Organizing Committee
Abstract

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 women and nonbinary researchers at various career stages in these mathematical areas to collaborate in groups on projects designed and led by female leaders in the field. See below for information regarding project leaders and topics.

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 7, 2019.

Image Credits: Ailsa Keating, Jo Nelson, Ina Petkova, and collective work of organizers.

Confirmed Speakers & Participants

Application Information

ICERM welcomes applications from faculty, postdocs, graduate students, industry scientists, and other researchers who wish to participate. Please fill out the application form to register for the event.

Your Visit to ICERM

ICERM Facilities
ICERM is located on the 10th & 11th floors of 121 South Main Street in Providence, Rhode Island. ICERM's business hours are 8:30am - 5:00pm during this event. See our facilities page for more info about ICERM and Brown's available facilities.
Traveling to ICERM
ICERM is located at Brown University in Providence, Rhode Island. Providence's T.F. Green Airport (15 minutes south) and Boston's Logan Airport (1 hour north) are the closest airports. Providence is also on Amtrak's Northeast Corridor. In-depth directions and transportation information are available on our travel page.
Lodging
ICERM's special rate will soon be made available via this page for our preferred hotel, the Hampton Inn & Suites Providence Downtown. ICERM also regularly works with the Providence Biltmore and Hilton Garden Inn who both have discounted rates available. Contact housing@icerm.brown.edu before booking anything.
The only way ICERM participants should book a room is through the hotel reservation links located on this page or through links emailed to them from an ICERM email address (first_last@icerm.brown.edu). ICERM never works with any conference booking vendors and never collects credit card information.
Childcare/Schools
Those traveling with family who are interested in information about childcare and/or schools should contact housing@icerm.brown.edu.
Family Support
With a recent gift from Microsoft Research, we are pleased to announce that there are fellowships available to assist in defraying dependent care costs. A separate application for this fellowship will be provided to those that have accepted an invitation to participate in an NSF-funded research program. Fellowship decisions will be made based on the justification and the availability of funds.
Technology Resources
Wireless internet access ("Brown-Guest") and wireless printing is available for all ICERM visitors. Eduroam is available for members of participating institutions. Thin clients in all offices and common areas provide open access to a web browser, SSH terminal, and printing capability. See our Technology Resources page for setup instructions and to learn about all available technology.
Discrimination and Harassment Policy
ICERM is committed to creating a safe, professional, and welcoming environment that benefits from the diversity and experiences of all its participants. The Brown University "Discrimination and Workplace Harassment Policy" applies to all ICERM participants and staff. Participants with concerns or requests for assistance on a discrimination or harassment issue should contact the ICERM Director, who is the responsible employee at ICERM under this policy.
Exploring Providence
Providence's world-renowned culinary scene provides ample options for lunch and dinner. Neighborhoods near campus, including College Hill Historic District, have many local attractions. Check out the map on our Explore Providence page to see what's near ICERM.

Visa Information

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Financial Support

Acceptable Costs
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Reimbursement Request Form

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Reimbursement Tips
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Reimbursement Timing

6 - 8 weeks after all documentation is sent to ICERM. All reimbursement requests are reviewed by numerous central offices at Brown who may request additional documentation.

Reimbursement Deadline

Submissions must be received within 30 days of ICERM departure to avoid applicable taxes. Submissions after thirty days will incur applicable taxes. No submissions are accepted more than six months after the program end.

Project Descriptions

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 categorified 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. Local approach to Khovanov homology involves working with complexes of modules over a family of rings. Extending from the sl(2) to sl(n) case requires matrix factorizations and their convolutions. Recent construction by L.H. Robert and E. Wagner is based on foam evaluation and points to a remarkable generalization of the state sum models in dimension one to higher dimensions. In our project, we will investigate these approaches to link homology, 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 Plamenevskayas 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.

Preferred background: Basic Khovanov homology, a bit of contact topology and/or homological algebra.

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), 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!).