Proposal Title: Surfaces in knot exteriors: The Lopez Conjecture.
Principal Investigator: Dr. William H. Jaco
Abstract: This project involves a Special Semester at Stanford University and The American Institute of Mathematics bringing together a team of four senior investigators, two postdoctoral fellows and two advanced graduate students for an intense and collaborative study of essential surfaces in knot exteriors. Two of the senior investigators have been working on a program over the past two decades to detect certain knots by properties of essential surfaces in the knot exterior. Their methods have been very successful in identifying such properties but do not lead to methods that will enable one to realize such knots. The other two senior investigators have in the last three years developed new methods for realizing knots having interesting properties. This project will bring these two methods together to address one of the most outstanding problems in this area of mathematical research.
Low-dimensional topology brings together many areas of mathematical research and provides a common ground for interaction and advance across all of mathematics. It provides a natural geometric model of most physical phenomena. Research in this area is making significant contribution to computational geometry and topology and complexity theory. It has consequences in physics, computer visualization and medical modeling.
