# MSI 2017 mid year Honours Conference

## Date & time

## Location

John Dedman 27, Room G35

## Speakers

## Event series

## Contacts

*Bai-Ling Wang*

Time: 9:30-10:20

Speaker: Yu Zheng

Title: On the integrable dynamical systems in statistical mechanics

Abstract: One-dimensional Ising model is introduced and solved statistical mechanically by partition function, quantum mechanically by Bethe ansatz and algebraic Bethe ansatz, and stochastically by master equation. Bethe ansatz are also applied to mass transport models including asymmetric simple exclusion process and zero range process. Several two-dimensional surface growth models and their corresponding universality classes are then discussed.

**10:20-11:00 Morning Tea**

Time: 11:00-11:50

Speaker: Zhengyuan Dong

Title: Weak Solution of p-Laplacian equation

Abstract: I will be giving a talk about property of the weak solutions of the $p$-Laplacian equation:

$$\text{div}(|Du|^{p-2}Du)=0$$ for $1<p<2$, including existence, uniqueness theory, differentiability and regularity results.

Time: 12:00-12:50

Speaker: David Quarel

Title: On a numerical upper bound for the extended Goldbach conjecture

Abstract: The Goldbach conjecture states that every even number can be decomposed as the sum of two primes. Let $D(N)$ denote the number of such prime decompositions for an even $N$. It is known that $D(N)$ can be bounded above. This bound includes a constant, $C^*$, called Chen's constant. It is conjectured that $C^*=2$. In 2004, Wu showed that $C^* \leq 7.8209$. We attempted to replicate his work in computing Chen's constant, and in doing so we provide an improved approximation of the Buchstab function. We also provide motivation for improving $C^*$, by discussing other results that are conditional improvements of $C^*$. We numerically extended Wu's results, and provide empirical evidence that any extension using Wu's work would likely provide a negligible improvement to $C^*$.

Time: 14:00-14:50

Speaker: Zack Wei

Title: Mutation and Genetic Drift in the Wright-Fisher & Galton-Watson model of population genetics

Abstract: Allele frequencies in a population may change due to four fundamental forces of evolution: Natural Selection, Genetic Drift, Mutations and Gene Flow. Population genetics studies that explored these processes have traditionally assumed a Wright-Fisher population model. Yet due to its constraint of a fixed population size, an arguably more realistic alternative is to consider a population governed by a Galton-Watson branching process, which allows for dynamic stochastic growth. From scratch, we construct both the 2-allele Wright-Fisher and 2-allele Galton-Watson models in this study, with only the effects of mutations and genetic drift assumed to be in force. Analytic solutions of the probability density function corresponding to the population are derived and verified against numerical simulations. A comparison between the models reveals a similarity between the two when mutations are switched off, yet a significant difference in their long term behaviour can be observed when mutations are introduced. We subsequently apply the Galton-Watson model to validate previous findings in regards to the dating of the Mitochondrial Eve, which is estimated to be around 125000 BC, with scarcely over 1000 people alive at the time.

Time: 15:00-15:50

Speaker: Yanbai Zhang

Title: From the Temperley-Lieb categories to topic code

Abstract: In this thesis our aim is to show the ground states of Hamiltonians on 2-dimensional surfaces, which is made up by putting together stabiliser operators corresponding to edges(vertices) and faces, coincide with topology of surfaces. we first introduce the Temperley-Lieb categories and then define the skein module of TL( \delta= 1), which is the specialization of the Temperley-Lieb category at \delta= 1. Then we introduce the Levin-Wen models, which is the generalization of the toric code, with general defined projections (operators in quantum physics) corresponding to edges and vertices. We then prove the kernel of Hamiltonian in the toric code is isomorphic to the skein module of TL( \delta= 1) and therefore only depends on the topology of surface it bases on.