I will explain how tame topology seems the natural setting for variational Hodge theory. As an instance I will sketch a new proof of the algebraicity of the components of the Hodge locus, a celebrated result of Cattani-Deligne-Kaplan (joint work with Bakker and Tsimerman).
Let k be an algebraically closed complete rank 1 non-trivially valued field. Let X be an algebraic curve over k and let X^an be its analytification in the sense of Berkovich. We functorially associate to X^an a definable set X^S in a natural language. As a corollary, we obtain an alternative proof of a result of Hrushovski-Loeser about the iso-definability of curves. Our association being explicit allows us to provide a concrete description of the definable subsets of X^S: they correspond to radial sets. This is a joint work with […]
There is a well-known analogy between the arithmetic of rational numbers and the theory of meromorphic functions over a normed field. It is a classical result of Julia Robinson that the first order theory of the field of rational numbers is undecidable, and one would expect such a result in the meromorphic setting. In this talk I'll give an outline of the proof of undecidability for rigid meromorphic functions in positive characteristic
Dans cet exposé je discuterai de la conjecture de cristallisation, qui consiste à prouver que des points se mettent automatiquement sur un réseau périodique, dans certaines situations. Je parlerai principalement du cas particulier des gaz de Riesz et de Coulomb qui sont omniprésents dans de nombreuses branches des mathématiques. Ils décrivent par exemple la répartition des zéros de solutions d'équations aux dérivées partielles, la statistique des valeurs propres des matrices aléatoires, le placement optimal de points de discrétisation sur des surfaces, et sont reliés à la fonction Zeta de Epstein.
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 […]
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.
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.
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.
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
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 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.
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
