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Invited Paper Session Abstracts - Low Dimensional Symplectic and Contact Topology

Friday, July 28, 1:00 p.m. - 4:00 p.m., Continental Ballroom B

The origins of symplectic and contact topology can be traced back to classical mechanical systems. Since then, both symplectic and contact topology have become very robust fields of study in their own right. The aim of this session will be to highlight techniques and recent results in the areas of low-dimensional symplectic and contact topology ranging from applications in knot theory to the theory of planar arrangements and singularities. Most of this work uses some version of Floer theory (such as contact homology or Heegaard Floer homology), which is an infinite-dimensional analog of Morse homology. We will aim to make this session understandable to nonexperts.

Organizers:
Dusa McDuff, Barnard College, Columbia University
Whitney George, University of Wisconsin LaCrosse

Constructing Interlocking Solid Tori in Contact 3-Manifolds

1:00 p.m. - 1:20 p.m.
Doug LaFountain, Western Illinois University

This talk will be hands-on using models with which participants can experiment. We will see how to construct interlocking solid tori, which have interesting applications in contact topology, and demonstrate how every positive braid which is not an obvious stabilization supports interlocking solid tori. Applications and open questions will be described as well; anyone is welcome, no previous knowledge will be required.

The Weinstein Conjecture

1:30 p.m. - 1:50 p.m.
Bahar Acu, University of Southern California and UCLA

The Weinstein conjecture asserts that the Reeb vector field of every contact form carries at least one closed orbit. The conjecture was proven for all closed 3-dimensional manifolds by Taubes. Despite considerable progress, it is still open in higher dimensions. In this talk, we will talk about its history and show that a \((2n+1)\)-dimensional "iterated planar” contact manifolds satisfy the Weinstein conjecture.

Contact Invariants and Reeb Dynamics

2:00 p.m. - 2:20 p.m.
Jo Nelson, Barnard College and Columbia University

Contact geometry is the study of certain geometric structures on odd dimensional smooth manifolds. A contact structure is a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability. The associated one form is called a contact form and uniquely determines a vector field called the Reeb vector field on the manifold. I will explain how to make use of J-holomorphic curves to obtain a Floer theoretic contact invariant whose chain complex is generated by closed Reeb orbits. In particular, I will explain the pitfalls in defining contact homology and discuss my work (in part joint with Michael Hutchings) which gives a rigorous construction of cylindrical contact homology via geometric methods. This talk will feature numerous graphics to acclimate people to the realm of contact geometry.

The Flexibility and Rigidity of Lagrangian Cobordisms

2:30 p.m. - 2:50 p.m.
Lisa Traynor, Bryn Mawr College

Cobordisms are common objects of study in topology. I will discuss cobordisms that have additional geometric constraints imposed by symplectic and contact structures. These Lagrangian cobordisms between Legendrian submanifolds arise in a relative version of Symplectic Field Theory. I will discuss results that show that sometimes Lagrangian cobordisms are flexible, in that they behave like topological cobordisms, while at other times Lagrangian cobordisms are rigid, in that they have properties very different than those seen in the topological setting. This is joint work with Joshua M. Sabloff.

A New Approach to the Symplectic Isotopy Problem

3:00 p.m. - 3:20 p.m.
Laura Starkston, Stanford University

One of the simplest closed symplectic manifolds is the complex projective plane, but we still have yet to answer one of the most basic questions about it: what is the classification of symplectic surfaces in CP2 up to symplectic isotopy? The adjunction formula determines the genus of such a symplectic surface from its homology class, and complex algebraic curves provide representatives of each of these homology classes. The symplectic isotopy problem asks if every symplectic surface is symplectically isotopic to one of these complex algebraic representatives. This problem has been solved affirmatively up to degree 17, but further progress has been halted by difficulties in the analysis of pseudoholomorphic curves. We present a new line of attack on this problem which translates it into a problem of finding certain Lagrangians with boundary.

Year: 
2017

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