Overview poster for the Master Classes in 2007
Abstract:
According to Atiyah and Segal's axioms, a Topological Quantum Field Theory (TQFT) describes how to compute quantum invariants of 3-dimensional manifolds by cutting and pasting. It involves in particular finite-dimensional representations of surface mapping class groups.
In this course we will focus on the Witten-Reshetikhin-Turaev SO(3)-TQFTs at odd primes. We will see that they admit a natural integral structure, meaning, among other things, that each mapping class group representation preserves a lattice defined over a ring of algebraic integers. This can be used to extract finer topological information from the invariants, and we will see examples of this.
While usual TQFT's can be described by semi-simple modular categories in the sense of Turaev, integral TQFT's require a generalization of this concept. We will review the skein-theoretical approach to TQFT and then give an explicit graphical description of the integral lattices using lollipop trees. This is joint work with Patrick Gilmer.
As another application, we will show that our integral TQFT's admit a 'perturbative' expansion as the order of the quantum parameter goes to infinity. This is expected to be related (in some not yet understood way) to Witten's asymptotic expansion conjecture. We will explain how this gives in particular a purely skein-theoretical construction of Ohtsuki's power series invariant of homology spheres.
No lectures on Thursday and Friday due to the Nielsen Retreat.
Lecture 2 Small size (161M) Larger size (713M)
Lecture 3 Small size (209M) Larger size (795M)
Lecture 4 Small size (154M) Larger size (679M)
Lecture 2 Small size (162M) Larger size (718M)
Lecture 2 Small size (157M) Larger size (697M)
Supplementary funding for this master class is provided by "Forskerskole i Matematik og anvendelser" based at the Department of Mathematics, University of Copenhagen.
Revised 22.10.2007
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