Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems

We study the problem of placing effective upper bounds for the number of zeros of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of given dimension n having m singular points. As a function of n,m, this bound turns out to be double exponential in the precise sense explained in the paper. As a corollary, we obtain a solution of the so called restricted infinitesimal Hilbert 16th problem, an explicit upper bound for the number of isolated zeros of Abelian integrals which is polynomially growing as the Hamiltonian tends to the degeneracy locus. This improves the exponential bounds recently established by A. Glutsyuk and Yu. Ilyashenko.Comment: Will appear in Annales de l'institut Fourier vol. 60 (2010

An invitation to 2D TQFT and quantization of Hitchin spectral curves

This article consists of two parts. In Part 1, we present a formulation of two-dimensional topological quantum field theories in terms of a functor from a category of Ribbon graphs to the endofuntor category of a monoidal category. The key point is that the category of ribbon graphs produces all Frobenius objects. Necessary backgrounds from Frobenius algebras, topological quantum field theories, and cohomological field theories are reviewed. A result on Frobenius algebra twisted topological recursion is included at the end of Part 1. In Part 2, we explain a geometric theory of quantum curves. The focus is placed on the process of quantization as a passage from families of Hitchin spectral curves to families of opers. To make the presentation simpler, we unfold the story using SL_2(\mathbb{C})-opers and rank 2 Higgs bundles defined on a compact Riemann surface $C$ of genus greater than $1$. In this case, quantum curves, opers, and projective structures in $C$ all become the same notion. Background materials on projective coordinate systems, Higgs bundles, opers, and non-Abelian Hodge correspondence are explained.Comment: 53 pages, 6 figure

Fock factorizations, and decompositions of the L2L^2 spaces over general Levy processes

We explicitly construct and study an isometry between the spaces of square integrable functionals of an arbitrary Levy process and a vector-valued Gaussian white noise. In particular, we obtain explicit formulas for this isometry at the level of multiplicative functionals and at the level of orthogonal decompositions, as well as find its kernel. We consider in detail the central special case: the isometry between the $L^2$ spaces over a Poisson process and the corresponding white noise. The key role in our considerations is played by the notion of measure and Hilbert factorizations and related notions of multiplicative and additive functionals and logarithm. The obtained results allow us to introduce a canonical Fock structure (an analogue of the Wiener--Ito decomposition) in the $L^2$ space over an arbitrary Levy process. An application to the representation theory of current groups is considered. An example of a non-Fock factorization is given.Comment: 35 pages; LaTeX; to appear in Russian Math. Survey

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

A unified mode decomposition method for physical fields in homogeneous cosmology

The methods of mode decomposition and Fourier analysis of classical and quantum fields on curved spacetimes previously available mainly for the scalar field on Friedman- Robertson-Walker (FRW) spacetimes are extended to arbitrary vector bundle fields on general spatially homogeneous spacetimes. This is done by developing a rigorous unified framework which incorporates mode decomposition, harmonic analysis and Fourier anal- ysis. The limits of applicability and uniqueness of mode decomposition by separation of the time variable in the field equation are found. It is shown how mode decomposition can be naturally extended to weak solutions of the field equation under some analytical assumptions. It is further shown that these assumptions can always be fulfilled if the vector bundle under consideration is analytic. The propagator of the field equation is explicitly mode decomposed. A short survey on the geometry of the models considered in mathematical cosmology is given and it is concluded that practically all of them can be represented by a semidirect homogeneous vector bundle. Abstract harmonic analytical Fourier transform is introduced in semidirect homogeneous spaces and it is explained how it can be related to the spectral Fourier transform. The general form of invariant bi-distributions on semidirect homogeneous spaces is found in the Fourier space which generalizes earlier results for the homogeneous states of the scalar field on FRW spacetimes