Â鶹AV

Brief Introduction

Since 1993, I have been on the faculty of Â鶹AV, 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 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 between Legendrians, Sabloff, J. and Traynor, L., Selecta Mathematica New Series, 23: 1419-1448, 2017.
• Non-Orientable Lagrangian Cobordisms between Legendrian Knots, Capovilla-Searle, O. and Traynor, L., Pacific Journal of Mathematics, Vol 285, No. 2, 319--334. 2016; DOI 10.2140/pjm.2016.285.319.  
• Lagrangian Cobordisms via Generating Families: Construction and Geography, Bourgeois, F., Sabloff, J. and Traynor, L., Algebraic and Geometric Topology, Vol 15, 2439-2477, 2015.
• Obstructions to Lagrangian Cobordisms between Legendrians via Generating Families, Sabloff, J. and Traynor, L., Algebraic and Geometric Topology, Vol 13, issue 5, 2733-2798, 2013.
• The surgery unknotting number of Legendrian links, B. Boranda, L. Traynor, S. Yan, Involve, Vol 6, No. 3, 273-299, 2013.
• Obstructions to the Existence and Squeezing of Lagrangian Cobordisms, Sabloff, J. and Traynor, L., Journal of Topology and Analysis, Vol 2, No. 2, 203 - 232, 2010.
• A Partial Ordering on Slices of Planar Lagrangians, Eisman, P., Lima, J., Sabloff, J., and Traynor, L., Journal of Fixed Point Theory and Applications, Volume 3, Number 2, 431-447, September 2008.
• Generating family invariants for Legendrian links of unknots, Jordan, J. and Traynor, L., Algebraic and Geometric Topology, Volume 6, 763-807, 2006.
• Legendrian Solid-Torus Links, Ng, L. and Traynor, L., Journal of Symplectic Geometry, Volume 2, Number 3, 411-443, 2004.
• Generating Function Polynomials for Legendrian Links, Traynor, L., Geometry and Topology, Vol. 5 (2001) Paper no. 23, pages 719--760.
• Symplectic Packings of Cotangent Bundles of Tori, Maley, F. M., Mastrangeli, J. and Traynor, L., Experimental Mathematics, Vol 9, No. 3, 2000, 435-455.
• A Legendrian Stratification of Rational Tangles, Traynor, L., Journal of Knot Theory and its Ramifications, Vol 7, No. 5, 1998, 659-700.
• Legendrian Circular Helix Links, Traynor, L., Mathematical Proceedings of the Cambridge Philosophical Society, Vol 122, 1997, pp. 301 - 314.
• Symplectic Packing Constructions, Traynor, L., Journal of Differential Geometry, Vol 42, No. 2, 1995, 411 - 429.
• Symplectic Homology via Generating Functions, Traynor, L., Geometric and Functional Analysis, Vol. 4, No. 6, 1994, pp. 718 - 748.
• Mathematics Journal, Vol. 72, No. 3, December 1993, pp. 573 - 594.
• The 4-dimensional Symplectic Camel and Related Results, McDuff, D. and Traynor, L., in Symplectic Geometry, D. Salamon (ed.), London Mathematical Society Lecture Note Series, Cambridge Univ. Press, 1993, pp. 169 - 182.

Teaching

I greatly enjoy teaching at Bryn Mawr. I teach a variety of courses through the curriculum.  Recently, I have taught Multivariable Calculus, Transition to Higher Mathematics, Topology, Complex Analysis, Â鶹AV Differential Topology, and Senior Conference.  Since COVID struck, I have started to teach many of my classes in a flipped style: I record approximately 30 minute videos for students to watch before coming to class, and then during class we work through worksheets to help the material sink in.  All my class materials are on Moodle.

Research with Students

I love doing research, and I truly enjoy doing research with both undergraduate and graduate students.  I have had 6 students complete their PhD under my direction; see . I have also supervised 8 MA theses and many undergraduate honors thesis projects. Some honors projects have been tied to my own research in symplectic and contact topology, but I have also supervised students on projects in knot theory, game theory, optimal control, and geometry topics connected to computer science.

Professional Service and Outreach

I have held a number of leadership positions in national professional organizations: in 2000-2003, I served as Member-At-Large of the American Mathematical Society (AMS) Council, and in 2008-2012, I served on the Executive Committee of the Association for Women in Mathematics (AWM). 

I truly enjoy working with  young mathematicians to help them find their own way in the mathematical community. Throughout my career, I have enthusiastically organized activities around professional development.  At Bryn Mawr, I have organized seminars for graduate students on numerous  topics including how to effectively give a talk and how to write abstracts, grants, CVs,  and teaching and diversity statements.  Three times I have been a mentor for the national held at Georgia Tech.  I have often been on career/advice panels for younger mathematicians during conferences. 

I have also welcomed opportunities to organize programs that promote women in mathematics: for many years, I was involved in the (Princeton). I also was a team leader for (in Banff, Canada) and (in Oaxaca, Mexico). I was also an organizer and project leader for the program , which took place in July 2019 at ICERM.  At Bryn Mawr. 

More details about my service can be found on my CV.

Additional Interests

I enjoy spending time with my family: my husband, Paul Hintz, and I have been together since our undergraduate days at Beloit College, and we have a son, Emmett, who is starting his second year of college at Colgate University.  I enjoy a variety of outdoor activities including hiking, biking, and canoeing and kayaking.  Since the pandemic started, I have been hooked on noontime walks + audio books. On a daily basis, I enjoy doing YouTube workouts:  I am a big fan of Heather Robertson's channel.  I also enjoy cooking and eating good food.  For many years, we have been members of a great CSA near our home: Pennypack Farm.  They have a vast selection of organic vegetables, and seasonal salads are my lunch of choice most days.  I also enjoy good wine, chocolate, and soft-serve ice cream!