Singularities and symplectic topology
Lead Research Organisation:
Lancaster University
Department Name: Mathematics and Statistics
Abstract
Singularities are everywhere we look: from the cuspy caustic curve that forms when you shine light into your coffee cup to the microscopic black holes that we expect to form in extra-dimensions in the theory of strings and branes.
The mathematical study of singularities involves algebraic geometry. This is a branch of geometry which involves writing equations for the geometrical spaces of interest. For example, the cusp curve is described by the equation y^2=x^3 in the plane. For algebraic geometers, singularities appear naturally when you try to classify all the possible spaces you can write down via equations (the efforts to carry out this classification go by the name of "the minimal model program").
I am proposing a new way to study singularities. If you deform the equation (for example you study y^2=x^3+t for some constant t) you can sometimes smooth out the singularity, but the smoothed space can have highly nontrivial topology (you can see this happen if you tilt your coffee cup into the light). This new piece of topology, which is crushed back down to the singular point as t goes to zero, is called a vanishing cycle. Let's suppose you want to show that two different singularities cannot form at the same time. I will try to do this by showing that the vanishing cycles cannot be moved apart from one another ("displaced"). If the vanishing cycles are nondisplaceable then one or other singularity can form, but both singularities cannot form at the same time.
The difficulty in making this kind of argument rigorous is that the vanishing cycles can themselves be singular! To prove such results, I will need to develop existing techniques for proving nondisplaceability ("Floer theory") to the situation where the vanishing cycles have singularities. These techniques should then have applications in other parts of mathematics where singular cycles play a role, for example the "singular SYZ fibres" which arise in studying the mysterious geometric duality called Mirror Symmetry (predicted by string theorists in the early 1990s and still not fully understood) or the singular limits of Lagrangian mean curvature flows in geometric analysis.
The mathematical study of singularities involves algebraic geometry. This is a branch of geometry which involves writing equations for the geometrical spaces of interest. For example, the cusp curve is described by the equation y^2=x^3 in the plane. For algebraic geometers, singularities appear naturally when you try to classify all the possible spaces you can write down via equations (the efforts to carry out this classification go by the name of "the minimal model program").
I am proposing a new way to study singularities. If you deform the equation (for example you study y^2=x^3+t for some constant t) you can sometimes smooth out the singularity, but the smoothed space can have highly nontrivial topology (you can see this happen if you tilt your coffee cup into the light). This new piece of topology, which is crushed back down to the singular point as t goes to zero, is called a vanishing cycle. Let's suppose you want to show that two different singularities cannot form at the same time. I will try to do this by showing that the vanishing cycles cannot be moved apart from one another ("displaced"). If the vanishing cycles are nondisplaceable then one or other singularity can form, but both singularities cannot form at the same time.
The difficulty in making this kind of argument rigorous is that the vanishing cycles can themselves be singular! To prove such results, I will need to develop existing techniques for proving nondisplaceability ("Floer theory") to the situation where the vanishing cycles have singularities. These techniques should then have applications in other parts of mathematics where singular cycles play a role, for example the "singular SYZ fibres" which arise in studying the mysterious geometric duality called Mirror Symmetry (predicted by string theorists in the early 1990s and still not fully understood) or the singular limits of Lagrangian mean curvature flows in geometric analysis.
Planned Impact
The principal impact of the proposed research will be academic. It will have a transformative effect on the field, introducing new directions in symplectic topology: exploring links with birational geometry and pioneering new methods in Floer theory. Hopefully it will also have significant impact on the ongoing effort to understand Mirror Symmetry. Of course, the wider long-term impact of foundational research is hard to predict.
There will be an economic impact on the UK's research competitiveness. Symplectic topology is a growth area in pure mathematics, with a lot of young graduate students entering the subject. Europe and the US are effectively capitalising on this growth, but the UK is still lagging behind in terms of number of full-time researchers and postdoctoral positions in the subject.
The recent award of two $3M Breakthrough Prizes to Donaldson (who has worked on Floer theory) and Kontsevich (who made the homological mirror symmetry conjecture) have highlighted modern geometry research in the public eye. This is a great opportunity for those of us in the field to capitalise on the publicity to produce quality expository material for the general public, explaining recent developments in geometry centred around mirror symmetry and Floer theory. Part of the output of this grant will be an outreach website, including videos and blog posts explaining the background to the proposed research and why it is an important direction in current mathematics. I hope this will improve awareness and encourage young people to explore mathematics as a career option; low uptake of mathematics as a degree option is a serious issue for the UK's competitiveness both nationally and internationally.
I intend to move towards an Open Notebook model, making available much of the "unwritten" material produced in the course of research. The server hosting the outreach website will also host auxiliary output of the research (tables of data, short expository notes) which will be useful for many researchers in the field. I hope this will encourage other researchers (not only in mathematics) to adopt an open notebook model. This could have a significant impact on the way we do research.
There will be an economic impact on the UK's research competitiveness. Symplectic topology is a growth area in pure mathematics, with a lot of young graduate students entering the subject. Europe and the US are effectively capitalising on this growth, but the UK is still lagging behind in terms of number of full-time researchers and postdoctoral positions in the subject.
The recent award of two $3M Breakthrough Prizes to Donaldson (who has worked on Floer theory) and Kontsevich (who made the homological mirror symmetry conjecture) have highlighted modern geometry research in the public eye. This is a great opportunity for those of us in the field to capitalise on the publicity to produce quality expository material for the general public, explaining recent developments in geometry centred around mirror symmetry and Floer theory. Part of the output of this grant will be an outreach website, including videos and blog posts explaining the background to the proposed research and why it is an important direction in current mathematics. I hope this will improve awareness and encourage young people to explore mathematics as a career option; low uptake of mathematics as a degree option is a serious issue for the UK's competitiveness both nationally and internationally.
I intend to move towards an Open Notebook model, making available much of the "unwritten" material produced in the course of research. The server hosting the outreach website will also host auxiliary output of the research (tables of data, short expository notes) which will be useful for many researchers in the field. I hope this will encourage other researchers (not only in mathematics) to adopt an open notebook model. This could have a significant impact on the way we do research.
Organisations
Publications
Evans J
(2022)
A Lagrangian Klein bottle you can't squeeze
in Journal of Fixed Point Theory and Applications
Evans J
(2021)
Symplectic cohomology of compound Du Val singularities
Evans J
(2021)
Constructing local models for Lagrangian torus fibrations
in Annales Henri Lebesgue
Evans J
(2022)
Antiflips, mutations, and unbounded symplectic embeddings of rational homology balls
in Annales de l'Institut Fourier
Evans J. D.
(2020)
Bounds on Wahl singularities from symplectic topology
in Algebraic Geometry
Habermann M
(2020)
Homological Berglund-Hübsch mirror symmetry for curve singularities
in Journal of Symplectic Geometry
KONSTANTINOV M
(2020)
Monotone Lagrangians in of minimal Maslov number n + 1
in Mathematical Proceedings of the Cambridge Philosophical Society
Smith J
(2020)
Quantum Cohomology and Closed-String Mirror Symmetry for Toric Varieties
in The Quarterly Journal of Mathematics
Smith J
(2022)
Superfiltered A8$A_\infty$-deformations of the exterior algebra, and local mirror symmetry
in Journal of the London Mathematical Society
| Description | In March 2019, the PI (J. Evans) moved from UCL to Lancaster, bringing his grant EP/P02095X/1 with him. The grant therefore split between two codes (even though it's the same project) EP/P02095X/1 and EP/P02095X/2. The report below is essentially the same as that report. - J. Evans (PI) and I. Smith managed to complete Task C.2 from the proposal by finding an explicit bound on the length of a Wahl singularities which arise on surfaces at the boundary of the KSBA moduli space. This bound was proved using purely symplectic methods (though not the ones I had originally imagined in the proposal!). It also improved enormously on the known bounds from algebraic geometry. The resulting paper has been accepted for publication in the new open access journal "Algebraic Geometry". I specifically wanted to publish in a venue which was both open access and aimed at algebraic (rather than symplectic) geometers, both to maximise impact and to facilitate further knowledge interchange between the areas. - J. Evans (PI) initiated a collaboration with Giancarlo Urzua, an algebraic geometer from Chile, after realising that one of Urzua's papers could have ramifications in symplectic geometry. This was inspired by the bounds proved in Goal C.2 and has led to another paper (UPDATE: now accepted in Annales de l'Institut Fourier, another open access journal) and further collaboration regarding the boundary of the KSBA moduli space. - Following the research visit of P. Hacking (funded by the grant), J. Evans (PI) had an idea of how to solve an old problem about singular Lagrangians which emerged from the work of M. Gross in the 1990s/2000s (namely how to write down an explicit SYZ fibration with a fibre of negative Euler characteristic on a local Calabi-Yau 3-fold). This problem is closely related to Task A.1 in the grant. I discovered that another young algebraic geometer (M. Mauri, Imperial College) was thinking about similar questions, and suggested that we collaborate. This has led to a paper (https://arxiv.org/abs/1905.09229, UPDATE: now accepted in Annales Henri Lebesgue), in which we solve this old problem and construct many interesting examples of SYZ fibrations using a blend of techniques from birational geometry and symplectic geometry. Inspired by conversations with Urzua and some of our joint results, J. Evans started two further projects: - In our joint paper we observed that the "pinwheel content" of a surface of general type (which dictates the kinds of singularities that can develop) depend strongly on the cohomology class of its symplectic form (polarisation). The precise nature of this dependence seems complicated, so I thought first about the simpler case of Klein bottles in Del Pezzo surfaces. This has now resulted in the paper https://arxiv.org/abs/2009.01546 (accepted for publication in Journal of Fixed Point Theory and its Applications). - It is an open question whether the KSBA moduli space of surfaces with K^2 = 1, p_g = 1 is connected or not. If it were connected, it would give an easy proof that the known surfaces with these invariants are diffeomorphic. I proposed to show directly that they are diffeomorphic by computing the vanishing cycles for the singularities of a Lefschetz pencil and comparing. Steady progress has been made on this question (this is ongoing; you can see some progress in my open research notebook). This kind of fruitful interaction with algebraic geometers, leading to pioneering crossover collaborations, was exactly what I had in mind when I wrote in the academic impact statement for my proposal that "one of my life goals is to develop symplectic methods to the point where they are part of every algebraic geometer's toolkit of transcendental techniques for proving useful theorems". Meanwhile, the PDRA, J. Smith, has been delving deep into the technical heart of Lagrangian Floer theory, in an effort to push forward Strand B of the research. This strand was predicated on (amongst other things) getting a deeper understanding of the works of Fukaya-Oh-Ohta-Ono (FOOO), and J. Smith has now written five papers on this subject, which he has submitted to excellent journals. - In "Quantum cohomology and closed-string mirror symmetry for toric varieties", he has used new ideas from algebra to greatly simplify and elucidate FOOO: in one, he establishes the isomorphism between quantum cohomology and the Jacobian ring of the mirror superpotential. - In "Generating the Fukaya categories of compact toric varieties", he proves that the barycentric Lagrangian torus in a toric variety generates the Fukaya category, going beyond what FOOO could prove. - In "Superfiltered A8-deformations of the exterior algebra, and local mirror symmetry", he proves a local version of mirror symmetry (that the subcategory of the Fukaya category split-generated by a monotone Lagrangian torus is quasiequivalent to the category of matrix factorisations of its superpotential). - In "Homological Berglund-Hübsch mirror symmetry for curve singularities", joint with Matt Habermann, they prove mirror symmetry for some singular curves. - In "A monotone Lagrangian casebook", he gives a wide array of useful examples of computations. - J. Smith has also written a paper jointly with M. Konstantinov (a PhD student of J. Evans) in which they solve an old conjecture about Lagrangian submanifolds of CP^n (showing that any monotone Lagrangian with maximal minimal Maslov number is homotopy equivalent to RP^n). These RP^ns arise as vanishing cycles of a certain singular degeneration of CP^n. Understanding degenerations of CP^n by studying Lagrangian vanishing cycles was one of the goals of Task C.3, and this paper represents unexpected progress in the right direction. When all of these papers appear, I will add them to the "outputs" page for this grant. After 2 years, J. Smith got a 5-year position at Johns College Cambridge, and I was able to reopen his post. I hired Daniel Cavey to replace him. Daniel is an algebraic geometer, fresh out of the PhD from Nottingham University. He has been working on the question of mutations for P^1 x P^1 (essentially Task C.1 of the grant) and making good progress. We have also started collaborating on a new joint project for finding new surfaces of general type by taking minimal resolutions of singular surfaces. This uses his expertise in algebraic geometry and computer algebra systems in a crucial way, and is exactly the kind of collaboration between symplectic and algebraic geometry which this grant was supposed to foster. In addition to this, Cavey proved a classification result for Fano toric 3-folds with certain singularities (closely related to Strand A of the project). J. Evans was awarded a 2021 LMS Whitehead Prize for the work accomplished in this grant. |
| Exploitation Route | The ideas used in our paper on Task C.2 are likely to be taken on by algebraic and symplectic geometers and low-dimensional topologists to establish a deeper understanding of the KSBA moduli spaces using transcendental methods. It is a fundamental question in geometry to classify 4-dimensional spaces, and understanding KSBA moduli spaces represents a big step in that direction. The research from strand A (Lagrangian torus fibrations) has led to my joint work with Urzua and Mauri, and also a book on Lagrangian torus fibrations which is now complete and has been submitted for publication in the LMS Student Texts series (full text can be found here: https://arxiv.org/abs/2110.08643) |
| Sectors | Other |
| Description | The insights into Lagrangian torus fibrations which I developed over the course of this project have been set out in a book which has been accepted for publication by Cambridge University Press, as part of their LMS Student Texts series. This is aimed at strong undergraduate students (so within academia, but below the research level), giving them some intuition about 4-dimensional geometry from 2-dimensional pictures. |
| Sector | Other |
| Impact Types | Cultural |