Signs, figures and time: Cavaillès on "intuition" in mathematics

This paper is concerned with Cavaillès¿ account of ¿intuition¿ in mathematics. Cavaillès starts from Kant¿s theory of constructions in intuition and then relies on various remarks by Hilbert to apply it to modern mathematics. In this context, ¿intuition¿ includes the drawing of geometrical figures, the use of algebraic or logical signs and the generation of numbers as, for example, described by Brouwer. Cavaillès argues that mathematical practice can indeed be described as ¿constructions in intuition¿ but that these constructions are not imbedded in the space and in the time of our Sensibility, as Kant believed: They take place in other structures which are engendered in the history of mathematics. This leads Cavaillès to a critical discussion of both Hilbert¿s and Brouwer¿s foundational program

Spectral analysis and zeta determinant on the deformed spheres

We consider a class of singular Riemannian manifolds, the deformed spheres $S^N_k$, defined as the classical spheres with a one parameter family $g[k]$ of singular Riemannian structures, that reduces for $k=1$ to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian $\Delta_{S^N_k}$, we study the associated zeta functions $\zeta(s,\Delta_{S^N_k})$. We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in $\zeta(s,\Delta_{S^N_k})$. An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular $\zeta(0,\Delta_{S^N_k})$ and $\zeta'(0,\Delta_{S^N_k})$. We give explicit formulas for the zeta regularized determinant in the low dimensional cases, $N=2,3$, thus generalizing a result of Dowker \cite{Dow1}, and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter $k$.Comment: 1 figur

Polynôme d'Alexander à l'infini d'un polynôme à deux variables

In this work, we compute the Alexander invariants at infinity of a complex polynomial in two variables by means of its resolution and also by means of the Eisenbud-Neumann diagram of the generic link at infinity of the polynomial

v-Quasi-ordinary power series: Factorisation, Newton trees and resultants

This article is devoted to the study of -quasiordinary singularities. They were introduced by H. Hironaka in order to study quasiordinary singularities from the point of view of the Newton polygon. We de ne Newton trees associated to power series, and the -quasiordinary singularities are the simplest case where the Newton tree is not trivial. New- ton trees generalise splice diagrams introduced by Eisenbud and Neumann for curves. We will show that for -quasiordinary singularities, there is a factorisation associated to its Newton polygon as in the case of curves. We de ne the bi-coloured Newton tree of a product fg and give a sucient condition onthis coloured Newton tree so that the resultant is a monomial times a unit and we compute this resultant from the decorations of the tree. This is a generalisation of the intersection multiplicity of curves and its computation from the splice diagram