All of the results of the project described below have been widely disseminated through preprints, publications, and conference and seminar talks.
Building on ideas of Seidel, I have developed a detailed plan for proving HMS for a broad class of spaces. A key part of the plan is to endow the candidate mirror pair (X,Y) with additional structure: X gets equipped with a 'divisor' D, and Y gets equipped with a 'maximal degeneration'. The idea is to start by showing that if we remove D from X, then the result is mirror to the degenerate version of Y; then study what happens when we plug the divisor back in on the symplectic side, and 'reverse the degeneration' on Y.
With Borman, El Alami* and Varolgunes*, we showed that the symplectic invariants of X are a deformation of those when D is removed, which precisely mirrors the behaviour of the algebraic invariants of Y. These ideas have influenced McLean in his proof of birational invariance of quantum cohomology, Pomerleano and Seidel in their proof of the exponential type conjecture, and others. Varolgunes* has also made important progress on a different approach to proving HMS; we are studying the relationship between the approaches.
With Ganatra, Hanlon, Hicks*, and Pomerleano, we implemented the approach to prove HMS for the broadest class of mirror pairs to date (Batyrev pairs). We used this result to prove the strongest result to date concerning 'integrality of mirror maps', a question in number theory. Kozevnikov* has established the first step towards generalizing this to the next-largest class of examples which is within reach (Batyrev-Borisov pairs). Our results and the approach used are established as the state of the art; e.g. the approach has been taken up by Hacking and Keating, who used it to attack another important class of examples (K3 surfaces).
With Di Dedda*, Gugiatti*, and Kozevnikov*, we are in the process of adapting this approach to prove HMS for Kuznetsov components of Fano varieties, allowing applications to birational geometry.
Hugtenburg* has proved that HMS implies Hodge-theoretic mirror symmetry, in the context of Fano manifolds (under some restrictive hypotheses). It is complemented by my work with Ganatra and Perutz, establishing this implication much more broadly. These results allow one to deduce results about counting curves from HMS, and represents the state of the art in this direction. Hugtenburg also proved similar results for counting curves with boundary, and this has been taken up by Haney, who combined it with our proofs of HMS to give new computations of curve counts which were previously inaccessible.
Kartal* has used ideas imported via mirror symmetry to study dynamics in symplectic topology, and has also laid technical foundations for spectral refinements of mirror symmetry, allowing applications to arithmetic geometry.
Zivanovic* has proved new results about the symplectic geometry of spaces arising in another area of maths, geometric representation theory, paving the way for applications of mirror symmetry in that new direction.
Hanlon, Hicks*, and Lazarev used ideas from mirror symmetry to establish an important conjecture (Orlov's conjecture) in algebraic geometry; their results are the strongest to date on Orlov's conjecture, and there was a workshop at the American Institute of Mathematics devoted to disseminating their ideas to the community.
Hicks* has also made diverse contributions to the study of Lagrangian cobordism groups via mirror symmetry, including using them to study the phenomena of 'wall-crossing', 'realizability', and 'displaceability'; with Smith, I showed that cobordism groups are typically too complicated to compute explicitly; while Muniz Brea* has discovered new structures in the cobordism group via mirror symmetry, and given the most sophisticated analysis of a specific cobordism group to date.
* = funded by the grant
