The Principal Investigator plans to study algebraic structures that arise naturally from invariants of Legendrian submanifolds in contact manifolds as well as arbitrary submanifolds in smooth manifolds. More specifically, the invariants for a Legendrian submanifold come from holomorphic curves in certain symplectic manifolds, while the invariants for a smooth submanifold arise from open string topology which is based on the intersection theory of the manifold's path space. Both theories have algebraic structures guided by topological field theory. Part of this project is to refine the language of differential graded operad algebras to express the two theories in the similar algebraic language. There is a growing body of evidence that suggests a connection between the string topology of a manifold and the holomorphic invariants for its lift in the symplectic cotangent bundle or contact unit cotangent bundle. The Principal Investigator plans to define and compute some of these invariants, hopefully leading to a connection between the two theories when the smooth submanifold is a knot in Euclidean 3-space. The Principal Investigator also proposes several more computational projects related to knots in 3-space and smooth surfaces in 4-space, also defined using holomorphic curves.
Contact geometry makes many appearances in physics, from optics to thermodynamics to classical mechanics. For example, particles obeying the Least Action Principal from mechanics translate into objects in contact geometry (or its closely related field, symplectic geometry) known as holomorphic curves. Studying these holomorphic curves have led to some powerful and sometimes surprising discoveries about contact rigidity and contact dynamics. Knot theory has applications in understanding large and small aspects of the universe, as well as long DNA strands confined to small space. A central question in knot theory is determining the complexity of knots which in turn requires developing computable non-trivial knot invariants. Again holomorphic curves has recently provided a plethora of such useful knot invariants. The Principal Investigator plans to develop other knot invariants, motivated by string theory, that should be connected to these invariants based on holomorphic curves.
Holomorphic curve invariants in symplectic and contact geometry lie at an interface of many subjects in math: geometry, which provide the world in which these invariants live; algebra, which provides the language for the structure of invariants; and analysis, which provides the means to rigorously prove things about the invariants. There are also connections to physics: the invariants appear in string theory in modern theoretical physics; and, they embody particles in phase space obeying the Least Action Principle, such as conservation of energy and momentum. In more concrete terms, symplectic/contact geometry is a way to describe an orbiting satellite, light passing through water, or a driver parallel parking. As interconnected as these invariants are in theory, one of the greatest difficulties is actually computing them. How does one know if the invariants are useful and interesting? A major part of this project is to consider one class of these invariants, and reduce the involved geometry and analysis to combinatorics such that a personal computer can calculate them. Another part of this project is to take these computations, connect them to invariants in other fields of mathematics, and use them to solve interesting and unknown problems in these other fields. The PI mentors graduate students (doctoral theses, reading courses) and undergraduate students (honors senior theses, REUs). He has organized, since its inception, a weekly research seminar that encourages interaction between graduate students and faculty. He recently began an outreach seminar to a local community college, encouraging other faculty to present as well. He has run, since its inception, the undergraduate actuarial program.
