911ºÚÁÏÍø

Brief Introduction

Since 1993, I have been on the faculty of 911ºÚÁÏÍø, where I greatly enjoy doing research and teaching students. My mathematical research is very visually motivated: I enjoy looking at some basic shapes of space, such as a knotted loop of string or a round ball, and seeing how this shape can evolve under mathematical equations that have deep roots in classical physics.  I teach a variety of courses throughout the curriculum, from calculus at the 100-level to graduate courses in topology at the 500-level.

I received my Ph.D. in Mathematics in 1992 from the State University of New York at Stony Brook; Dusa McDuff was my research advisor. I pursued postdoctoral work at the Mathematical Sciences Research Institute in Berkeley, CA, at Stanford University, at Centre Emile Borel in the Institut Henri Poincaré in Paris, France and at the Isaac Newton Institute in Cambridge, England.  I have spent sabbaticals at the Mathematical Sciences Research Institute (MSRI), Berkeley, CA, at the Institute for Advanced Study (IAS), Princeton, NJ, and at the American Institute of Mathematics (AIM), Palo Alto, CA.  I was on sabbatical for the 2022-2023 academic year and spent the fall as a Research Professor for the program at MSRI/SLMath in Berkeley, CA.

Research

My research interests are in geometry and topology. More specifically, I work in symplectic and contact topology, where we study many questions similar to those studied in topology except under additional constraints given by the geometry of a symplectic or contact structure. I am particularly interested in studying the flexibility and rigidity of Lagrangian submanifolds of standard symplectic spaces and of Legendrian submanifolds of contact spaces.  A motivating question is to understand the boundary between flexibility (when the Lagrangian or Legendrian submanifold behaves like their smooth counterparts) and rigidity (when the behavior of these geometric submanifolds is more restrictive). I enjoy exploring the edge between flexible and rigid phenomena. 

I am a co-organizer of the weekly Math PACT (Philadelphia Area (Contact) Topology) seminar and of the monthly PATCH (Philadelphia Area Topology: Contact & Hyperbolic) seminar.  

Over the years, I have worked on a variety of problems including the symplectic camel problem, symplectic homology, symplectic packings, Legendrian knots, and Lagrangian cobordisms between Legendrian knots. I have employed a variety of techniques including J-holomorphic curves, generating families of functions, and convex surfaces.  Below you can find a list of my research publications.

Research Publications

• Obstructions to Reversing Lagrangian Surgery in Lagrangian Fillings, Capovilla-Searle, O., Legout, N., Limouzineau, M., Murphy, E., Pan, Y., and Traynor, L., to appear in J. Symplectic Geometry, 
• Legendrian Torus and Cable Links, Dalton, J., Etnyre, J., and Traynor, L., to appear in J. Symplectic Geometry,
• Constructions of Lagrangian Cobordisms, Blackwell, S., Legout N. Leverson, C., Limouzineau, M., Myer, Z., Pan, Y., Pezzimenti, S., Suárez, L.S., and Traynor, L. in  Research Directions in Symplectic and Contact Geometry and Topology (Association for Women in Mathematics Series 27), Springer 2021;
• An Introduction to the World of Legendrian and Transverse Knots, Traynor, L., in A Concise Encyclopedia of Knot Theory, CRC Press, 2021.
• The Relative Gromow Width of Lagrangian Cobordisms between Legendrians, Sabloff, J. and Traynor, L., Journal of Symplectic Geometry, Vol 18, No 1, 217-250, 2020.
• The Minimal Length of a Lagrangian Cobordism bet