From Subfactors to Quantum Topology - In memory of Vaughan Jones

Contribution List

1. Zeta cycles and the scaling site

Alain Connes

6/27/22, 10:00 AM

2. Planar algebras, noncommutativity and hyperfinite subfactors

Dietmar Bisch

6/27/22, 11:30 AM

The hyperfinite II_1 factor contains a wealth of subfactors that give rise to many new and fascinating mathematical structures.
Vaughan Jones discovered that the unitary tensor category generated by the standard representation of a subfactor has a planar structure and can be described as what he called a ``planar algebra''. It is a complete invariant for amenable subfactors by a deep result...

3. The search for the exotic in Subfactors and Conformal Field Theory

David Evans

6/27/22, 2:30 PM

Groups can act as symmetries of physical systems and on their mathematical models as in conformal field theory.
Vaughan's subfactor theory provides a framework for quantum symmetries beyond those arising from groups or their deformations as quantum groups or loop groups. The accepted position was that the Haagerup system, associated with the a subfactor at index $(5+ \sqrt(13))/2$, was...

4. From Jones relation to representations of Mapping Class Groups

Christian Blanchet

6/28/22, 9:30 AM

We will review fundamental contributions of Vaughan Jones in the genesis of Quantum Topology. Then we will focus on representations of Mapping Class Groups highlighting a contribution of Vaughan Jones in genus 2. We will finally discuss homological models producing new representations.

5. Skein algebra of a punctured surface

Helen Wong

6/28/22, 11:00 AM

In the case of a closed surface, there is a rich body of work describing how the Kauffman bracket skein algebra can be regarded as a quantization of Teichmuller space. In order to generalize to a surface with punctures, Roger and Yang defined a skein algebra with extra generators and relations that they conjectured to be a quantization of Penner's decorated Teichmuller space. In joint work...

15. Cars, Interchanges, Traffic Counters, and a Pretty Darned Good Knot Invariant

Dror Bar-Natan

6/28/22, 1:30 PM

Reporting on joint work with Roland van der Veen, I'll tell you some stories about ρ1, an easy to define, strong, fast to compute, homomorphic, and well-connected knot invariant. ρ1 was first studied by Rozansky and Overbay, it is dominated by the coloured Jones polynomial (but it isn't lesser!), it has far-reaching generalizations, and I wish I understood it.

drorbn.net

7. Skein modules for generic quantum parameters”

Pavel Safronov

6/28/22, 3:00 PM

Skein modules were defined by Przytycki and Turaev as a way to generalize the Jones polynomial and the Kauffman bracket to links in manifolds other than the 3-sphere. In this talk I will review some recent structural results, such as the fact that the skein module of a closed 3-manifold is finite-dimensional for generic quantum parameters. I will also describe a work in progress joint with...

10. The cell dispensibility problem for spaces and manifolds

Jean-Claude Hausmann

6/29/22, 9:30 AM

We consider the following problem: when is a CW-space X homotopy equivalent to a CW-complex without j-cells for k<j<r ?
We show that this is equivalent to some cohomology condition together with the vanishing of an algebraic K-theory "cell-dispensability obstruction", analogous but not equal to the Wall finiteness obstruction. A similar theory holds for closed manifolds, replacing ``cells''...

11. Knot my problem: looking for rigidity in group theory

Martin Bridson

6/29/22, 11:00 AM

13. On skein theory in dimension four

Paul Wedrich

6/30/22, 9:30 AM

The Temperley-Lieb algebra describes the local behaviour of the Jones polynomial and gives rise to the Kauffman bracket skein modules of 3-manifolds. Going up by one dimension, Bar-Natan's dotted cobordisms describe the local behaviour of Khovanov homology and, likewise, give rise to skein modules of 4-manifolds. I will describe the construction of these skein modules and methods to compute...

14. 4-manifolds with boundary and fundamental group Z

Lisa Piccirillo

6/30/22, 11:00 AM

In this talk I will discuss a classification of topological 4-manifolds with boundary and fundamental group Z, under some mild assumptions on the boundary. I will apply this classification classify surfaces in simply-connected 4-manifolds with 3-sphere boundary, where the fundamental group on the surface complement is Z. I will also compare these homeomorphism classifications with the smooth...

6. The ADE link conjecture

Cameron Gordon

6/30/22, 1:30 PM

It is well known that the ADE graphs arise in many classification problems in mathematics. In 2019 Michel Boileau, Steve Boyer and I conjectured a modest addition to this list: the fibered links that induce the tight contact structure on S^3 and have a cyclic branched cover whose fundamental group is left-orderable. We will describe the conjecture, its background, and some recent results that...

8. Braids, Dualities and more subfactors

Hans Wenzl

6/30/22, 3:00 PM

{Abstract:}\textit{The famous Schur-Weyl duality states that the commutant of the action of
$Gl(V)$ on $V^{\otimes n}$ is generated by the obvious action of the symmetric group $S_n$
on $V^{\otimes n}$. We will first give a survey of quantum groups $U_q{\mathfrak g}$ and representations $V$,
where the commutant of the action of $U_q{\mathfrak g}$ on $V^{\otimes n}$ is (almost) generated by...

17. Facets of Temperley-Lieb algebra” 10:30-11:00 Coffee Break

Mikhail Khovanov

7/1/22, 9:30 AM

Skein modules were defined by Przytycki and Turaev as a way to generalize the Jones polynomial and the Kauffman bracket to links in manifolds other than the 3-sphere. In this talk I will review some recent structural results, such as the fact that the skein module of a closed 3-manifold is finite-dimensional for generic quantum parameters. I will also describe a work in progress joint with...

18. Lattice models arising from non-semisimple TQFT

Aaron Lauda

7/1/22, 11:00 AM

There is a rich interplay between two-dimensional topological phases in quantum mechanical systems and topological quantum field theory. This interaction is further enriched as topological structures inherent in TQFT lead to novel features, such as non-abelian braiding statistics for low energy excitations, when expressed in the corresponding quantum mechanical models. In this talk, we will...