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Administrator of this website: Title and abstract of Noah Snyder were added. An example of gauging the symmetry of modular tensor categories Abstract: Given a unitary modular category with a symmetry, there is a way to construct new unitary modular categories via the gauging procedure.
During this talk, we will define gauging and show some interesting properties of this construction. We will focus on one specific example, 2-copies of the Fibonacci category and the swap action, and we will show how the procedure works in this case.
Reconstructing Spinor Categories Abstract: The 0-part is known from the classification of orthogonal categories, assuming braiding. Its module action is then classified using a type B version of the so-called BMW algebra. This allows the reconstruction of the whole category. Introduction to Coherence in Monoidal Categories Abstract: Categorical versions of notions such as associativity of a product often involve additional assumptions needed to obtain "coherence" results.
I will explain the idea of coherence, and motivate the need for coherence theorems. I will then give a detailed account of the proof of a coherence theorem in the case of monoidal categories, with a focus quantum three box problem algebra how the techniques involved can be adapted to other types of categories.
Modular representation associated to modular categories Abstract: We will show that modular categories give rise to projective representations of SL 2, Z by graphical calculus.
Time permitted, we will discuss the resolution of the anomaly and the Galois action on the modular data. On gauging symmetry of modular categories. Whenever there are group actions, interesting things happen.
In this talk, we will focus on group actions on unitary modular categories, which are mathematical descriptions of symmetries of two dimensional topological phases of matter.
We will introduce the concept of gauging on unitary modular categories following the paper by Cui, Galindo, Plavnik and Wang and work with simple examples. Fusion categories generalize the representation categories of quantum groups, and we think of them as objects which encode quantum symmetry.
All currently known fusion categories fit into 4 families: This might take 2 weeks to explain, so perhaps this will be part 1 of 2. The multi-variable Alexander polynomial is a generalization of the Alexander polynomial to links where each strand is colored by a representation in a different parameter.
After noting some properties of these representations, we will explore their tensor product structure. More specifically, we provide the fusion rules for decomposable tensor products and discuss consequences of this decomposition. Enriched module categories and internal hom Abstract: I will introduce the notion of a V-module category for a monoidal category V, and I will explain the basics of the adjunction calculations used in understanding such categories.
I will review a few basic facts about integrals of Hopf algebras and sketch their use in the construction of 3-manifolds invariants. Particularly, the defining axiom of an integral can be directly understood as an algebraic translation of handle-slide moves both for Heegaard presentations a la Kuperberg and surgery presentations a la Hennings.
Other basic Hopf algebra relations correspond, for example, to handle cancellations. Introduction to the volume conjecture Abstract: The volume conjecture is a conjecture that relates an algebraically defined knot invariant, the colored Jones polynomial, to a geometric knot invariant, the hyperbolic volume of the knot complement.
We will define the hyperbolic volume of a knot complement and look at the example of the figure eight knot for which the volume conjecture has been proven.
On Generalized Metaplectic Modular Categories. Metaplectic modular categories are modular tensor categories whose fusion rules are given by the Verlinde fusing rules of Spin n at level 2.
One can generalize these fusion rules by replacing the cyclic group of order n with an arbitrary finite abelian group A.box. The Hydrogen Atom Series solution for energy eigenstates. The scale of the world.
Quantum Mechanics as Linear Algebra Review of vectors and matrices. Linear algebra in bra-ket notation. Linear algebra The Problem of Measurement Mixtures and pure .
Erwin Schrodinger was one of the key figures in quantum physics, even before his famous "Schrodinger's Cat" thought experiment.
He had created the quantum wave function, which was now the defining equation of motion in the universe, but the problem is . The Three-Body Problem is one of the oldest unsolved problems of classical mechanics. It arose as a natural extension of the Two-Body Prob- Quantum mechanics: Deals with phenomenon at the subatomic scale, vari-ables that are discrete and is probabilistic in nature.
The problem is that most of the mysteries of the SM have to do with what appear to be weakly coupled degrees of freedom, where perturbation theory is a very effective computational tool and I don’t see how quantum computers would help.
Some problems are so complex that, although we know how to solve them in theory, it would take billions of years to find a solution even on the fastest supercomputer.
Come on a trip through the world of complexity and meet the famous P versus NP problem. See this GLL post, and maybe change the problem to that of making a rigorous version of my sketch of the == direction given there.
Page 21, diagram of 4-cycle: Labels '3' and '4' should be switched. Page 25, ex.
The commutators are group commutators, not the ring commutators of the form AB - BA which are fundamental elsewhere in quantum.