Quantum Groups and Knot Theory (GQT)

Credits	8 credit points
Instructors	Stokman, J.V. (Universiteit van Amsterdam), Cohen, A.M. (Technische Universiteit Eindhoven), Opdam, E.M. (Universiteit van Amsterdam)
E-mail	j.v.stokman@uva.nl, A.M.Cohen@tue.nl, e.m.opdam@uva.nl
Aim	The aim of Cohen's part of the course is to introduce the students to knot theory and familiarize them with algorithmic aspects and known software. The aim of the second part is to develop the foundations of the theory on quantum groups and on quantum invariants of ribbon links and knots.
Description	The core problem of knot theory is to decide whether two knots are equal. Partial solutions by means of knot invariants are known, the most common of which will be discussed in the course by Arjeh Cohen. We will focus on algorithmic aspects and work and use software for visualization and computation of invariants.In the second part of the course we discuss a general procedure to assigninvariants to colored oriented framed links and knots, and we treat some of the basics on the theory of quantum groups. Key examples of quantum groups are deformations of universel enveloping algebras of semisimple Lie algebras.They give rise to an abundance of admissable colorings of the oriented ribbon links. The associated quantum invariants are shown to encompass important knot and link invariants, such as the (colored) Jones polynomial. All notions mentioned above will be defined in the course and their properties, as needed, developed, with the exception of the basic material mentioned below (see `Prerequisites').
Organization	The introduction to knot theory will be given during the first three meetings, 12, 19, and 26 September 2007. If needed, a fourth meeting later in the course will be devoted to knot theory. The precise schedule will be announced in due course. Typically, there will be two hours of lectures on each Wednesday, and one hour devoted to homework problems, each hour lasting 45 minutes, with intermissions in between. The responsibility for the homework problem sessions for the first part of the course lies with Dan">http://www.win.tue.nl/~droozemo">Dan
Examination	Every week approximately five to ten homework problems will be handed out and published on the websites. Students who wish to get credit for the course should, every week, hand in solutions to four of these problems, of their own choice. Cooperation between students is allowed (and indeed encouraged), but each students should write down the solution in his/her own words; solutions that are verbally identical are not acceptable. There will not be a final examination. The students' grades are determined by their performance on the homeworks. Details about due dates for homeworks and how and where to hand them in, will be announced in due course.
Literature	The basic material on knot theory can be found in "Knots and Links" by D. Rolfsen (Publish or Perish, 1990), more advanced material in "Knot theory" by V. Manturov (Chapman and Hall/CRC, 2004).These books will not be followed in detail. The following book is used for the second part of the course: C. Kassel, M. Rosso, V. Turaev, "Quantum groups and knot invariants" (Panoramas et Syntheses, no. 5, 1997). Other basic references are V. Turaev, "Quantum invariants of knots and 3-manifolds" (Studies in Mathematics 18, Walter de Gruyter, 1994), C. Kassel, "Quantum Groups" (Graduate Texts in Math. 155, Springer Verlag, 1995).
Prerequisites	It is supposed that the students had some previous exposure to algebra and linear algebra, including the basic properties of groups, rings, and fields. Much more than is needed from algebra can be found in the book "Algebra" by S. Lang (Springer-Verlag, 2002) as well as in the Dutch course notes Algebra 1, 2, 3 by P. Stevenhagen (Universiteit Leiden), see http://websites.math.leidenuniv.nl/algebra/. A minimum of what is needed can be found in the book "Algebra interactive" by A.M. Cohen, H.Cuypers, and H. Sterk (Springer-Verlag, 1999). More useful algebra can be found at http://www.win.tue.nl/~sterk/algebra3/hoofd.pdf
Remarks	Websites: http://www.win.tue.nl/~droozemo/quantum/ http://staff.science.uva.nl/~jstokman/quantum/

Last changed: 15-03-2012 15:37