Frobenius structure conjecture and application to cluster algebras

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.