- Lecture 1: Knots and Groups
- Lecture 2: Lecture 2: Von Neumann Algebra and Physics
- Lecture 3: Do all Subfactors arise in Conformal Field Theory?

*Abstract:* Knots are among the more concrete features in the mathematical landscape. Groups are more pervasive and more abstract. But the two subjects have been intimately connected since the early days of the study of both. After defining knots and groups we will give the first such connection-the “fundamental group” of the knot. This group is known to determine the knot but a construction is not immediate. The braid group is a concrete group with some structural resemblance to knots. We will show how all knots arise from elements of the braid group and how to learn things about the knot from its braid. In particular a family of “knot polynomials” appears from this study.

*Abstract:* The states of a quantum system are given by vectors in a Hilbert space with inner product \(\langle \xi,\eta \rangle\). Observables are self-adjoint operators on that Hilbert space. The fundamental formula connecting the two is that if \(a\) is an operator/observable and \(\xi\) is a unit vector/state then \(\langle a\xi,\xi \rangle\) is a real number giving the average value of repeated measurements of the observable \(a\) if the system is prepared each time in the state \(\xi\). Von Neumann introduced the algebras that bear his name in large part to help understand the mathematical structure of quantum theory. His prophetic ideas have been very fruitful in low dimensional quantum field theory and are intimately related to the knot polynomials of the first lecture.

*Abstract:* A subfactor is a pair of von Neumann algebras with trivial center (factors) one included in the other. A subfactor \(N \subset M\) has an index \([M : N]\) which is a real number defined by von Neumann’s theory. For the most obvious examples of factors \([M : N]\) is actually an integer but in fact it can be any number in the set \(\{4\cos^2(\frac\pi n) \colon n = 3, 4, 5, 6, ...\} \cup [4, \infty]\). Subfactors realising these values can be constructed from algebras of observables as in the second lecture. It is an open and intriguing question whether or not ALL subfactors (of finite index) can be obtained from quantum field theory. An attempt to take a continuum limit from the data of a subfactor has led to a new construction of knots and links from certain groups of homeomorphisms of the unit interval known as the Thompson groups.

Virginia Mathematics Lectures archive