This book highlights a number of recent research advances in the field of symplectic and contact geometry and topology, and related areas in low-dimensional topology. This field has experienced significant and exciting growth in the past few decades, and this volume provides an accessible introduction into many active research problems in this area. The papers were written with a broad audience in mind so as to reach a wide range of mathematicians at various levels. Aside from teaching readers about developing research areas, this book will inspire researchers to ask further questions to continue to advance the field.
The volume contains both original results and survey articles, presenting the results of collaborative research on a wide range of topics. These projects began at the Research Collaboration Conference for Women in Symplectic and Contact Geometry and Topology (WiSCon) in July 2019 at ICERM, Brown University. Each group of authors included female and nonbinary mathematicians at different career levels in mathematics and with varying areas of expertise. This paved the way for new connections between mathematicians at all career levels, spanning multiple continents, and resulted in the new collaborations and directions that are featured in this work.
Author(s): Bahar Acu, Catherine Cannizzo, Dusa McDuff, Ziva Myer, Yu Pan, Lisa Traynor
Series: Association for Women in Mathematics Series, 27
Publisher: Springer
Year: 2022
Language: English
Pages: 346
City: Cham
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
