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