Mini-Courses

Before the conference during May 6-10, 2013, Dave Futer, Kazuo Habiro, and Gregor Masbaum will each give a 5-lecture mini-course at the Institute of Mathematics in Hanoi (the capital, around 1200 km north of Nha Trang).
Schedule

May 6-10, 2013, Ha Noi Institute of Mathematics

• Dave Futer: (5 lectures, 50 minutes each)
  Topic: Hyperbolic knot theory
  Abstract: Around 1980, William Thurston proved that every knot in S^3 falls into exactly one of the following three families: torus knots, satellite knots, and hyperbolic knots (whose complements admit complete metrics of constant curvature -1). Hyperbolic knots are the dominant family, and the metrics on these knot complements are unique. This means that geometric measurements such as volume and the lengths of geodesics provide powerful knot-theoretic invariants. Until the early 2000s, mathematicians understood very little about how exactly the geometric invariants of knot complements relate to the combinatorics of diagrams, as well as to combinatorial polynomial invariants. By contrast, in the few years the relationship between geometric and combinatorial invariants has started to come into focus. I will start at the beginning of the subject, and survey some of what we know about this relationship.

   

• Kazuo Habiro: (5 lectures, 50 minutes each)
  Topic: Category-theoretic structures in 3-dimensional topology
  Abstract: After the discovery of the Jones polynomial link invariant and then Witten-Reshetikhin-Turaev invariants of 3-manifolds, many new invariants of links and 3-manifolds have been constructed and studied intensively. These invariants are usually defined as a functor from a category of topological structures to a category of algebraic structures. For example, quantum link invariants, including the Jones polynomial, can be defined as a functor from the category of tangles to the category of modules over a quantum group, and the Witten-Reshetikhin-Turaev invariant can be defined as a functor from the category of 3-dimensional cobordisms to the category of vector spaces. In the lectures, I plan to explain the structures of such topological categories and the functorial invariants. I also plan to explain quantum fundamental groups and quantum representation varieties which can be defined as a functor defined on the category whose objects are 3-manifolds and whose morphisms are isotopy classes of embeddings.

   

• Gregor Masbaum: (5 lectures, 60 minutes each)
  Topic: Skein-theoretic construction of TQFT and Integral TQFT
  Abstract: The concept of Topological Quantum Field Theory (TQFT) comes from physics: Witten used it to give a physicist's interpretation of the famous Jones polynomial of knots. The first mathematically rigorous construction of an invariant generalizing the Jones polynomial and satisfying the Atiyah-Segal axioms of TQFT was given by Reshetikhin and Turaev using quantum groups. In my lectures in Hanoi, I plan to explain a technically simpler construction of the same TQFT using skein theory, which was given later by Blanchet, Habegger, Vogel, and myself. This approach is based on the Kauffman bracket, which is a simple yet powerful combinatorial invariant of tangle diagrams in the plane. I will mainly focus on one particular aspect of TQFTs, namely that they provide finite-dimensional matrix representations of mapping class groups of surfaces. Skein theory allows one to construct these representations very concretely. I also plan to discuss integrality properties of these representations, which follow from the theory of Integral TQFT developed in my joint work with P. Gilmer. Some applications of this integrality will be discussed in my talk in Nha Trang.