Journée du 30/03/2021 [avec SLIDES et VIDÉOS]

Bon ton du Séminaire différentiel en ligne : merci de vous connecter en utilisant vos nom et prénom réels et complets. À défaut de voir le visage des personnes qui suivent l'exposé, nous pensons que l'orateur doit au moins savoir qui sont les collègues connectés. 

Organisateurs : Alin Bostan et Lucia Di Vizio

Programme

10h45 : accueil

11h00-12h00 : Masha Vlasenko (Institute of Mathematics of the Polish Academy of Sciences, Varsovie, Pologne) 

Titre  : Integrality of instanton numbers [AFFICHER LES SLIDES]  

{vimeo}538429131{/vimeo}

Résumé :  Instanton numbers of Calabi--Yau threefolds are defined by Gromov--Witten theory. They 'count' curves of fixed degree on the manifold. The actual definition involves integration over the moduli space of curves, which gives a priori rational numbers. Integrality of instanton numbers is an arithmetic counterpart of the mirror symmetry conjecture. Mirror theorem allows to express them in terms of solutions of a differential equation on the dual manifold. However, the integrality of instanton numbers is not clear from this expression. In 2002 Jan Stienstra outlined an approach to integrality using the p-adic Frobenius structure on the differential equation. In this talk we will explain an explicit and rather elementary construction of the Frobenius structure, which allows us to prove integrality of instanton numbers in some key examples of mirror symmetry. This is joint work with Frits Beukers.

12h15-14h15  : déjeuner  

14h30-15h30 : Duco van Straten (Johannes Gutenberg-Universität Mainz, Mayence, Allemagne)

{vimeo}541144658{/vimeo}

Titre : Congruences, Kernels and Calabi-Yau operators

Résumé : In the talk I will report on older work and work in progress with V. Golyshev, A. Mellit and V. Roubtsov on multiplication kernels for differential operators and a classification approach for arithmetic differential operators of Calabi-Yau type.

15h45-16h45 : Jason Bell (University of Waterloo, Ontario, Canada)

Titre : A height gap theorem for coefficients of Mahler functions [AFFICHER LES SLIDES] 

{vimeo}541215421{/vimeo}

Résumé :  Given a positive integer $k$, a power series $F(z)$ is said to be $k$-Mahler if it satisfies a non-trivial linear difference equation of the form
$$\sum_{i=0}^d p_i(z) F(z^{k^i}) = 0,$$ with $p_0,\ldots ,p_d$ polynomials. We consider $k$-Mahler series with coefficients in the field of algebraic numbers. The most fundamental way of understanding the complexity of algebraic numbers is arguably via their (logarithmic Weil) height, and we consider the asymptotic behaviour of the heights of coefficients of Mahler series, and show that their behaviour can be completely understood in terms and that all series fall into certain classes. In particular, we show that there are five different growth behaviours which can occur (with each of them occurring). We give an overview of the five classes and characterizations of when Mahler series fall into various classes. This is joint work with Boris Adamczewski and Daniel Smertnig.