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.

In this talk, we will discussion these fundamental arithmetic theorems of modular tensor categories. In addition, if time allows, we will mention some classification results and open problems related with gauging.

Such functions are called weight systems. Two constructions of the Jones polynomial, part 2 Abstract: This will be an introductory talk. During this talk, we will define gauging and show some interesting properties of this construction.

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.

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. In this talk, we will start by introducing some of the basic definitions and properties of modular categories, and we will also give some basic examples to have a better understanding of their structures.

It is a general principle of Quantum Mechanics that there is an operator for every physical observable. Vassiliev ascribes to a Lie algebra knot invariant a function on chord diagrams.

Given a unitary modular category with a symmetry, there is a way to construct new unitary modular categories via the gauging procedure. The 0-part is known from the classification of orthogonal categories, assuming braiding.

Gauging is a well-known theoretical tool used in physics to promote a global symmetry to a local gauge symmetry.

Its module action is then classified using a type B version of the so-called BMW algebra. The talk will explain to the audience the current state of the theory. In particular, a construction due to V. 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.

Time permitted, we will discuss the resolution of the anomaly and the Galois action on the modular data. I will explain the idea of coherence, and motivate the need for coherence theorems. Administrator of this website: We will use properties of the Jones Wentzl idempotent to create colored trivalent graphs called quantum spin networks which will lead to methods for calculating the colored Jones polynomial.

Whenever there are group actions, interesting things happen. All currently known fusion categories fit into 4 families: Fusion tensor categories have a rich and fascinating structure.

I will discuss the classification of modular categories with such fusion rules in the case that A is of odd order. A synoptic chart of tensor categories Abstract: Eigenstate, Eigenvalues, Wavefunctions, Measurables and Observables Often in discussions of quantum mechanics, the terms eigenstate and wavefunction are used interchangeably.

The study of Frobenius-Schur indicators has provided new insights on the arithmetic properties of spherical fusion categories. Solving eigenvalue problems is a key objective from linear algebra courses.

The value of the observable for the system is the eigenvalue, and the system is said to be in an eigenstate. Gauging the symmetry of modular categories Abstract: Introduction to Skein Theory and the colored Jones polynomial Abstract: Two constructions of the Jones polynomial Abstract: Introduction to Coherence in Monoidal Categories Abstract: A physical observable is anything that can be measured.

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.

On Generalized Metaplectic Modular Categories. 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. If the wavefunction that describes a system is an eigenfunction of an operator, then the value of the associated observable is extracted from the eigenfunction by operating on the eigenfunction with the appropriate operator.

Reconstructing Spinor Categories Abstract: In particular, the congruence subgroup theorem, Cauchy theorem, and the conjectural congruence properties of modular categories were established via the generalized Frobenius-Schur indicators.

In addition to the mathematical interest, a motivation for pursuing a classification of modular categories comes from their application in condensed matter physics and quantum computing.The very first problem you will solve in quantum mechanics is a particle in a box.

Suppose there is a one dimensional box with super stiff walls. There is a ball in that box that can bounce back. problem can be solved in logarithmic number of quantum queries to the oper- ation table if it is a quasigroup resp.

group. In Section 7 we consider the distributive problem, given a set S and two binary. Quantum Mechanics Made Simple: Lecture Notes Weng Cho CHEW1 September 23, 1The author is with U of Illinois, Urbana-Champaign.

He works part time at Hong Kong U this summer. 3/29 Title, abstract, references for Julia Plavnik's talk (4/4 Tuesday) were added. 3/21 Title and abstract of Diana Hubbard's talk on 3/30 were added.

Below is 12/11 Slides for Eric Rowell's talk were added. 11/17 Slides for Yilong Wang's talk were added. 11/14 Title, abstract, references for Yilong Wang's (11/17) talk were added.

Solving eigenvalue problems is a key objective from linear algebra courses. To every dynamical variable \(a\) in quantum mechanics, there corresponds an eigenvalue.

Because Box A has two of these coins while Box B only has one, it is double as probable that the gold coin came from Box A. (2 gold coins)*(1/3 possibility of being the one chosen) = 2/3.

Therefore the probability that the coin left in the box is a gold one is 2/3.

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