Spring 2018 Seminar
Organizers: Stephen Bigelow, Colleen Delaney, James Tener
May 30 | Alan Tran (UCSB)
Fermionic topological phases and modular transformations
We develop the algebraic theory of fermionic topological phases in (2+1) dimensions and study their modular transformations. In order to allow for consistently defined spinors, fermionic systems necessarily reside on manifolds with spin structures. This must be incorporated in the structure of the low-energy effective theory of a fermionic topological phase, which is, therefore, a ``spin topological quantum field theory"(spinTQFT). We show how a (2+1)D spinTQFT can be associated to a modular tensor category (MTC) containing a distinguished object corresponding to the physical fermions of the system. The physical fermion provides a Z2-grading on the MTC that partitions the topological charges into quasiparticles and vortices.
May 9 | Stephen Bigelow (UCSB)
A recipe for spiders
This talk is mostly a definition and a conjecture. A spider diagrammatically encodes the representation theory of a quantum Lie algebra. I will describe an all-purpose recipe to go from the root system to a diagrammatic algebra that I conjecture contains the corresponding spider. I can prove a lot of cases by direct computation, and I have some ideas about why it works in general, but much remains mysterious.
April 18 | Sherilyn Tamagawa (UCSB)
Quandles and Friends
Quandles encode knots as algebraic structures. In this introductory talk, I will describe quandles and some of the ways we use them to study knots. I will also describe some algebraic structures closely related to quandles.
April 4 | Wade Bloomquist (UCSB)
Quantum Representations of Mapping Class Groups and Their Applications
A ``quantum" representation of the mapping class group of a surface is one coming from a 2+1 TQFT. Certain quantum representations can be built using colored ribbon graph invariants. For special families of these representations a second meaning arises for ``quantum". In particular, for these families, in the limit of a parameter classical topological information can be recovered (a property called asymptotic faithfulness). After introducing these representations and discussing asymptotic faithfulness we will dive into some applications both within mathematics and topological quantum computing.