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Uncountable categoricity of structures based on Banach spaces

Sophie Germain salle 2015

A continuous theory T of bounded metric structures is said to be kappa-categorical if T has a unique model of density kappa. Work of Ben Yaacov and Shelah+Usvyatsov shows that Morley's Theorem holds in this context: if T has a countable signature and is kappa-categorical for some uncountable kappa, then T is kappa-categorical for all uncountable kappa. In classical (discrete) model theory, there are several characterizations of uncountable categoricity. For example, there is a structure theorem for uncountably categorical theories T, due to Baldwin+Lachlan: there is a strongly minimal set […]

Coloriages de graphes, nombre d’or et topologie quantique

ENS (amphithéâtre Galois sous la bibliothèque de mathématique)

En partant d'objets simples comme les polynômes chromatiques, liés au fameux théorème des 4 couleurs, on va explorer quelques invariants - dans lesquels le nombre d'or est omniprésent - associés à des graphes noués dans l'espace. Ce sera l'occasion d'introduire les idées et les problèmes de la topologie dite quantique.

Une méthode constructive pour résoudre des EDP et des EDO / Existence et stabilité d’une solution explosive pour une équation de la chaleur avec un terme non linéaire critique fonction du gradient

ENS Salle W

Dans la première séance, on présentera une méthode de construction de solutions d'équations aux dérivées partielles (EDP) avec comportement prescris. La méthode sera illustrée au moyen du cas de l'équation semilinéaire de la chaleur avec nonlinéarité polynomiale. On tâchera d'expliquer la méthode en recourant à des exemples plus simples tirés des équations différentielles ordinaires (EDO).Lors de la seconde séance, on mettra en oeuvre la méthode sur une EDP plus élaborée, celle présentant une double source polynomiale, fonction de la solution et de son gradient, en proportion critique.

Strongly NIP almost real closed fields

Sophie Germain salle 1016

The following conjecture is due to Shelah--Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non-trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of strongly NIP almost real closed fields. As a result, we obtain a complete characterisation of this class.

Après-midi de théorie des groupes

Salle W

14.00-14.45 Emmanuel Militon (Nice), Groups of diffeomorphisms of a Cantor set15.00-15.45 Simon André (Rennes), Hyperbolicity is preserved under elementary equivalence15.45-16.15 coffee break(CANCELED) 16.15-17.00 Nikolay Nikolov (Oxford), On conjugacy classes in compact groups

Density of compressiblity in NIP theories

Sophie Germain salle 2015

Joint with Itay Kaplan and Pierre Simon.Distal theories are NIP theories which are ?Roewholly unstable?R. Chernikov and Simon's ?Roestrong honest definitions?R characterise distal theories as those in which every type is compressible. Adapting recent work in machine learning of Chen, Cheng, and Tang on bounds on the ?Roerecursive teaching dimension?R of a finite concept class, we find that compressibility is dense in NIP structures, i.e. any formula can be completed to a compressible type in S(A). Considering compressibility as an isolation notion (which specialises to l-isolation in stable theories), we […]

The word and conjugacy problems in finitely generated groups

Sophie Germain salle 2015

The word and conjugacy problems are central decision problems associated with finitely generated groups. In particular, there are deep results which bridge some of the main concepts of the theories of computability and computational complexity with group theoretical invariants through the word problem in groups. In this talk I will recall some of the well-known facts about the word and conjugacy problems in groups as well as discuss new results concerning the relationship between them.

Independence of CM points in elliptic curves

ENS Salle W

I will speak about joint work with Jacob Tsimerman. Let E be an elliptic curve parameterized by a modular (or Shimura) curve. There are a number of results (..., Buium-Poonen, Kuhne) to the effect that the images of CM points are (under suitable hypotheses) linearly independent in E. We consider this issue in the setting of the Zilber-Pink conjecture and prove a result which improves previous results in some aspects

Counting rational points with the determinant method

ENS Salle W

The determinant method gives upper bounds for the number of rational points of bounded height on or near algebraic varieties defined over global fields. There is a real-analytic version of the method due to Bombieri and Pila and a p-adic version due to Heath-Brown. The aim of our talk is to describe a global refinement of the p-adic method and some applications like a uniform bound for non-singular cubic curves which improves upon earlier bounds of Ellenberg-Venkatesh and Heath-Brown.

Patching over Berkovich Curves

ENS Salle W

Patching was first introduced as an approach to the Inverse Galois Problem. The technique was then extended to a more algebraic setting and used to prove a local-global principle by D. Harbater, J. Hartmann and D. Krashen. I will present an adaptation of the method of patching to the setting of Berkovich analytic curves. This will then be used to prove a local-global principle for function fields of curves that generalizes that of the above mentioned authors.

Nombres p-adiques et espaces de Berkovich

ENS (amphithéâtre Galois sous la bibliothèque de mathématique)

Étant donné un nombre premier p, nous expliquerons tout d’abord comment construire une valeur absolue sur Q, dite p-adique, puis un complété, de la même façon que l’on construit R à partir de Q. Ce complété, le corps des nombres p-adiques, possède des propriétés arithmétiques intéressantes, mais présente de nombreuses pathologies topologiques. Nous expliquerons comment y remédier en le plongeant dans un espace plus grand, un espace de Berkovich, et exposerons quelques applications.

Computability, orders, and groups

Sophie Germain, salle 2015

Orderable groups are extensively studied by logicians and group theorists. In my talk I will address aspects of left- or bi-orderable groups that are connected with computability theory. In particular, I will talk about constructions of bi-orderable computable groups that cannot be embedded into groups with computable bi-order. I will also discuss our recent work in progress with M. Steenbock about simplicity and computably left-orderability.