# MSI Honours Conference - November 2017

## Date & time

## Location

CBE Building 26C, Lecture Theatre 3

## Speakers

## Event series

## Contacts

*Bryan Wang*

The MSI Honours Conference is one of the exciting annual events in MSI. It is the time for the Honours students to showcase their research projects and share with us their discoveries.

The event is open and free to all members of MSI.

## Program:

#### November 3

9:30 — 10:20

Speaker: Owen Cameron

Title: Universal Compression/Prediction of iid Processes

Abstract: The Shannon entropy of a random data source gives a meaningful limit on the compressibility of its random output. Consider the problem of compressing a sequence of iid variables when the `true' distribution is not known. One algorithm is shown to compress 'nearly optimally' (with respect to Shannon's notion of optimality), even in the absence of any knowledge of the true parameters. In my thesis, we investigated whether this algorithm also performs 'near optimally' if the source parameters change (a form of robustness). An original result gives a guarantee that it does.

**10:30-11:00 Morning tea (MSI common room)**

11:00 — 11:50

Speaker: Joel Martin

Title: TBA

Abstract: TBA

**12:00 — 13:30 Lunch Break**

13:30 — 13:55

Speaker: Alexandra Grant

Title: Tilings from Graph-Directed Iterated Function Systems.

Abstract: This talk explains how to produce infinite families of tilings associated with certain graph-directed iterated function systems. These tilings satisfy nice properties including being quasiperiodic and having a finite prototile set. Additionally, we will explain when a conjugacy exists from a symbolic dynamical system with the shift operator to the tiling dynamical system.

14:00 — 14:50

Speaker: Qiuyi Li

Title: Generalized linear mixed models for count data

Abstract: As an important generalization of linear statistical models, the generalized linear mixed models (GLMMs) have become increasingly popular over the recent decades. Given an unobserved vector of random effects, the response variables are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function in GLMMs. GLMMs are one of the best tools for analyzing non-normal data. However GLMMs are surprisingly challenging to use due to the difficulty to obtain the maximum likelihood estimates. In the first part of this thesis, we will explore the estimation methods of GLMMs (especially for count data) and a few generalizations will be introduced to improve the accuracy or the efficiency of the estimation. The methods of modeling overdispersion in the Poisson mixed models will be discussed in the second part of the thesis. This thesis aims to provide a rigorous mathematical basis for the general techniques for parameter estimation and modeling the potential overdispersion in GLMMs and to explore the features for each method.

#### November 6

9:30 — 10:20

Speaker: An Ran Chen

Title: Generalized Frobenius-Schur indicators for spherical fusion categories

Abstract: In this talk we discuss generalized Frobenius-Schur(FS) indicators for spherical fusion categories. Generalized FS indicators have proven to be a useful tool for analyzing fusion categories and have many applications. The aim of this talk is to first give an introduction to these indicators and then highlight their application in computing torus link invariants.

**10:30-11:00 Morning tea (MSI common room)**

11:00— 11:50

Speaker: Samuel Quinn

Title: TBA

Abstract: TBA

**12:00 — 14:00 Lunch Break**

14:00— 14:50

Speaker: Caitlin Mattner

Title: Prime Numbers in Short Intervals

Abstract: Although primes are simple to define, their distribution is not well understood. A classical, and as of yet unresolved, question about primes is Legendre's conjecture, which states that there is a prime between every pair of consecutive squares. In my thesis I explored the work that has been done on the slightly easier problem of primes between cubes and more generally mth powers. The value of m such that there is a prime between n^m and (n+1)^m for all n greater than or equal to 1 was reduced.

**15:00 — 15:30 Afternoon tea (MSI common room)**

15:30 — 16:20

Speaker: Weiqiong Zheng

Title: TBA

Abstract: TBA

#### November 7

9:30 — 10:20

Speaker: Angus Gruen

Title: Computing Modular Data using Representation Theory of Hopf Algebras

Abstract:

Both fusion categories and modular tensor categories are important areas of study in category theory due to their frequent occurrence and their links to physics.

This talk will explain the what a ribbon fusion category is and develop a graphical calculus that can be used with such a category to simplify proofs. From this graphical calculus we will explain what the modular data is and give a definition of a modular tensor category. Then we will explain a construction called the Drinfeld centre, one of the main methods for producing modular tensor categories from fusion categories. We then describe methods used to calculate the modular data when the modular tensor category is given by representations of a Hopf algebra. As a case example we analyse the representation category of a Hopf algebra known as the quantum double of a finite group.

**10:30-11:00 Morning tea (MSI common room)**

11:00— 11:50

Speaker: Kie Seng Nge

Title: Symmetric Group $S_n$ and Linear Braids

Abstract: The braid group $\mathcal{B}_n$ is a group generated by Artin generators $\sigma_i$ for $1 \leq i \leq n-1$ with relations: $\sigma_i\sigma_{i+1}\sigma_i= \sigma_{i+1}\sigma_i \sigma_{i+1} and \sigma_i\sigma_j=\sigma_j\sigma_i \text{ for } |i-j|>1$. There is a canonical surjective homomorphism $\pi$ from the braid group $\mathcal{B}_n$ to the symmetric group $S_n$ . Linear braids are certain distinguished lifts from the symmetric group $S_n$ to the braid group $\mathcal{B}_n$, determined by a splitting of the root set of the symmetric group element. The notion of a linear braid comes from braid group actions on categories.

In this talk, we would like to prove a core theorem – A braid $\beta$ is linear if and only if $\beta$ is the product of a Garside generator and a negative Garside generator. Here, Garside generator refers to an element in the braid group, which has length identical to the length of its image under $\pi$ and it has either a positive or a negative expression. Furthermore, we will give two new presentations to the braid group with respect to the Garside generators and the linear generators by imposing appropriate relations. In fact, we will see that there is a clear relationship between the length function in the Garside generators and the length function in the linear generators.

**12:00 — 14:00 Lunch Break**

14:00— 14:50

Speaker: Edmund Heng

Title: Algebraic Envelope of Groups and their Properties

Abstract: The constant group scheme G, constructed from a given group G is possibly one of the simplest and most typical example of a group scheme. For G finite, the constant group scheme constructed is affine. For G infinite, it is not affine. Nevertheless, we can take the affinisation of such G, which is called the algebraic envelope of G. Despite so, these affine group schemes usually turn out to be non-algebraic (but instead pro-algebraic) and most of the studies on affine group schemes focused on the algebraic ones. We will use these objects as a starting point to dive into the world of non-algebraic affine group schemes. In particular, we will try and understand the algebraic envelope of Z over an algebraically closed field k concretely, which turns out to be the product of the additive group (unipotent part) and the diagonalisable group given by the multiplicative group of k.

**15:00 — 15:30 Afternoon tea (MSI common room)**

15:30 — 16:20

Speaker: James Bailie

Title: Vector Fields on Spheres

Abstract: In 1962, Adams determined the maximum number of linearly independent, tangent vector fields on S^n. In this talk, we will discuss his proof, which involves stable homotopy theory and K theory.