7 avril 2021

Mickaël Latocca (ENS)
Probabilistic Well-Posedness for the Schrödinger equation posed for the Gruchin Laplacian

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Lieu : https://bbb.imo.universite-paris-saclay.fr/b/nic-jog-f24-z6j

Résumé : In the first part of the talk I will introduce the random initial data which we will consider.
Then I will explain why randomisation helps to lower the well-posedness threshold : this is a general argument in study of dispersive equations with random data. Then I will explain how bilinear random estimates relate to our probabilistic well-posedness quest, which finally, if time permits, we will prove.
This talk is based on a recent joint work with Louise Gassot

Probabilistic Well-Posedness for the Schrödinger equation posed for the Gruchin Laplacian  Version PDF

Tony Yue Yu (Orsay)
Frobenius structure conjecture and application to cluster algebras

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Lieu : Demander le lien Zoom à jean-michel.bismut@universite-paris-saclay.fr

Résumé : I will explain the Frobenius structure conjecture of Gross-Hacking-Keel in mirror symmetry, and an application towards cluster algebras. Let U be an affine log Calabi-Yau variety containing an open algebraic torus. We show that the naive counts of rational curves in U uniquely determine a commutative associative algebra equipped with a compatible multilinear form. Although the statement of the theorem involves only elementary algebraic geometry, the proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. I will explain various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, our algebra generalizes, and gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. The comparison is proved via a canonical scattering diagram defined by counting infinitesimal non-archimedean analytic cylinders, without using the Kontsevich-Soibelman algorithm. Several combinatorial conjectures of GHKK, as well as the positivity in the Laurent phenomenon, follow readily from the geometric description. This is joint work with S. Keel, arXiv:1908.09861. If time permits, I will mention another application towards the moduli space of KSBA (Kollár-Shepherd-Barron-Alexeev) stable pairs, joint with P. Hacking and S. Keel, arXiv : 2008.02299.

Frobenius structure conjecture and application to cluster algebras  Version PDF