Periodic Reporting for period 4 - HMS (Homological mirror symmetry, Hodge theory, and symplectic topology)
Période du rapport: 2024-09-01 au 2025-08-31
The Project is built around a deep and mysterious connection between two very different areas of mathematics: symplectic geometry and algebraic geometry.
Geometry is the study of spaces equipped with additional structure: for example, 'Riemannian' geometry is about spaces in which you can measure lengths and angles. 'Symplectic' geometry is about spaces in which you can measure the area of two-dimensional surfaces. It has its roots in celestial mechanics. In the 1980s, it was discovered that it is equally relevant to modern particle physics: given a symplectic space, one can define a set of mathematical rules for particles to follow (let's call them 'symplectic laws of physics').
'Algebraic' geometry is about spaces which are defined by algebraic equations. It has its roots in the work of the ancient Greeks on conic sections. It turns out that you can also define 'algebraic laws of physics'.
In the 1990s some physicists discovered a phenomenon called 'mirror symmetry'. They observed that there are lots of pairs of spaces X and Y, where X is symplectic and Y is algebraic, such that the rules of physics defined by X and Y are completely equivalent.
When physicists used mirror symmetry to solve a mathematical problem that had previously been intractable to mathematicians, mathematicians started to pay attention, and set about trying to understand mirror symmetry in mathematical terms. The problem the physicists solved was to count the number of "curves" on X. Since mirror symmetry says that the symplectic laws on X are equivalent to the algebraic laws on Y, the number of curves on X should be equal to the number of a certain other type of object on Y, which they knew how to count using algebraic geometry. This argument was predicated on the validity of mirror symmetry, so needed to be verified mathematically, which took several years.
Decades later, mathematicians have made huge strides in understanding mirror symmetry, but we're still nowhere near understanding it completely. We've realized that it reflects a much deeper relationship between the two types of geometry than was originally understood, but there is no single definitive 'Mirror Symmetry Conjecture', but rather a collection of related ideas. Some of the key contributions of the project have been to develop new aspects and applications of mirror symmetry.
The other focus of the Project has to do with proving precise predictions of mirror symmetry, and establishing how they are related to each other. One of the most important predictions of mirror symmetry is Kontsevich's Homological Mirror Symmetry (HMS) conjecture, which predicts that if X and Y are mirror, then the Fukaya category of X is equivalent to the derived category of Y. A key achievement of the project has been to prove this conjecture for the broadest class of examples to date (known as Batyrev mirror pairs), in the process laying the technical foundations for an even more general proof.
The mathematical structures involved in HMS ('categories') are quite abstract, but powerful. We have established that HMS implies various other predictions of mirror symmetry, such as the validity of the physicsts' approach to counting curves in X via algebraic computations on Y, which started the mathematical interest in mirror symmetry, and which I described above. Establishing these results shows that HMS is the 'key' to unlocking much of the power of mirror symmetry.
Geometry is the study of spaces equipped with additional structure: for example, 'Riemannian' geometry is about spaces in which you can measure lengths and angles. 'Symplectic' geometry is about spaces in which you can measure the area of two-dimensional surfaces. It has its roots in celestial mechanics. In the 1980s, it was discovered that it is equally relevant to modern particle physics: given a symplectic space, one can define a set of mathematical rules for particles to follow (let's call them 'symplectic laws of physics').
'Algebraic' geometry is about spaces which are defined by algebraic equations. It has its roots in the work of the ancient Greeks on conic sections. It turns out that you can also define 'algebraic laws of physics'.
In the 1990s some physicists discovered a phenomenon called 'mirror symmetry'. They observed that there are lots of pairs of spaces X and Y, where X is symplectic and Y is algebraic, such that the rules of physics defined by X and Y are completely equivalent.
When physicists used mirror symmetry to solve a mathematical problem that had previously been intractable to mathematicians, mathematicians started to pay attention, and set about trying to understand mirror symmetry in mathematical terms. The problem the physicists solved was to count the number of "curves" on X. Since mirror symmetry says that the symplectic laws on X are equivalent to the algebraic laws on Y, the number of curves on X should be equal to the number of a certain other type of object on Y, which they knew how to count using algebraic geometry. This argument was predicated on the validity of mirror symmetry, so needed to be verified mathematically, which took several years.
Decades later, mathematicians have made huge strides in understanding mirror symmetry, but we're still nowhere near understanding it completely. We've realized that it reflects a much deeper relationship between the two types of geometry than was originally understood, but there is no single definitive 'Mirror Symmetry Conjecture', but rather a collection of related ideas. Some of the key contributions of the project have been to develop new aspects and applications of mirror symmetry.
The other focus of the Project has to do with proving precise predictions of mirror symmetry, and establishing how they are related to each other. One of the most important predictions of mirror symmetry is Kontsevich's Homological Mirror Symmetry (HMS) conjecture, which predicts that if X and Y are mirror, then the Fukaya category of X is equivalent to the derived category of Y. A key achievement of the project has been to prove this conjecture for the broadest class of examples to date (known as Batyrev mirror pairs), in the process laying the technical foundations for an even more general proof.
The mathematical structures involved in HMS ('categories') are quite abstract, but powerful. We have established that HMS implies various other predictions of mirror symmetry, such as the validity of the physicsts' approach to counting curves in X via algebraic computations on Y, which started the mathematical interest in mirror symmetry, and which I described above. Establishing these results shows that HMS is the 'key' to unlocking much of the power of mirror symmetry.
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
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