EDUCATION

State University of New York at Stony Brook,
Ph.D. in Mathematics, 1992.

Beloit College, Beloit, Wisconsin,
B.S. in Mathematics, Summa Cum Laude, 1987.

PROFESSIONAL APPOINTMENTS

Assistant Professor of Mathematics, Bryn Mawr College,
1993 - Present.

Mathematical Sciences Research Institute Member,
Low Dimensional Topology Program, Berkeley, 1996-1997.

Isaac Newton Institute of Mathematical Sciences Member,
Program on Symplectic Geometry, Cambridge, England,
Fall 1994.

Centre Emile Borel, Institut Henri Poincare Fellow,
Symplectic Topology Semester, Paris, France,
February - July 1994.

NSF Postdoctoral Fellow at Stanford University,
September 1993 - February 1994.

Mathematical Sciences Research Institute Postdoctoral Fellow,
Berkeley, 1992 - 1993.

Teaching Assistant and Research Fellow, SUNY Stony Brook,
1987- 1992.

RESEARCH FUNDING AWARDS

National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship, 1993-1997.

Fellowship through Ministere des Affaires Etrang¸res, France, to attend Symplectic Topology Program at Institut Henri Poincare, Spring 1994.

Funded Participant for the Seminaire de Mathematiques Superieures NATO Advanced Study Institute on Gauge Theory and Symplectic Geometry, Montreal, Canada, July 1995.

Funded Participant for the NSF Regional Geometry Institute on Algebraic Geometry, Park City, Utah, June - July 1993.

Alfred P. Sloan Doctoral Dissertation Fellowship, 1991-1992.

• Coorganizer with Y. Eliashberg of the Research Program for the Institute for Advanced Study/ Park City Mathematics Institute on Symplectic Geometry and Topology, 1997.

• Guest Lecturer for High School Teacher Program at the Institute for Advanced Study/Park City Mathematics Institute, July 1997.

• Organizing Committee Member of the Institute for Advanced Study/Park City Mathematics Institute Women's Program, 1996-Present.

• Graduate Program Coordinator and Lecturer for the Institute for Advanced Study/Park City Mathematics Institute Women's Program on Symplectic Geometry, May 1997.

• Panelist for Girls and Women in Science Conference, Beloit College,
  Beloit, WI, April 1997.

• Author of the article The Compatibility of Research and a Liberal Arts College to the AWM (Association for Women in Mathematics) newsletter, November-December, 1996.

• Mentor for Women in Algebraic Geometry Workshop, Berkeley, May 1993.

• UC-Berkeley and Mills College Summer Program Colloquium, July 1993.

Ph.D. DISSERTATION SUPERVISION

I directed the Ph.D. dissertation research of Jean Mastrangeli who completed her degree at Bryn Mawr in May 1997. Mastrangeli now has a tenure track position at Immaculata College, Immaculata, PA.

A Legendrian Stratification of Rational Tangles, Traynor, L., Journal of Knot Theory and its Ramifications,To Appear, 1998, (45 pages).

Abstract: A subset of legendrian 2-string tangles are defined to be minimal if the strands realize the minimum absolute value of the Bennequin invariant. The restrictiveness of this condition is examined by studying which topological rational tangles have minimal representatives. It is shown that a rational of finite parity has a minimal representative if and only if its standard continued fraction expansion contains only non-negative entries, is of odd length, and has every horizontal entry even. This is proved by applying recent results of Fuchs and Tabachnikov or Chmutov and Goryunov that give an upperbound for the Bennequin invariant of links in terms of the minimal exponent of the framing variable of the Kauffman polynomial and Yokota's precise formula for this topological invariant. A second geometric stratum of rational tangles with infinite parity is defined and it is then shown a positive rational has a minimal representative if and only if its continued fraction expansion is a particular extension of a minimal of finite parity and a negative rational of infinite parity has a minimal representative if and only if it has even length with every vertical entry even.

Abstract: Examples are given of legendrian links in the manifold of cooriented contact elements of the plane, or equivalently, in the 1-jet space of the circle, which are not equivalent via an isotopy of contact diffeomorphisms. These examples have generalizations to linked legendrian spheres in contact manifolds diffeomorphic to R^n x S^n-1. These links are distinguished by applying the theory of generating functions to contact manifolds.

Abstract: Let B(r) be the closed 4-dimensional ball of radius r with the standard symplectic structure. A symplectic packing of B(1) with k balls of packing radius r is a set of k symplectic embeddings of B(r) into the interior of B(1) with disjoint images. For a fixed k, there is an upper bound to the set of possible packing radii since symplectic maps preserve volume. Via the theory of pseudo-holomorphic curves, Gromov proved that for certain values of k, there are obstructions to obtaining this upper bound of the packing radii. By combining pseudo-holomorphic curve theory with the theory of symplectic blow-ups, McDuff and Polterovich precisely determined the least upper bound of the set of packing radii for k = 1,..., 9 embeddings. In the following, explicit and elementary constructions are given for these maximal packings when k = 1,...,6,9. Constructions are also given for full packings of the 2n dimensional ball by kn balls. In addition, the techniques are applied to study alternate packing problems.

Abstract: A finite dimensional version of Floer -Hofer symplectic homology is developed for subsets of standard symplectic R2n using the theory of generating functions. The Floer-Hofer symplectic homology groups are constructed by studying the action functional on the loop space of a symplectic manifold. The generating function symplectic homology groups are first defined for a symplectomorphism as the relative homology groups of the sublevel sets of the generating function for the lagrangian associated to the diffeomorphism. Symplectic homology groups for an open set U are constructed via a limit of homology groups associated to symplectomorphisms supported on U. These groups are invariants of U under the action of symplectic diffeomorphisms of R2n and satisfy isotopy invariance and numerous functorial properties shared by Floer-Hofer homology. Generating function homology groups are calculated for ellipsoids and for some symplectically non-convex subsets of R2n.

Symplectic Embedding Trees for Generalized Camel Spaces, Traynor, L., Duke Mathematics Journal, Vol. 72, No. 3, December 1993, pp. 573 - 594.

Abstract: In previous work, it was shown that varying the size of the ''hole'' in the ''camel space'', the space underlying the symplectic camel theorem, produces symplectically nonequivalent sets. In the following, generalizations of the camel space are examined. First it is shown that there are many perturbations of the shape of the hole that do not change the symplectic type of the space. In addition, spaces with multiple 1 holed walls, W spaces, and spaces with one multiple holed wall, Z spaces, are explored. Invariants for these W and Z spaces are constructed by studying associated spaces of symplectic embeddings of balls. For the W spaces, ''embedding trees'' are constructed in order to prove that different orderings of holes can lead to symplectically different spaces.

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.

Abstract: Consider R4 with the standard symplectic structure and then consider the subset E(s) consisting of the union of two disjoint half spaces separated by a hyperplane and the open 3-ball of radius s in this dividing hyperplane. We give details of Eliashberg and GromovÕs proof of the symplectic camel theorem which says it is not possible to move a ball of radius r from one half space of E(s) to the other when r ³ s. In addition, it is proved that the ''camel spaces'', E(s) and E(t) are not symplectically equivalent when s ­ t. These statements are proved using GromovÕs technique of pseudo-holomorphic curves and, in particular, EliashbergÕs technique of filling by holomorphic discs.

PRESENTATIONS OF PAPERS (SELECT)

• Colloquium at University of Arkansas, Fayetteville, March 1998.

• American Mathematical Society Special Session on Symplectic Topology and Quantum Cohomology, Milwaukee, Wisconsin, October 1997.

• American Mathematical Society Special Session on Symplectic Geometry and Mechanics, Western Division Meeting, Corvallis, Oregon, April 1997.

• Seventh Midwest Geometry Conference, Lawrence, Kansas, April 1997.

• Conference on Symplectic Topology and Four- and Three Dimensional Manifolds, Monte Verita, Ascona, Switzerland, July 1996.

• American Mathematical Society Special Session on Geometry, Topology, and Analysis of Noncompact Manifolds, Joint Mathematics Meeting, Orlando, January 1996.

• NATO Advanced Study Institute on Gauge Theory and Symplectic Geometry, Sˇminaire de Mathˇmatiques Supˇrieures, Universitˇ de Montrˇal, July 1995.

• Workshop on Geometry, Topology, and Dynamics, Centre de Recherches Mathematiques, Universite de Montreal, June 1995.

• Workshop on Floer Homology, Newton Institute, England, September 1994.

• Seminare Sud Rhodanien de Geometry, Aussois, France, June 1994.

• American Mathematical Society Special Session on Symplectic Geometry, Joint Mathematics Meeting, Vancouver, August 1993.

• Midwest Several Complex Variables Conference, University of Michigan, March 1993.

• Colloquium at University of California at Berkeley, February 1993.

• American Mathematical Society Special Session on Symplectic Topology, Joint Mathematics Meeting, Baltimore, January 1992.