# Project Details

### Description

Project Summary

The proposed grant is intended to fund research surrounding the question: if X is a contractible

symplectic manifold, under what assumptions on the Fukaya category can we ensure that X is symplectomorphic to Cn? The PI outlines a program for approaching this question, whose ingredients involve wrapped Fukaya categories, microlocal sheaf theory, arboreal Lagrangian singularities, and h–principles in contact geometry.

Intellectual Merit

One of the most important classes of exact symplectic manifolds are Weinstein manifolds. A

significant amount of progress has been made in recent years as to our understanding of their

geometry, on both the flexible and rigid side. On the rigid side, the state-of-the-art tool associates

to any Weinstein manifold X its wrapped Fukaya category W(X), which is a type of algebraic

invariant (specifically an A∞ category). This category (and associated tools such as symplectic

cohomology) has extremely rich structure. One important application is to establish the wide

breadth of possible Weinstein manifolds: when n > 2 there are infinity many symplectically distinct

Weinstein manifolds X, realizing any diffeomorphism type.

A major limitation, however, is that we have a poor understanding as to what extent W(X) is

a complete invariant. Even in the most basic case, when we assume that X is diffeomorphic to R2n

and W(X)∼=0, it is an open question whether this characterizes X∼=Cn, or at the other extreme

if there could be infinitely many X satisfying these constraints. Naturally this is a fundamental

question to symplectic geometry

The PI has recently proven some partial results in the positive direction, and this grant proposal

is intended to further her research on this topic. The research program, outlined in the Project

Description, involves a number of tools established only very recently: the existence of an arboreal

Lagrangian skeleton inside any Weinstein manifold, the h–principle for loose Legendrians and

flexible Weinstein manifolds, and the relationship between the wrapped Fukaya category and sheaf theory.

Broader Impacts

The PI has a career interest in working with young mathematicians. For instance she has organized a graduate summer school on the h–principle through MSRI two years in a row (2017 and 2018), she will be a project leader for twoWomen in Geometry conferences this upcoming summer (WiSCon at ICERM and Women in Geometry 2 at CMO), and she was the scientific committee chair and NSF PI for the first Kylerec conference, which has become an annual conference organized by and for graduate students in symplectic geometry. She has additionally organized two other conferences in the past two years: “Engel structures” at AIM and “Geometric methods in symplectic and contact topology” at Stanford.

This focus on the mathematical community dovetails nicely with the proposed research topic,

because it involves many smaller subprojects which are challenging but relatively straightforward.

This gives excellent opportunities for learning contemporary tools, and many aspects of the program come from different fields of math, some more algebraic or more topological, which allows younger mathematicians the opportunity to contribute in the aspects they’re most interested in.

The proposed grant is intended to fund research surrounding the question: if X is a contractible

symplectic manifold, under what assumptions on the Fukaya category can we ensure that X is symplectomorphic to Cn? The PI outlines a program for approaching this question, whose ingredients involve wrapped Fukaya categories, microlocal sheaf theory, arboreal Lagrangian singularities, and h–principles in contact geometry.

Intellectual Merit

One of the most important classes of exact symplectic manifolds are Weinstein manifolds. A

significant amount of progress has been made in recent years as to our understanding of their

geometry, on both the flexible and rigid side. On the rigid side, the state-of-the-art tool associates

to any Weinstein manifold X its wrapped Fukaya category W(X), which is a type of algebraic

invariant (specifically an A∞ category). This category (and associated tools such as symplectic

cohomology) has extremely rich structure. One important application is to establish the wide

breadth of possible Weinstein manifolds: when n > 2 there are infinity many symplectically distinct

Weinstein manifolds X, realizing any diffeomorphism type.

A major limitation, however, is that we have a poor understanding as to what extent W(X) is

a complete invariant. Even in the most basic case, when we assume that X is diffeomorphic to R2n

and W(X)∼=0, it is an open question whether this characterizes X∼=Cn, or at the other extreme

if there could be infinitely many X satisfying these constraints. Naturally this is a fundamental

question to symplectic geometry

The PI has recently proven some partial results in the positive direction, and this grant proposal

is intended to further her research on this topic. The research program, outlined in the Project

Description, involves a number of tools established only very recently: the existence of an arboreal

Lagrangian skeleton inside any Weinstein manifold, the h–principle for loose Legendrians and

flexible Weinstein manifolds, and the relationship between the wrapped Fukaya category and sheaf theory.

Broader Impacts

The PI has a career interest in working with young mathematicians. For instance she has organized a graduate summer school on the h–principle through MSRI two years in a row (2017 and 2018), she will be a project leader for twoWomen in Geometry conferences this upcoming summer (WiSCon at ICERM and Women in Geometry 2 at CMO), and she was the scientific committee chair and NSF PI for the first Kylerec conference, which has become an annual conference organized by and for graduate students in symplectic geometry. She has additionally organized two other conferences in the past two years: “Engel structures” at AIM and “Geometric methods in symplectic and contact topology” at Stanford.

This focus on the mathematical community dovetails nicely with the proposed research topic,

because it involves many smaller subprojects which are challenging but relatively straightforward.

This gives excellent opportunities for learning contemporary tools, and many aspects of the program come from different fields of math, some more algebraic or more topological, which allows younger mathematicians the opportunity to contribute in the aspects they’re most interested in.

Status | Active |
---|---|

Effective start/end date | 8/1/19 → 7/31/22 |

### Funding

- National Science Foundation (DMS-1906564)