Courbes de semi-groupe donné

Sin resume

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

Numerical evidence toward a 2-adic equivariant ''Main Conjecture''

International audienceWe test a conjectural non abelian refinement of the classical 2-adic Main Conjecture of Iwasawa theory. In the first part, we show how, in the special case that we study, the validity of this refinement is equivalent to a congruence condition on the coefficients of some power series. Then, in the second part, we explain how to compute the first coefficients of this power series and thus numerically check the conjecture in that setting

An exact solution method for 1D polynomial Schr\"odinger equations

Stationary 1D Schr\"odinger equations with polynomial potentials are reduced to explicit countable closed systems of exact quantization conditions, which are selfconsistent constraints upon the zeros of zeta-regularized spectral determinants, complementing the usual asymptotic (Bohr--Sommerfeld) constraints. (This reduction is currently completed under a certain vanishing condition.) In particular, the symmetric quartic oscillators are admissible systems, and the formalism is tested upon them. Enforcing the exact and asymptotic constraints by suitable iterative schemes, we numerically observe geometric convergence to the correct eigenvalues/functions in some test cases, suggesting that the output of the reduction should define a contractive fixed-point problem (at least in some vicinity of the pure $q^4$ case).Comment: flatex text.tex, 4 file

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

On the Milnor formula in arbitrary characteristic

The Milnor formula $\mu=2\delta-r+1$ relates the Milnor number $\mu$, the double point number $\delta$ and the number $r$ of branches of a plane curve singularity. It holds over the fields of characteristic zero. Melle and Wall based on a result by Deligne proved the inequality $\mu\geq 2\delta-r+1$ in arbitrary characteristic and showed that the equality $\mu=2\delta-r+1$ characterizes the singularities with no wild vanishing cycles. In this note we give an account of results on the Milnor formula in characteristic $p$. It holds if the plane singularity is Newton non-degenerate (Boubakri et al. Rev. Mat. Complut. (2010) 25) or if $p$ is greater than the intersection number of the singularity with its generic polar (Nguyen H.D., Annales de l'Institut Fourier, Tome 66 (5) (2016)). Then we improve our result on the Milnor number of irreducible singularities (Bull. London Math. Soc. 48 (2016)). Our considerations are based on the properties of polars of plane singularities in characteristic $p$.Comment: 18 page

The combinatorics of plane curve singularities. How Newton polygons blossom into lotuses

This survey may be seen as an introduction to the use of toric and tropical geometry in the analysis of plane curve singularities, which are germs $(C,o)$ of complex analytic curves contained in a smooth complex analytic surface $S$. The embedded topological type of such a pair $(S, C)$ is usually defined to be that of the oriented link obtained by intersecting $C$ with a sufficiently small oriented Euclidean sphere centered at the point $o$, defined once a system of local coordinates $(x,y)$ was chosen on the germ $(S,o)$. If one works more generally over an arbitrary algebraically closed field of characteristic zero, one speaks instead of the combinatorial type of $(S, C)$. One may define it by looking either at the Newton-Puiseux series associated to $C$ relative to a generic local coordinate system $(x,y)$, or at the set of infinitely near points which have to be blown up in order to get the minimal embedded resolution of the germ $(C,o)$ or, thirdly, at the preimage of this germ by the resolution. Each point of view leads to a different encoding of the combinatorial type by a decorated tree: an Eggers-Wall tree, an Enriques diagram, or a weighted dual graph. The three trees contain the same information, which in the complex setting is equivalent to the knowledge of the embedded topological type. There are known algorithms for transforming one tree into another. In this paper we explain how a special type of two-dimensional simplicial complex called a lotus allows to think geometrically about the relations between the three types of trees. Namely, all of them embed in a natural lotus, their numerical decorations appearing as invariants of it. This lotus is constructed from the finite set of Newton polygons created during any process of resolution of $(C,o)$ by successive toric modifications.Comment: 104 pages, 58 figures. Compared to the previous version, section 2 is new. The historical information, contained before in subsection 6.2, is distributed now throughout the paper in the subsections called "Historical comments''. More details are also added at various places of the paper. To appear in the Handbook of Geometry and Topology of Singularities I, Springer, 202

On the non-abelian Brumer-Stark conjecture and the equivariant Iwasawa main conjecture

We show that for an odd prime p, the p-primary parts of refinements of the (imprimitive) non-abelian Brumer and Brumer-Stark conjectures are implied by the equivariant Iwasawa main conjecture (EIMC) for totally real fields. Crucially, this result does not depend on the vanishing of the relevant Iwasawa mu-invariant. In combination with the authors' previous work on the EIMC, this leads to unconditional proofs of the non-abelian Brumer and Brumer-Stark conjectures in many new cases.Comment: 33 pages; to appear in Mathematische Zeitschrift; v3 many minor updates including new title; v2 some cohomological arguments simplified; v1 is a revised version of the second half of arXiv:1408.4934v