On étudie la version réelle suivante d'un théorème célèbre d'Abhyankar-Moh : quelles applications rationnelles de la droite affine dans le plan affine, dont le lieu réel est un plongement fermé non singulier de R dans R^2, sont équivalentes, à difféomorphisme birationnel du plan près, au plongement trivial ? Dans ce cadre, on montre qu’il existe des plongements non équivalents. Certains d’entre eux sont détectés pas la non-négativité de la dimension de Kodaira réelle du complémentaire de leur image. Ce nouvel invariant est dérivé des propriétés topologiques de « faux plans réels » particuliers associés à ces plongements. (Travail en commun avec Adrien Dubouloz.)
Many of the most important results in mathematics are based on some inequality, of geometric or analytic nature. On the other hand, this separation between geometry and analysis is not sharp and the most intriguing inequalities are indeed the ones that have a mixed nature and enhance the interplay of the two realms. Moreover, many apparently purely geometric inequalities have some powerful functional counterpart, like for instance the Isoperimetric Inequality and Sobolev Inequality. I will try to give some general overview on geometric-analytic inequalities and will concentrate on one of them, precisely the Brunn-Minkowski inequality, an apparently geometric inequality which is at the core of modern convex geometry, and on its functional counterpart, the Borell-Brascamp-Lieb inequality. And also possibly show some applications to PDEs.
Let $R$ be a real closed field. We prove that if $R$ is uncountable, then any separately Nash (resp. arc-Nash) function defined over $R$ is semialgebraic (resp. continuous semialgebraic). To complete the picture, we provide an example showing that the assumption on $R$ to be uncountable cannot be dropped. Moreover, even if $R$ is uncountable but non-Archimedean then the shape of the domain of a separately Nash function matters for the conclusion. For $R = R$ we prove that arc-Nash functions coincide with arc-analytic semialgebraic functions. Joint work with W. Kucharz and A. El-Siblani.
The speculative ambition of replacing the old theory of program approximation based on syntactic continuity with the theory of resource consumption based on Taylor expansion and originating from the differential λ-calculus is nowadays at hand. Using this resource sensitive theory, we provide simple proofs of important results in λ-calculus that are usually demonstrated by exploiting Scott’s continuity, Berry’s stability or Kahn and Plotkin’s sequentiality theory. A paradigmatic example is given by the Perpendicular Lines Lemma for the Böhm tree semantics, which is proved here simply by induction, but relying on the main properties of resource approximants: strong normalization, confluence and linearity.
In 1979 O. Zariski proposed a general theory of equisingularity for algebraic or algebroid hypersurfaces over an algebraically closed field of characteristic zero. It is based on the notion of dimensionality type that is defined recursively by considering the discriminants loci of subsequent ``generic'' projections. The singularities of dimensionality type 1 are isomorphic to the equisingular families of plane curve singularities. In this talk we consider the case of dimensionality type 2, the Zariski equisingular families of surface singularities in 3-space. Using an approach going back to Briançon and Henry, we show that in this case generic linear projections are generic in the sense of Zariski (this is still open for dimensionality type greater than 2). Over the field of complex numbers, we show that such families are bi-Lipschitz trivial, by construction of an explicit Lipschitz stratification. (Based on joint work with L. Paunescu.)
TBA
In this talk we consider the Cauchy problem for the 2D Euler equations for incompressible inviscid fluids. It is well-known that given a smooth initial datum, the Cauchy problem is well-posed and in particular the energy is conserved and the vorticity is transported by the flow of the velocity. When we consider weak solutions this might not be the case anymore. We will review some recent results obtained in collaboration with Gianluca Crippa and Gennaro Ciampa where we extend those properties to the case of irregular vorticities. In particular, under low integrability assumptions on the vorticity we show that certain approximations important from a physical and a numerical point of view converge to solutions satisfying those properties.
Quantum hydrodynamic (QHD) systems arise in the effective description of phenomena where quantistic behavior can be seen also at a macroscopic scale. This is the case for instance in Bose-Einstein condensation, superfluidity or in the modeling of semiconductor devices. Standard results for global existence of finite energy weak solutions to the QHD system often exploit the analogy with a nonlinear Schrödinger equation; by using the Madelung transform it is possible to define a solution to the QHD by considering the momenta (mass and current density) associated to a wave function. In particular this argument requires the initial data to be determined by a given wave function. This usual approach hence shows the existence of solutions but can not be used to study their stability properties in a general framework. In this talk I will present some recent developments that overcome those difficulties for the one dimensional QHD system. First of all I will provide an existence result for a large class of initial data, without requiring them to be generated by a wave function. Furthermore, I will prove a stability result for weak solutions. This exploits a novel functional which formally controls the L^2 norm of the chemical potential, weighted with the particle density. This is a joint work with P. Marcati and H. Zheng.
(Joint work with P. LeFanu Lumsdaine.)
Lawvere theories and (coloured) operads provide particularly nice representations for suitable algebraic theories with a given set of sorts, as monoids in certain categories of collections.
We extend this to dependent type theories: For an inverse category C, we show how suitable “C-sorted type theories” may be viewed (1) as monoids in a category of collections, and (2) as generalised Lawvere theories in the sense of Berger–Melliès–Weber. Moreover, (essentially) every dependent type theory arises in this way.
Inverse categories are known from homotopy theory, where they (or their opposite categories) provide a good notion of a category of ``cells''. Examples are the category of semi-simplices, the category of globes, the category of opetopes, etc.
Cet exposé est une introduction élémentaire aux espaces de Finsler qui constituent la généralisation la plus naturelle de la géométrie riemannienne. Après une présentation des principales propriétés de la géométrie de Finsler, nous verrons à travers quelques exemples comment celle-ci apparaît dans diverses situations
Hypergroups are objects like groups but with addition taking possibly many values. Likewise, hyperrings and hyperfields are objects similar to rings and fields, but with multivalued addition. Hyperfields provide a convenient tool in axiomatizing the algebraic theory of quadratic forms and in this talk we shall focus on three such applications. Firstly, we shall show how Witt equivalence of fields can be conveniently expressed in the language of hyperfields and will present some recent results on Witt equivalence of function fields over global and local fields. Secondly, we shall show how orderings of higher level can be defined for hyperrings and hyperfields, and, consequently, how they can be used to provide an axiomatic framework to study forms of higher order. Finally, we shall define the category of, so called, presentable fields and define their Witt rings, thus providing yet another machinery to study quadratic forms over fields. The results presented in this talk were obtained jointly with Murray Marshall and Krzysztof Worytkiewicz.
En 1933 Derrick Lehmer propose une méthode pour trouver de grands nombres premiers dans des suites récurrentes linéaires ayant comme ingrédients principaux des polynômes à coefficients entiers de très petite mesure de Mahler. Le problème de Lehmer est de trouver de tels polynômes de mesure < 1.1762. Ce nombre, dit nombre de Lehmer, est resté la plus petite valeur connue. La Conjecture de Lehmer stipule qu'il n'en existe pas de mesure arbitrairement proche de un. On présentera les fonctions analytiques (déterminants de Fredholm généralisés, fonctions zêta dynamiques) du système dynamique de numération de Rényi-Parry en base entier algébrique variable (beta-shift), pour montrer comment certaines propriétés de ces fonctions donnent des points d'attaque de cette Conjecture, et la fracturabilité des polynômes minimaux de ces bases. On fera le lien avec l'algorithme de Schur développé par Dufresnoy, Pisot, Amara, Bertin, pour les plus petits nombres de Pisot. On évoquera le problème des valeurs d'adhérence (Deninger, Rodriguez-Villegas) de l'ensemble de mesures de Mahler de nombres algbériques dans ce contexte.
Le but de l'exposé sera de présenter une preuve alternative de la stabilité asymptotique d'équilibres spatialement homogènes pour des perturbations localisées d'équations de Vlasov posées dans l'espace entier. La preuve originale due à Bedrossian-Mouhot-Masmoudi, est inspirée de la preuve du Landau damping pour des solutions périodiques et utilise les propriétés dispersives du transport libre en Fourier. On présentera une approche basée sur la méthode des caractéristiques et une étude des propriétés dispersives du linéarisé dans l'espace physique. (collaboration avec D. Han-Kwan (Polytechnique) et T. Nguyen (Penn-State))
Dans un article de 1996, van den Dries et Miller conjecturent que la structure R_{an, x^R} obtenue par adjonction des puissances réelles aux sous-analytiques (globaux) est maximale parmi les réduites polynomialement bornées de la structure R_{an, exp} obtenue en ajoutant l'exponentielle (non restreinte) aux sous-analytiques. On montre un analogue polynomialement borné de cette conjecture. Plus précisément, étant donné L un sous corps de R et S une structure polynomialement bornée, la structure S_{x^L} (obtenue en ajoutant à S les puissances réelles avec exposant dans L) est maximale parmi les réduites de S_{x^R} (obtenue en ajoutant toutes les puissances réelles à S) ayant L comme corps d'exposants. Autrement dit, si X est un ensemble que l'on peut définir à l'aide de S et de puissances réelles, seules les puissances qui se redéfinissent depuis S et X sont nécessaires pour définir X; les autres exposants, cachés dans la définition de X, peuvent s'éviter. On déduit le corollaire suivant qui répond, au niveau des fonctions d'une variable, à une généralisation de la conjecture de van den Dries et Miller: si f:R->R est définissable dans S_{exp} et si S_{f} est polynomialement bornée de corps d'exposants L, alors f est en fait définissable dans S_{x^L}. (Travail commun avec G. Jones)