The Women in Symplectic and Contact Geometry and Topology workshop (WiSCon) is a Research Collaboration Conference for Women (RCCW) in the fields of contact and symplectic geometry/topology and related areas of low-dimensional topology. The goal of this workshop is to bring together researchers at various career stages in these mathematical areas to collaborate in groups on projects designed and led by leaders in the field.
The mathematical fields of symplectic and contact geometry/topology, rooted in concepts from classical physics, have experienced huge growth in the past few decades. This growth has come in many forms, including multiple flavors of homology theories, symplectic embedding problems, techniques for regularizing spaces of pseudoholomorphic curves, and examples of mirror symmetry, to name a few. This workshop aims to generate research collaborations which build on the growing momentum in these topics, while fostering a network for the traditionally underrepresented groups of women and nonbinary mathematicians. Successful applicants will be assigned to a research project based on their expertise. Participants in this workshop will form groups of 4-6 members, each led by 2 research leaders, and tackle open problems in a variety of such topics as described in the group descriptions below, perhaps incorporating computational techniques using ICERM’s exceptional computing resources.
Partially supported by NSF-HRD 1500481 - AWM ADVANCE grant.
Applications are now open. Applicants should rank their top 3 choices of projects in their personal statement. Project descriptions can be found below.
Application deadline is January 22, 2019.
Confirmed Speakers & Participants
Talks will be presented virtually or in-person as indicated in the schedule below.
- Poster Presenter
- Virtual Attendee
University of Iowa
Lahore University of Management Sciences
Ruhr University of Bochum
University of Georgia
University of California, Berkeley
California State University, Fresno
University of Leeds
Julia E Grigsby
Eötvös Loránd University
Uppsala University (Sweden)
University of California Los Angeles
Michigan State University
Georgia Institute of Technology
University of Iowa
University of Cambridge
Feride Ceren Kose
The University of Texas at Austin
University of Washington
Christine Ruey Shan Lee
University of South Alabama
Georgia Institute of Technology
Australian National University
Mathematical Institute of the University of Cologne
Chiu-Chu Melissa Liu
University of California, Davis
University of Belgrade
University College London
University of California, Davis
Instituto Superior Técnico
Massachusetts Institute of Technology
Penn State Brandywine
University of Texas
University of Notre Dame
Ana Rita Pires
The University of Edinburgh
Stony Brook University
Michigan State University
Max Planck Institute for Mathematics
North Carolina State University
University of Heidelberg, Germany
Lara Simone Suarez
Ruhr Universitat Bochum
Bryn Mawr College
University of Texas at Austin
University of Strasbourg – IRMA
University College London
University of Georgia
Monday, July 22, 2019
Tuesday, July 23, 2019
Wednesday, July 24, 2019
Thursday, July 25, 2019
Friday, July 26, 2019
|9:00 - 10:30am EDT||Group Work||Assigned Collaborative Spaces|
|10:30 - 11:00am EDT||Coffee/Tea Break||11th Floor Collaborative Space|
|11:00 - 12:00pm EDT||Group Work||11th Floor Lecture Hal|
|12:00 - 1:30pm EDT||Break for Lunch / Free Time|
|1:30 - 2:30pm EDT||Final Group Meeting||11th Floor Lecture Hal|
|2:30 - 3:00pm EDT||Coffee/Tea Break||11th Floor Collaborative Space|
|3:00 - 4:00pm EDT||Group Presentations (5mins/group)||11th Floor Lecture Hal|
|4:00 - 4:30pm EDT||Closing Remarks||11th Floor Lecture Hall|
Toric constructions of monotone Lagrangian submanifolds in the complex projective plane and the product of spheres
Project 1: Applications of Heegaard Floer homology to low-dimensional topology
Leadership: Jen Hom (Georgia Tech), Allison Moore (UC Davis)
Heegaard Floer homology, developed by Ozsváth-Szabó at the turn of this century, provides a package of invariants for 3- and 4-manifolds as well as knots and surfaces inside of them. The knot invariant, independently discovered by J. Rasmussen, categorifies the Alexander polynomial and provides information about the Heegaard Floer homology of any Dehn surgery along the knot. Our goal will be to use properties of Heegaard Floer homology to answer questions about knot theory (e.g., knot concordance, the cosmetic crossing conjecture) and 3-manifolds (e.g., homology cobordism, surgery questions).
Preferred background: General knowledge of low-dimensional topology (e.g., knot theory, 3-manifolds) and working knowledge of Heegaard Floer homology.
Project 2: Khovanov homology and related invariants: local and global approaches
Leadership: Radmila Sazdanovic (NC State), Christine Ruey Shan Lee (UT Austin)
The turn of the century brought along the plethora of categorifed knot invariants that lead to advances in knot theory, topology, and their relations with representation theory of quantum groups. Extracting the essential structure and topological and geometric information captured by these theories is still wide open.
Khovanov homology has a rich algebraic structure, including a range of spectral sequences such as Lee, Turner, and Bar Natan. Additionally, there are spectral sequences relating Khovanov to the Heegaard Floer knot theories (HFK), such as the ones constructed by Kronheimer and Mrowka, and more recently by Dowlin. In our project, we will use these structures to investigate Khovanov link homology and its torsion and relate it to HFK, as well as the contact-geometric content of the Plamenevskaya invariant. O. Plamenevskaya\’s invariant is defined for a link transverse to the standard contact structure of the 3-sphere based on its Khovanov homology. We will focus on the effectiveness of Plamenevskaya's invariant for twisted torus knots by considering the action of full twists on the link homology, using recent advances in the study of stable Khovanov homology of torus knots by E. Gorsky and M. Hogancamp.
Project 3: Bordered invariants in contact manifolds
Leadership: Ina Petkova (Dartmouth), Vera Vértesi (Université de Strasbourg)
The deep interplay of Heegaard Floer homology and contact geometry has been apparent since the beginning. The proposed project is to understand more of this connection by defining invariants for contact objects with boundary via bordered Floer homology. Bordered Floer homology, defined by Lipshitz, Ozsváth, and Thurston, is an invariant of 3-manifolds with parametrized boundary. The bordered contact invariants would be gluable, and could be used to make, cut and paste arguments for the contact invariants of closed contact objects. The simplest toy examples to start working with are Legendrian invariants for tangles in the bordered versions of knot Floer homology defined by Ozsváth and Szabó using Kauffmann states, or Petkova and Vértesi using generalized grid diagrams. There is an obvious candidate for a Legendrian invariant in the Petkova--Vértesi setup, due to its grid diagram flavor. As a first step, the participants will prove that this candidate indeed defines an invariant for Legendrian tangles and will determine its properties. Then the project can develop in several directions, for example defining a Legendrian invariant in the Ozsváth--Szabó setup or using the intuitions from the toy example to understand contact invariants for more general contact 3-manifolds with bordered boundary.
Preferred background: Familiarity with the following two areas, with working knowledge in at least one: contact topology, Heegaard Floer homology, including (some version of) bordered Floer homology.
Project 4: Polyfold Laboratory
Leadership: Katrin Wehrheim (UC Berkeley), Penka Georgieva (IMJ-PRG)
Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding commonalities in the analysis of a variety of geometric elliptic PDEs, in particular, pseudoholomorphic curves. It systematically addresses the common difficulties of compactification and transversality with a new notion of smoothness on Banach spaces, new local models for differential geometry, and a nonlinear Fredholm theory in the new context. By fall 2018, the polyfold-Fredholm property of Gromov-Witten, SFT, and Lagrangian boundary condition moduli spaces will be established, and the abstract theory has been formulated as black box that essentially allows one to argue as if moduli spaces are the zero sets of smooth sections of finite dimensional orbibundles. So it is timely for this project to serve as hands-on training in applying the polyfold framework to prove geometric applications of pseudoholomorphic curve moduli spaces. The choice of concrete applications will be made by participants and could range from elegant short proofs of classical results (e.g. Gromov nonsqueezing) via contributing to larger infrastructure (e.g. extending the work at www.polyfolds.org on Fukaya categories) to workshopping projects arising from participants' or other projects' interests, or a rather bold goal of casting adiabatic limits (e.g. between gauge theoretic equations and pseudoholomorphic curves) into the polyfold framework -- thus extending Atiyah-Floer type results beyond monotone settings.
Preferred background: You should have some experience with the analysis or geometric applications of moduli spaces of pseudoholomorphic curves or similar geometric PDEs. While we will not (need to) study the details of polyfold theory, we will in preparation for the workshop do a (guided, collaborative) reading of parts of https://arxiv.org/pdf/1210.6670.pdf, so e.g. the beginnings of sections 2.1 and 4.1 should look accessible to you.
Project 5: Mirror symmetry and symplectic geometry
Leadership: Chiu-Chu Melissa Liu (Columbia), Ailsa Keating (University of Cambridge)
Description: Mirror Symmetry relates the A-model defined by the symplectic structure on a manifold to the B-model defined by the complex structure of a mirror manifold. We will study homological mirror symmetry concerning Fukaya categories and Lagrangian Floer theory, as well as more traditional mirror symmetry concerning quantum cohomology and Gromov-Witten invariants counting J-holomorphic curves.
Preferred background: Familiar with basics of either Lagrangian Floer theory or Gromov-Witten theory.
Project 6: Homological invariants, braids, transverse links, and surfaces
Leadership: Eli Grigsby (Boston College), Diana Hubbard (Brooklyn College, CUNY), Keiko Kawamuro (University of Iowa), Olga Plamenevskaya (Stony Brook)
It is a classical theorem of Alexander that every link in the three-sphere can be realized as a closed braid, and Bennequin later proved a version of this theorem for transverse links with respect to the standard tight contact structure on the three-sphere. There are some explicit connections between dynamical properties of a braid conjugacy class (viewed as a mapping class of the punctured disk) and surfaces the braid closure (viewed either in its smooth or transverse isotopy class) can bound in the three-sphere and the four-ball. We will investigate questions related to this observation, using Khovanov homology and Heegaard-Floer homology. In particular, we will be interested in how geometric properties of closed braid representatives of a link are reflected in algebraic properties of these homological invariants.
Preferred background: Knot and braid theory and the basics of handlebody theory and contact geometry, as well as familiarity with homology theories for knots and links.
Project 7: Weinstein Kirby calculus and Fukaya categories
Leadership: Emmy Murphy (Northwestern), Laura Starkston (UC Davis)
Weinstein manifolds are a class of exact symplectic manifolds defined by Morse theory decompositions compatible inducing contact structures on level sets. This allows us to describe, for example, Weinstein 4-manifolds using Legendrian knots in S^3. This is analogous to Kirby calculus in smooth topology but strictly enriches it with the underlying geometries. Pseudoholomorphic invariants of these knots, such as Legendrian contact homology, also correspond to invariants of the Weinstein manifold, such as the wrapped Fukaya category. Similar to the smooth story, any two decompositions of a given Weinstein manifold can be induced by Kirby moves transforming the Legendrian knots into new knots. The interactions between these Kirby moves and the corresponding Fukaya invariants have many questions still to be explored, both in the general theory as well as many interesting examples.
Preferred background: Basic symplectic topology, Lagrangian Floer homology and smooth Morse/handlebody theory.
Project 8: Lagrangian cobordisms between Legendrian submanifolds
Leadership: Lisa Traynor (Bryn Mawr), Yu Pan (MIT)
A classic problem is to understand Legendrian submanifolds up to the equivalence given by Legendrian isotopy. Motivated by ideas from Symplectic Field Theory, a coarser relation is to understand when Legendrian submanifolds are related by Lagrangian cobordism. The classical Thurston-Bennequin and rotation number invariants of Legendrian submanifolds give obstructions to the existence of a Lagrangian cobordism. There are also non-classical obstructions coming from Legendrian contact homology defined through the techniques of holomorphic curves or generating families. This project will explore some of the many open questions about the existence and obstructions to Lagrangian cobordisms, fillings, and caps.
Preferred background: Basics of Legendrian and Lagrangian submanifolds, some familiarity with Legendrian contact homology and/or generating family homology.
Project 9: Symmetry and moment maps in symplectic geometry and topology
Leadership: Tara Holm (Cornell), Ana Rita Pires (Edinburgh)
Moment maps are a key tool in symplectic geometry. They are used to study global topological questions via Morse theory and to construct new symplectic manifolds from old via symplectic reduction. Toric symplectic manifolds are completely determined by their moment map image. Similarly, moment maps and symmetry data govern the behavior of complexity one spaces and of toric origami manifolds. Moment map data have also been used to resolve questions in symplectic topology. Because they provide local models for Hamiltonian T-spaces, one can use them to find lower bounds for Gromov width, for example. Similar "soft" techniques have also proved useful in computing upper bounds. In this group, we will probe questions about the global topological and symplectic invariants of these various manifolds. We expect to use some computer experimentation to formulate conjectures.
Preferred background: Working knowledge of symplectic geometry, Hamiltonian actions and momentum maps. General knowledge of classical algebraic topological invariants and/or ECH capacities. Some coding experience could be helpful (but not necessary!).
- Bahar Acu, Catherine Cannizzo, Dusa McDuff, Ziva Myer, Yu Pan, Lisa Traynor, Research Directions in Symplectic and Contact Geometry and Topology, Association for Women in Mathematics Series, Springer Cham, 2021.