Singularities, symplectic topology and mirror symmetry
Lead Research Organisation:
University of Cambridge
Department Name: Pure Maths and Mathematical Statistics
Abstract
Symplectic geometry is a rapidly developing field, with tools drawn from many different areas of mathematics. Modern geometry studies manifolds, smooth objects that at small enough scale look like the standard space of a fixed dimension. For instance, the surface of a ball is a 2D-manifold, standard space-time is a 4D-manifold, and the parameter space for a biological experiment might be an 18D-manifold. Symplectic manifolds are equipped with an extra structure that generalises conservation laws from classical mechanics. This makes them the natural formal framework for studying orbits of satellites or space probes. Also, some models in string theory, a branch of physics, allow any symplectic manifold in lieu of space-time. Duality ideas in physics have led to mirror-symmetry, a booming field that relates symplectic geometry with a very different looking part of mathematics: algebraic geometry, which studies solutions of polynomial equations in several variables.
This project is guided by the major open question: `What are the transformations (that is, global symmetries) of a symplectic manifold?' By transformation, we mean a rule for taking each point to another, which is smooth (no breaks), invertible (you can go backwards), and preserves the additional symmetries.
We don't understand symplectic transformations well: for a lot of spaces, the one real source is something called Dehn twists. Let me describe these for 2D surfaces. (2D surfaces are symplectic if they have orientations: the surface of a ball or of an inner tube does, a Mobius strip does not.) Start with a closed curve without self-intersections - for instance, a circle around the thin part of an inner tube. Cut the surface open along it: the inner tube is now a long annulus, with two boundary components, each a circle. Twist each of the boundaries to the right by 180 degrees and glue the edges together again. You have got the same surface back! This transformation is a Dehn twist. Circles on surfaces are 1D-spheres, and in general, we can define Dehn twists analogously in higher dimensions, by using higher dimensional spheres inside symplectic manifolds - for instance, copies of the usual sphere (the surface of a ball) in four-dimensional symplectic manifolds.
In 2D, all transformations can be decomposed into sequences of twists. A major goal of the project is to show that the higher-dimensional situation can be radically different, by constructing large families of new examples of transformations, inspired by mirror symmetry. These translate to a different sort of transformation in the world of algebraic geometry, where we propose to settle questions of independent interest.
A long-term goal is to compare dynamical properties of transformations of surfaces with the ones in higher dimensions. For instance, Dehn twists on surfaces have linear dynamics: the number of fixed points grows linearly with iteration. However, a generic surface transformation, called a pseudo-Anosov map, has exponential dynamics. For large families of examples, we will study the possible growth-rates of fixed points of transformations, and whether there is a generic behaviour.
Many of the objects that will be studied in the project arise naturally in singularity theory, a field tied to the parts of mathematics that explain discontinuities and abrupt changes - for instance, the cuspy caustic curve that appears when light shines through water. We also propose to use ideas from symplectic geometry to study classical structural questions about spaces of deformations of generalised caustics.
Lots of other geometric structures enter the project too: for instance, braid groups, which are mathematical formalisations of the braids you can make with hair or ribbons; and Coxeter groups, which are transformations of space generalising the ones you can obtain from reflections in configurations of (physical, light-reflecting) mirrors.
This project is guided by the major open question: `What are the transformations (that is, global symmetries) of a symplectic manifold?' By transformation, we mean a rule for taking each point to another, which is smooth (no breaks), invertible (you can go backwards), and preserves the additional symmetries.
We don't understand symplectic transformations well: for a lot of spaces, the one real source is something called Dehn twists. Let me describe these for 2D surfaces. (2D surfaces are symplectic if they have orientations: the surface of a ball or of an inner tube does, a Mobius strip does not.) Start with a closed curve without self-intersections - for instance, a circle around the thin part of an inner tube. Cut the surface open along it: the inner tube is now a long annulus, with two boundary components, each a circle. Twist each of the boundaries to the right by 180 degrees and glue the edges together again. You have got the same surface back! This transformation is a Dehn twist. Circles on surfaces are 1D-spheres, and in general, we can define Dehn twists analogously in higher dimensions, by using higher dimensional spheres inside symplectic manifolds - for instance, copies of the usual sphere (the surface of a ball) in four-dimensional symplectic manifolds.
In 2D, all transformations can be decomposed into sequences of twists. A major goal of the project is to show that the higher-dimensional situation can be radically different, by constructing large families of new examples of transformations, inspired by mirror symmetry. These translate to a different sort of transformation in the world of algebraic geometry, where we propose to settle questions of independent interest.
A long-term goal is to compare dynamical properties of transformations of surfaces with the ones in higher dimensions. For instance, Dehn twists on surfaces have linear dynamics: the number of fixed points grows linearly with iteration. However, a generic surface transformation, called a pseudo-Anosov map, has exponential dynamics. For large families of examples, we will study the possible growth-rates of fixed points of transformations, and whether there is a generic behaviour.
Many of the objects that will be studied in the project arise naturally in singularity theory, a field tied to the parts of mathematics that explain discontinuities and abrupt changes - for instance, the cuspy caustic curve that appears when light shines through water. We also propose to use ideas from symplectic geometry to study classical structural questions about spaces of deformations of generalised caustics.
Lots of other geometric structures enter the project too: for instance, braid groups, which are mathematical formalisations of the braids you can make with hair or ribbons; and Coxeter groups, which are transformations of space generalising the ones you can obtain from reflections in configurations of (physical, light-reflecting) mirrors.
Description | FSMP Distinguishe Professor visitor funding |
Amount | € 9,200 (EUR) |
Organisation | Paris Mathematical Sciences Foundation |
Sector | Charity/Non Profit |
Country | France |
Start | 02/2023 |
End | 04/2023 |
Description | Research Membership for Fall 2022 "Floer homotopy theory" program |
Amount | $16,000 (USD) |
Organisation | Mathematical Sciences Research Institute |
Sector | Charity/Non Profit |
Country | United States |
Start | 08/2022 |
End | 12/2023 |
Description | Hacking |
Organisation | University of Massachusetts Amherst |
Country | United States |
Sector | Academic/University |
PI Contribution | This is a reseach collaborator with Prof. Paul Hacking at UMass Amherst. He was a named proposed collaborator in the EPSRC proposal. We are collaborating on parts of Objective A from the research proposal, including some exciting developments which go beyond what was hoped for in Objective A. Loosely, the project is in the area of homological mirror symmetry; this requires technical expertise in two different areas of geometry & topology: symplectic topology and algebraic geometry. I contribute the expertise in symplectic geometry (and we work jointly at the interface). |
Collaborator Contribution | Prof. Hacking contributes the expertise in algebraic geometry (and we work jointly at the mirror symmetry interface). |
Impact | Two journal articles to date: Symplectomorphisms of mirrors to log Calabi-Yau surfaces https://arxiv.org/abs/2112.06797 Homological mirror symmetry for log Calabi-Yau surfaces https://arxiv.org/abs/2005.05010, accepted, Geometry & Topology Additionally, several week-long graduate summer schools have prominently featured our work, including: - Mirror symmetry for Looijenga pairs and beyond, July 2022 https://umutvg.github.io/Looijenga.html - Mirror symmetry in the log Calabi-Yau setting, May-June 2023 https://sites.google.com/view/hms-workshop2023 - Homological Mirror Symmetry and Symplectomorphisms, June 2023 https://kylerec.wordpress.com/ |
Start Year | 2018 |
Description | Ward |
Organisation | Massachusetts Institute of Technology |
Country | United States |
Sector | Academic/University |
PI Contribution | I am collaborating with Dr. Abigail Ward (MIT) on a project directly addresses Problem C.2 from the Case for Support, and goes significantly beyond it. Dr. Ward is a very promising postdoctoral resarcher at MIT. This collaboration started as a direct result of the EPSRC application process. It started a little after I was interviewed for the fellowship. (There was a significant delay before I heard of the final decision, though for mentorship reasons it seems important to start working on the project straight away.) I contributed the problem, the overall strategy, and technical expertise on certain aspects (including some of my preliminary work with Hacking). |
Collaborator Contribution | Ward has strong technical expertise on the geometry of almost-toric fibrations, partly stemming from a project she did with Hanlon. She also has stronger expertise than me in some of the tools we need from algebraic geometry. |
Impact | We are in the process of finalising a preprint with our results, which we expect to post to the arxiv, and submit for publication in a journal, sometime in Spring 2023. Additionally, Dr. Ward has accepted a PDRA position in the UK next year, funded by a UKRI grant help by my colleague Prof. Ivan Smith. She has made it clear that her increased interest in mathematics in the UK comes in part as a direct result of this collaboration, and that she now has interest in applying for permanent posts in the UK (lectureships, Royal Society fellowship, etc). As Ward was US-trained until this point, this is clearly a very positive development for UK maths, and a direct consequence of UK funding agency support (both EPSRC and UKRI). |
Start Year | 2021 |
Description | MSRI Connections |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | The PI co-organised "Connections" [formerly Connections for Women] at MSRI, as part of the Floer homotopy theory special semester. The two-day event is a primarly network-building is opportunity of Ph.D and early career female & non-binary mathematicians. It also gives them the opportunity to find female mentors & role models within their field. As well as short mini-courses and talks by female mathematicians, there was an advice panel [detailed in a different entry], and mentorship pairing groups, which met over lunch & coffee breaks both days. |
Year(s) Of Engagement Activity | 2022 |
URL | https://www.msri.org/programs/335 |
Description | PIMS summer school |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | The PI co-organised a major, two-week summer summer school for graduate students, on algebraic topology and symplectic geometry. This consisted of approx. eight intensive 3-4h mini-courses, 4 "hot topics" talks to introduce them to current research topics, and 8 problem sessions. There were approx. 90 participants (heavily oversubscribed: there were more than twice as many applications as that). This was also very important for community building for graduate students, many of whom were meeting peers working on similar topics at other institutions for the first time, because of the effect of covid-restrictions earlier. Several collaborations between student teams were started as a result of the conference. |
Year(s) Of Engagement Activity | 2022 |
URL | https://www.pims.math.ca/scientific-event/220711-sdms2fht |
Description | Panel Chair, MSRI connections workshop |
Form Of Engagement Activity | A formal working group, expert panel or dialogue |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | The PI chaired a career & work-life balance panel held at the Mathematical Sciences Research Institute as part of their "Connections" [formerly Connections for Women] workshop. This was held in person. The audience consisted primarily of female & non-binary graduate students, with a few more senior female & non-binary mathematicians also in attendance. Anonymised feedback forms were collected from participants, with overall very positive feedback. |
Year(s) Of Engagement Activity | 2022 |
URL | https://www.msri.org/workshops/974 |
Description | Panel Chair, PIMS workshop |
Form Of Engagement Activity | A formal working group, expert panel or dialogue |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | The PI chaired an expert advice panel on Academic Career Paths at the PIMS graduate summer school on symplectic & algebraic topology. The panel was organised by themes, and audience members had the opportunity to submit questions anonymously ahead of time, or live during the panel. We received very positive feedback from participants, including via anonymised feedback forms. |
Year(s) Of Engagement Activity | 2022 |
Description | Panel Member, European Society in Maths event |
Form Of Engagement Activity | A formal working group, expert panel or dialogue |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Undergraduate students |
Results and Impact | The PI was a panel member for an online event organised by the European Women in Maths association: https://sites.google.com/view/a2wim-get-together/home The theme was career advice for young mathematicians. The discussions were very positive and open, and I received very good feedback from the organisers afterwards. |
Year(s) Of Engagement Activity | 2022 |
URL | https://sites.google.com/view/a2wim-get-together/home |