A canonical Frobenius structure

We show that it makes sense to speak of THE Frobenius manifold attached to a convenient and nondegenerate Laurent polynomialComment: 24 page

Homological perturbation theory for nonperturbative integrals

We use the homological perturbation lemma to produce explicit formulas computing the class in the twisted de Rham complex represented by an arbitrary polynomial. This is a non-asymptotic version of the method of Feynman diagrams. In particular, we explain that phenomena usually thought of as particular to asymptotic integrals in fact also occur exactly: integrals of the type appearing in quantum field theory can be reduced in a totally algebraic fashion to integrals over an Euler--Lagrange locus, provided this locus is understood in the scheme-theoretic sense, so that imaginary critical points and multiplicities of degenerate critical points contribute.Comment: 22 pages. Minor revisions from previous versio

Multiplicity of complex hypersurface singularities, Rouche' satellites and Zariski's problem

Soient $f,g\colon (\hbox{\aa C}^n,0) \to (\hbox{\aa C},0)$ des germes de fonctions holomorphes r\'eduits. Nous montrons que $f$ et $g$ ont la m\^eme multiplicit\'e en 0 si et seulement s'il existe des germes r\'eduits $f'$ et $g'$ analytiquement \'equivalents \`a $f$ et $g$, respectivement, tels que $f'$ et $g'$ satisfassent une in\'egalit\'e du type de Rouch\'e par rapport \`a un `petit' cercle g\'en\'erique autour de~0. Comme application, nous donnons une reformulation de la question de Zariski sur la multiplicit\'e et une r\'eponse partielle positive \`a celle--ci.Comment: Final versio

Suspending Lefschetz fibrations, with an application to Local Mirror Symmetry

We consider the suspension operation on Lefschetz fibrations, which takes p(x) to p(x)-y^2. This leaves the Fukaya category of the fibration invariant, and changes the category of the fibre (or more precisely, the subcategory consisting of a basis of vanishing cycles) in a specific way. As an application, we prove part of Homological Mirror Symmetry for the total spaces of canonical bundles over toric del Pezzo surfaces.Comment: v2: slightly expanded expositio

Motivic Milnor fibre for nondegenerate function germs on toric singularities

We study function germs on toric varieties which are nondegenerate for their Newton diagram. We express their motivic Milnor fibre in terms of their Newton diagram. We extend a formula for the motivic nearby fibre to the case of a toroidal degeneration. We illustrate this by some examples.Comment: 14 page

Computing zeta functions of sparse nondegenerate hypersurfaces

Using the cohomology theory of Dwork, as developed by Adolphson and Sperber, we exhibit a deterministic algorithm to compute the zeta function of a nondegenerate hypersurface defined over a finite field. This algorithm is particularly well-suited to work with polynomials in small characteristic that have few monomials (relative to their dimension). Our method covers toric, affine, and projective hypersurfaces and also can be used to compute the L-function of an exponential sum.Comment: 37 pages; minor revisio

On Sasaki-Einstein manifolds in dimension five

We prove the existence of Sasaki-Einstein metrics on certain simply connected 5-manifolds where until now existence was unknown. All of these manifolds have non-trivial torsion classes. On several of these we show that there are a countable infinity of deformation classes of Sasaki-Einstein structures.Comment: 18 pages, Exposition was expanded and a reference adde

Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

We give an expression for the {\L}ojasiewicz exponent of a wide class of n-tuples of ideals $(I_1,..., I_n)$ in \O_n using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of {\L}ojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions $(\mathbb{C}^n,0)\to (\mathbb{C},0)$ for which the {\L}ojasiewicz of its gradient map $\nabla f$ attains the maximum possible value.Comment: 25 pages. Updated with minor change

Dissipation time and decay of correlations

We consider the effect of noise on the dynamics generated by volume-preserving maps on a d-dimensional torus. The quantity we use to measure the irreversibility of the dynamics is the dissipation time. We focus on the asymptotic behaviour of this time in the limit of small noise. We derive universal lower and upper bounds for the dissipation time in terms of various properties of the map and its associated propagators: spectral properties, local expansivity, and global mixing properties. We show that the dissipation is slow for a general class of non-weakly-mixing maps; on the opposite, it is fast for a large class of exponentially mixing systems which include uniformly expanding maps and Anosov diffeomorphisms.Comment: 26 Pages, LaTex. Submitted to Nonlinearit

Generic linear sections of complex hypersurfaces and monomial ideals

Let f : (C^n, 0) -->(C, 0) be an analytic function germ. Under the hypothesis that f is Newton non-degenerate, we compute the \mu*-sequence of f in terms of the Newton polyhedron of f . This sequence was defined by Teissier in order to characterize the Whitney equisingularity of deformations of complex hypersurfaces.Work supported by DGICYT Grant MTM2009-08933.Bivià-Ausina, C. (2012). Generic linear sections of complex hypersurfaces and monomial ideals. Topology and its Applications. 159(2):414-419. https://doi.org/10.1016/j.topol.2011.09.015S414419159