Periodic Reporting for period 2 - HMS (Homological mirror symmetry, Hodge theory, and symplectic topology)
Okres sprawozdawczy: 2021-09-01 do 2023-02-28
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, and 'conformal' geometry is about spaces where you can only measure angles. 'Symplectic' geometry is about spaces in which you can measure the area of two-dimensional surfaces. It has its roots in mechanics, where it can be used to study the properties of systems such as a collection of planets orbiting a star. 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 completely baffling 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. In other words, if you lived inside space X, and were made of particles following the symplectic laws of physics, it would feel identical to living in space Y, and being made of particles following the algebraic laws of physics!
At first sight mathematicians dismissed it as being too bizarre to be true. But when physicists used it to solve a problem that had previously been intractable to mathematicians, the mathematical world was convinced, and set about trying to understand it in mathematical language. 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. Indeed one of the key objectives of the Project is to develop yet another aspect of mirror symmetry, which relates the subject of algebraic cycles (which have been intensively studied for many years, but about which many interesting questions remain open, for example the Hodge Conjecture) to the much more recent subject of Lagrangian cobordism groups.
The other objectives of the Project have 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 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 objective of the Project is to prove this conjecture in a broad class of examples.
The mathematical structures involved in Homological Mirror Symmetry ('categories') are quite abstract, but powerful. For example, the other objectives of the project are to show that Homological Mirror Symmetry 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 would show that Homological Mirror Symmetry 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, and 'conformal' geometry is about spaces where you can only measure angles. 'Symplectic' geometry is about spaces in which you can measure the area of two-dimensional surfaces. It has its roots in mechanics, where it can be used to study the properties of systems such as a collection of planets orbiting a star. 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 completely baffling 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. In other words, if you lived inside space X, and were made of particles following the symplectic laws of physics, it would feel identical to living in space Y, and being made of particles following the algebraic laws of physics!
At first sight mathematicians dismissed it as being too bizarre to be true. But when physicists used it to solve a problem that had previously been intractable to mathematicians, the mathematical world was convinced, and set about trying to understand it in mathematical language. 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. Indeed one of the key objectives of the Project is to develop yet another aspect of mirror symmetry, which relates the subject of algebraic cycles (which have been intensively studied for many years, but about which many interesting questions remain open, for example the Hodge Conjecture) to the much more recent subject of Lagrangian cobordism groups.
The other objectives of the Project have 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 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 objective of the Project is to prove this conjecture in a broad class of examples.
The mathematical structures involved in Homological Mirror Symmetry ('categories') are quite abstract, but powerful. For example, the other objectives of the project are to show that Homological Mirror Symmetry 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 would show that Homological Mirror Symmetry is the 'key' to unlocking much of the power of mirror symmetry.
I have developed a detailed plan for proving Homological Mirror Symmetry for a broad class of spaces; namely, the Gross-Siebert (toric degenerative) mirror pairs. 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.
I have carried out a key step in this plan, in joint work with Strom Borman and Umut Varolgunes (the latter being a postdoc funded by the grant); 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.
Varolgunes has also made important progress on a different approach to proving Homological Mirror Symmetry; we are studying the relationship between our approaches.
My PhD student, Kai Hugtenburg, has proved that Homological Mirror Symmetry implies another version of mirror symmetry, namely Hodge-theoretic mirror symmetry, in the context of Fano manifolds (under some restrictive hypotheses). This result allows one to deduce results about counting curves from Homological Mirror Symmetry.
I have made significant progress on a general framework for defining symplectic invariants, in the presence of a divisor D, which will be used to broaden the existing results (such as Hugtenburg's) which allow one to use Homological Mirror Symmetry to `count' curves of different types.
My postdoc, Filip 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.
Another key component of the project is to use mirror symmetry to study a fundamental object attached to any symplectic space, called the cobordism group. My postdoc, Jeff Hicks, has used cobordisms to prove new results relevant to mirror symmetry, including about the phenomena of 'wall-crossing', 'realizability', and 'displaceability'.
Another of my PhD students, Alvaro Muniz Brea, has discovered new structures in the cobordism group which were predicted to exist using mirror symmetry.
I have carried out a key step in this plan, in joint work with Strom Borman and Umut Varolgunes (the latter being a postdoc funded by the grant); 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.
Varolgunes has also made important progress on a different approach to proving Homological Mirror Symmetry; we are studying the relationship between our approaches.
My PhD student, Kai Hugtenburg, has proved that Homological Mirror Symmetry implies another version of mirror symmetry, namely Hodge-theoretic mirror symmetry, in the context of Fano manifolds (under some restrictive hypotheses). This result allows one to deduce results about counting curves from Homological Mirror Symmetry.
I have made significant progress on a general framework for defining symplectic invariants, in the presence of a divisor D, which will be used to broaden the existing results (such as Hugtenburg's) which allow one to use Homological Mirror Symmetry to `count' curves of different types.
My postdoc, Filip 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.
Another key component of the project is to use mirror symmetry to study a fundamental object attached to any symplectic space, called the cobordism group. My postdoc, Jeff Hicks, has used cobordisms to prove new results relevant to mirror symmetry, including about the phenomena of 'wall-crossing', 'realizability', and 'displaceability'.
Another of my PhD students, Alvaro Muniz Brea, has discovered new structures in the cobordism group which were predicted to exist using mirror symmetry.
By the end of the project, I expect to have proved Homological Mirror Symmetry for the above-mentioned Gross-Siebert mirror pairs; or at least, for the large sub-class of Batyrev mirror pairs. Either way, this will be by far the broadest class of examples for which Homological Mirror Symmetry has been established.
I also expect to have proved that Homological Mirror Symmetry implies certain other predictions of mirror symmetry: namely, a version of `enumerative' mirror symmetry, and the so-called `Gamma conjecture'.
Enumerative mirror symmetry allows one to count curves on one space by performing a computation on the mirror space; physicists' predictions in this direction were the original inspiration for mathematicians to try to understand mirror symmetry. I am working to show that Homological Mirror Symmetry implies a very general version of enumerative mirror symmetry, which allows one to count more complicated kinds of curves than in the physicsists' original predictions.
The Gamma conjecture is a prediction which allows one to compute `periods' of an algebraic space (which are, roughly speaking, volumes of the spaces sitting inside it) in terms of computations on the mirror symplectic space. Periods are of central interest in other areas of mathematics, such as number theory, and the Gamma conjecture provides an unexpected angle from which to study them.
Since the first part of the project aims to establish Homological Mirror Symmetry in many examples, and the second part aims to show that it implies certain other predictions of mirror symmetry, when we put the two parts together we establish the validity of the latter predictions.
Finally, I expect to have made progress developing the mirror dictionary between algebraic cycles and Lagrangian cobordism groups, and using it to elucidate the structure of the latter.
I also expect to have proved that Homological Mirror Symmetry implies certain other predictions of mirror symmetry: namely, a version of `enumerative' mirror symmetry, and the so-called `Gamma conjecture'.
Enumerative mirror symmetry allows one to count curves on one space by performing a computation on the mirror space; physicists' predictions in this direction were the original inspiration for mathematicians to try to understand mirror symmetry. I am working to show that Homological Mirror Symmetry implies a very general version of enumerative mirror symmetry, which allows one to count more complicated kinds of curves than in the physicsists' original predictions.
The Gamma conjecture is a prediction which allows one to compute `periods' of an algebraic space (which are, roughly speaking, volumes of the spaces sitting inside it) in terms of computations on the mirror symplectic space. Periods are of central interest in other areas of mathematics, such as number theory, and the Gamma conjecture provides an unexpected angle from which to study them.
Since the first part of the project aims to establish Homological Mirror Symmetry in many examples, and the second part aims to show that it implies certain other predictions of mirror symmetry, when we put the two parts together we establish the validity of the latter predictions.
Finally, I expect to have made progress developing the mirror dictionary between algebraic cycles and Lagrangian cobordism groups, and using it to elucidate the structure of the latter.