Yau reported on work from the past three years focusing on the interactions among the categorical, analytic and metric aspects of mirror symmetry. Major progress here incudes new, precise understanding of the differential geometry of SYZ fibrations of a Calabi-Yau manifold, its connections to Bridgeland stability, and consequences for the quantum Weil-Petersen geometry on the moduli space for the B-model. On the categorical side, Yau described explicit computations of Floer-theoretic deformations of singular torus fibers and immersed Lagrangians in symplectic manifolds and the homological construction of rigid analytic neighborhoods of mirrors. He also discussed a newly discovered class of Calabi-Yau mirrors arising from fractional complete intersections which exhibit unexpected enumerative and arithmetic features. The new features suggest remarkable connections between mirror symmetry and other enumerative dualities, and Yau described several results establishing these connections in dimensions 2 and 3. In the last part of his talk, Yau outlined several important open problems for the HMS/SYZ program going forward.
Kontsevich and Soibelman presented a new, streamlined definition and construction of the Fukaya category based on the concept of a singular Lagrangian skeleton. A major advance reported by Kontsevich is a solution of the big outstanding problem of defining quantum corrections for skeleta. The solution uses tools from Symplectic Field Theory and builds on deep work in contact and symplectic topology by Ekholm-Lekili and Ganatra. It provides a purely algebraic formalism of Fukaya categories. The new definition has far-reaching ramifications and leads to unexpected insights into the structure of symplectic invariants. In particular, it gives new existence predictions for Bridgeland stabilities on Fukaya categories for symplectic manifolds with trivial first Chern class and provides a concrete description of interactions between coisotropic A-branes. A fascinating outcome is a nonlinear version of the Riemann-Hilbert correspondence which identifies coherent sheaves on a quantized complex symplectic manifold and Fukaya objects with singular Lagrangian supports. Soibelman discussed this new theory in the special case of symplectic surfaces and made a connection to the classical Riemann-Hilbert correspondence for differential, q-difference and elliptic difference equations, to non-abelian Hodge theory in dimension one and to periodic monopoles.
Ganatra presented a series of new structural results for partially wrapped Fukaya categories, and their role in the recent proof of an equivalence between the partially wrapped Fukaya category of a cotangent bundle stopped at any subanalytic singular isotropic Λ and a category of “large” sheaves on the zero section microsupported along the same Λ. The proof is a formal consequence of the structural results and ties up with a powerful novel axiomatic characterization for general wrapped Fukaya categories, which is currently being developed by Ganatra, Pardon and Shende.
Iritani described a remarkable progress in understanding the birational geometry of noncommutative Hodge theoretic invariants and specifically of the Gamma-integral structures of Katzarkov-Kontsevich-Pantev. He discussed the change of quantum cohomology of toric orbifolds under ips and explained how the Gamma-integral structure of quantum cohomology are functorially related under toric blow-ups. These results fit in a program jointly developed by Abouzaid, Ganatra, Iritani and Sheridan, which proposes a new method for computing asymptotics of periods using tropical geometry. In this framework, Riemann zeta values appear naturally as error terms in tropicalization. The method also explains how the Gamma class should arise from the SYZ conjecture and gives a new proof of the Gamma conjecture for Batyrev pairs of mirror Calabi-Yau hypersurfaces.
Abouzaid, Fukaya and Tonkonog presented important technical advances in Floer theory that are essential for the algebraic construction of mirrors. Fukaya described the relations of the various constructions of Floer homology and focused on the remaining technical problems for understanding the requisite bounding cochains and deformation theory for tropical or analytic disk counting in non-compact manifolds with a normal crossing boundary. Tonkonog reported on deep results relating the ad hoc SYZ gluing constructions of Landau-Ginzburg mirrors of Fano varieties and new, direct constructions in Floer theory. He also discussed the period integral formula for quantum periods of Fanos and the quantum Lefschetz formula. Abouzaid gave a comprehensive survey of the newly developed local-to-global proofs of mirror symmetry. He reviewed the progress made in the proof of mirror equivalences via families of Lagrangians, including the proof of the equivalence for local models of the degenerations studied by Gross and Siebert. He also formulated the precise challenges in the remaining steps for the proof of mirror symmetry in the case of Calabi-Yau families admitting toric degenerations and explained how many of these challenges can be solved by using his recent joint results with Ganatra and Sylvan as well as recent work of Groman and Varolgunes.