynman integrals, which is based on Leray coboundaries and, like
dimensional regularization, replaces a divergent integral with a Laurent series
in a complex parameter. The Connes--Kreimer formulation of renormalization can
be applied to this regularization method. We relate the dimensional
regularization of the Feynman integral to the Mellin transforms of certain
Gelfand--Leray forms and we show that, upon varying the external momenta, the
Feynman integrals for a given graph span a family of subspaces in the
cohomological Milnor fibration. We show how to pass from regular singular
Picard--Fuchs equations to irregular singular flat equisingular connections. In
the last section, which is more speculative in nature, we propose a geometric
model for dimensional regularization in terms of logarithmic motives and
motivic sheaves.Comment: LaTeX 43 pages, v3: final version to appea
KMS weights on higher rank buildings
We extend some of the results of Carey-Marcolli-Rennie on modular index
invariants of Mumford curves to the case of higher rank buildings: we discuss
notions of KMS weights on buildings, that generalize the construction of graph
weights over graph C*-algebras.Comment: 25 pages, LaTeX, 4 jpg figure
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