Research Directions in Symplectic and Contact Geometry and Topology

Preface
	About This Volume
	About the 2019 WiSCon Workshop
	Acknowledgments
Contents
Introduction
	Paper 1: A Polyfold Proof of Gromov's Nonsqueezing Theorem
	Paper 2: Infinite Staircases for Hirzebruch Surfaces
	Paper 3: Action-Angle and Complex Coordinates on Toric Manifolds
	Paper 4: An Introduction to Weinstein Handlebodies for Complements of Smooth Toric Divisors
	Paper 5: Constructions of Lagrangian Cobordisms
	Paper 6: On Khovanov Homology and Related Invariants
	Paper 7: Braids, Fibered Knots, and Concordance Questions
A Polyfold Proof of Gromov's Non-squeezing Theorem
	1 Introduction
		1.1 Polyfold Notions and Regularization Theorems
	2 Outline of the Proof
		2.1 Compactifying the Target Space
		2.2 The Unique J0-Holomorphic Curve
		2.3 Using the Monotonicity Lemma
		2.4 A Compact Moduli Space
		2.5 Applying the Polyfold Regularization Scheme
	3 Polyfold Setup
		3.1 The Gromov-Witten Space of Stable Curves
		3.2 Trivial Isotropy
		3.3 The Base Space
		3.4 The Bundle
		3.5 The Section
		3.6 Linearization
		3.7 Transversality at the Boundary
	Appendix 1: The Monotonicity Lemma for Pseudoholomorphic Maps
	References
Infinite Staircases for Hirzebruch Surfaces
	1 Introduction
		1.1 Overview of Results
		1.2 Organization of the Paper
	2 Embedding Obstructions from Exceptional Spheres
		2.1 The Role of Exceptional Spheres
		2.2 Characterizing Staircases
		2.3 Blocking Classes
		2.4 Pre-staircases and Blocking Classes
	3 The Fibonacci Stairs, Its Cognates, and Beyond
		3.1 The Main Theorems
		3.2 Proof of Theorems 56 and 1
		3.3 Proof of Theorems 54, 58, 2 and 5
		3.4 Cremona Reduction
	4 Obstructions from ECH Capacities
		4.1 Toric Domains
		4.2 ECH Capacities and Exceptional Classes
		4.3 There Is No Infinite Staircase for b=1/5
	5 Mathematica Code
		5.1 Computing Many ECH Capacities of Xb Quickly
		5.2 Obstructions from Single ECH Capacities and a Lower Bound for cHb
		5.3 Obstructions from Exceptional Classes
		5.4 Strategy for Finding Staircases
		5.5 Plots of Ellipsoid Embedding Functions
	References
Action-Angle and Complex Coordinates on Toric Manifolds
	1 Introduction
	2 Toric Manifolds as Symplectic Quotients
		2.1 Complex Geometric Quotients
		2.2 Symplectic Quotients
		2.3 Canonical Line Bundle KM of a Toric Manifold M
	3 Toric Actions and Moment Maps
		3.1 Toric Tn-action on M and Its Moment Map
		3.2 Canonical Bundle KM Continued
		3.3 Holomorphic Coordinate Charts for M
		3.4 Justification of Choices for KM
		3.5 Moment Map for KPn in Homogeneous Coordinates
	4 Kähler Potential
	5 Connection to Mirror Symmetry
		5.1 Mirror Symmetry for Calabi-Yau Manifolds
		5.2 Mirror Symmetry for Landau-Ginzburg Models
		5.3 Monodromy in Mirror Symmetry
	6 Notation
	References
An Introduction to Weinstein Handlebodies for Complements of Smoothed Toric Divisors
	1 Introduction
		1.1 Main Results
	2 Weinstein Handlebodies and Kirby Calculus
		2.1 Weinstein Handle Structure
		2.2 Weinstein Kirby Calculus
	3 The Local Model for Our Handle Attachment
	4 The Algorithm Through an Example
	5 A More Complicated Example: Smoothing a Toric Divisor in CP2 # 3CP2
	References
Constructions of Lagrangian Cobordisms
	1 Introduction
	2 Background
		2.1 Legendrian Knots and Links
		2.2 Lagrangian Cobordisms
		2.3 Obstructions to Lagrangian Cobordisms
	3 Combinatorial Constructions of Lagrangian Cobordisms
		3.1 Decomposable Moves
		3.2 Guadagni Moves
		3.3 Lagrangian Diagram Moves
	4 Geometrical Constructions of Lagrangian Cobordisms
		4.1 The Legendrian Satellite Construction
		4.2 Lagrangian Cobordisms for Satellites
		4.3 Obstructions to Cobordisms Through Satellites
	5 Candidates for Non-decomposable Lagrangian Cobordisms
		5.1 Candidates for Non-decomposable Lagrangian Cobordisms from Normal Rulings
		5.2 Candidates for Non-decomposable Lagrangian Concordances from Topology
		5.3 Candidates for Non-decomposable Lagrangian Cobordisms from GRID Invariants
		5.4 Non-decomposable Candidates Through Surgery
	6 Conclusion
	References
On Khovanov Homology and Related Invariants
	1 Introduction
	2 A Survey of Applications of Khovanov Homology
		2.1 Rasmussen's s-Invariant
		2.2 Mutants
		2.3 Ribbon Concordance
		2.4 Unknotting and Unlinking via Spectral Sequences
		2.5 sl(n) Homology and HOMFLY-PT Homology
	3 Link Homologies and Ribbon Concordance
	4 Gordian Distance and Spectral Sequences in Khovanov Homology
		4.1 Results
		4.2 Examples
		4.3 Proofs
	References
Braids, Fibered Knots, and Concordance Questions
	1 Introduction
	2 Background on Fractional Dehn Twist Coefficient and Braids
		2.1 The Braid Group
		2.2 Fractional Dehn Twist Coefficient
		2.3 Dehornoy's Braid Ordering
	3 Concordance Invariants and Genus Bounds
	4 Quasipositive Braids and the FDTC Bounds
	5 An Interesting Example
	6 Potential Bounds on Slice Genus from the Braid Perspective
	7 Fibered Knots and Knot Floer Stable Equivalence
	8 Fractional Dehn Twist Coefficient of Fibered Slice Knots
	References
Photographs