BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Tony Yue Yu (Université Paris-Sud) DTSTART:20210303T130000Z DTEND:20210303T140000Z DTSTAMP:20260423T004135Z UID:Rega/10 DESCRIPTION:Title: Fr obenius structure conjecture and application to cluster algebras\nby T ony Yue Yu (Université Paris-Sud) as part of RéGA (Réseau des étudiant s en Géométrie Algébrique)\n\n\nAbstract\nI will explain the Frobenius structure conjecture of Gross-Hacking-Keel in mirror symmetry\, and an app lication towards cluster algebras. Let $U$ be an affine log Calabi-Yau var iety containing an open algebraic torus. We show that the naive counts of rational curves in $U$ uniquely determine a commutative associative algebr a equipped with a compatible multilinear form. Although the statement of t he theorem involves only elementary algebraic geometry\, the proof employs Berkovich non-archimedean analytic methods. We construct the structure co nstants 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 $X$-clus ter variety\, our algebra generalizes\, and gives a direct geometric const ruction of\, the mirror algebra of Gross-Hacking-Keel-Kontsevich. The comp arison is proved via a canonical scattering diagram defined by counting in finitesimal non-archimedean analytic cylinders\, without using the Kontsev ich-Soibelman algorithm. Several combinatorial conjectures of GHKK\, as we ll as the positivity in the Laurent phenomenon\, follow readily from the g eometric description. This is joint work with S. Keel\, arXiv:1908.09861. If time permits\, I will mention another application towards the moduli sp ace of KSBA (Kollár-Shepherd-Barron-Alexeev) stable pairs\, joint with P. Hacking and S. Keel\, arXiv: 2008.02299.\n LOCATION:https://researchseminars.org/talk/Rega/10/ END:VEVENT END:VCALENDAR