BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Tony Yue Yu (Université Paris-Sud\, Paris-Saclay) DTSTART:20210217T190000Z DTEND:20210217T200000Z DTSTAMP:20260414T173525Z UID:BUGeom/16 DESCRIPTION:Title: Frobenius structure conjecture and application to cluster algebras\nby Tony Yue Yu (Université Paris-Sud\, Paris-Saclay) as part of Boston Univ ersity Geometry/Physics Seminar\n\nLecture held in Zoom meeting ID: 974 56 41 9902.\n\nAbstract\nI will explain the Frobenius structure conjecture of Gross-Hacking-Keel in mirror symmetry\, and an application towards cluste r algebras. Let U be an affine log Calabi-Yau variety containing an open a lgebraic torus. We show that the naive counts of rational curves in U uniq uely determine a commutative associative algebra equipped with a compatibl e multilinear form. Although the statement of the theorem involves only el ementary 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 wil l explain various properties of the counting\, notably deformation invaria nce\, symmetry\, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety\, our algebra general izes\, and gives a direct geometric construction of\, the mirror algebra o f Gross-Hacking-Keel-Kontsevich. The comparison is proved via a canonical scattering diagram defined by counting infinitesimal non-archimedean analy tic cylinders\, without using the Kontsevich-Soibelman algorithm. Several combinatorial conjectures of GHKK\, as well as the positivity in the Laure nt phenomenon\, follow readily from the geometric description. This is joi nt work with S. Keel\, arXiv:1908.09861. If time permits\, I will mention another application towards the moduli space of KSBA (Kollár-Shepherd-Bar ron-Alexeev) stable pairs\, joint with P. Hacking and S. Keel\, arXiv: 200 8.02299.\n LOCATION:https://researchseminars.org/talk/BUGeom/16/ END:VEVENT END:VCALENDAR