Spring 2017 Seminar
Organizers: Stephen Bigelow, Colleen Delaney, James Tener
May 31 | Wade Bloomquist (UCSB)
Skein quantum mapping class group representations
The bridge between classical topology and quantum topology is an exciting area of current research. In this talk, I will attempt to bring some light to a small corner of this world. In particular, the focus will be on a family of mapping class group representations arising from quantum topology (coming from Reshetikhin-Turaev TQFTs). I will explain asymptotic faithfulness as well as some of its implications, including recovering the Nielsen-Thurston classification of mapping classes. The end of the talk will include a more recent formulation of the volume conjecture by Tian Yang and Qingtao Chen.
May 17 | Julia Plavnik (Texas A&M)
On the classification of (super-) modular categories
In this talk we will continue the presentation of modular categories started in the Hypatian seminar. We will start recalling the main definitions and examples. Then we will give an overview about the current situation of the classification program for modular categories. We will explain some of the techniques that we found useful to push further the classification. We will also present some of the constructions that give rise to modular categories and some related open questions. If time allows, we will comment on the situation of the classification of pre-modular categories (specially super-modular categories).
May 3 | Pieter Naaijkens (UC Davis)
Total quantum dimension as an information rate
Consider a unitary modular tensor category. Such categories arise, for example, when one wants to describe the anyonic excitations in topologically ordered models. The total quantum dimension is an invariant of such categories, and for topologically ordered models is directly related to the anyons of the theory. On the other hand, in many models the total quantum dimension can be obtained as the Jones index of a particular subfactor. This picture allows one to interpret the quantum dimension as a rate of information that can be send securely in a particular quantum information setting.
This connection explains the physical mechanism at work behind the secure communication task, while on the other hand it hints at the power of using operator algebra in quantum information. In this talk I will try to explain give an overview of the main ideas, without assuming any background on subfactor theory. Partly based on joint work with Leander Fiedler and Tobias Osborne.
April 19 | Susama Agarwala (USNA)
Positive Grassmannians in Wilson Loop Diagrams
In this talk, I introduce a class of diagrams that appear in SYM N=4 theory. I then show that these diagrams represent points in the positive Grassmannian. For the rest of the talk, I discuss the combinatorics and geometry of positive Grassmannians.
April 12 | James Tener (UCSB)
Computing with planar diagrams
Planar algebras were first introduced in the late 90's by Vaughan Jones to axiomatize the standard invariant of a subfactor. Jones' idea was that the structure of standard invariants had a description in terms of planar diagrams, and that one could compute things about the subfactor by manipulating the pictures. In this talk I will demonstrate some of the features of calculating with pictures. I also plan to discuss the origins of planar algebras, and their role in the ongoing subfactor classification program. This talk will be a colloquium-style introduction, and should be accessible to a general mathematical audience.