# Mathematics

## Finding equiangular lines and their (projective) symmetries

### Supervisor

#### Faculty of Science

#### Project code: SCI017

There are three equiangular lines in R^2 (there cannot be more), and there are four equiangular lines in C^2 (there cannot be more). There is a conjecture that there are d^2 equiangular lines in d-dimensional complex space (these lines have been constructed numerically to high precision as an orbit of the Heisenberg group).

The aim of this project is to construct and analyse complex spherical t-designs for t>2 (spherical 2-designs are equiangular lines). These are "equally spaced" configurations of points on the complex sphere. Ideally, one would obtain an explicit construction of an unknown t-design as the orbit of one or more vectors.

## Useful redundancies - finite tight frames

### Supervisor

#### Faculty of Science

#### Project code: SCI018

In the game of "battleships'' the position of a ship is given by two coordinates. These Cartesian coordinates appeal to mathematicians because they use the minimum amount of information. It is possible to give three coordinates for a ship with the property that if one is lost, or changed, then the position is still known precisely.

Such representations are called finite tight frames, and are increasingly used in applications precisely because of this useful redundancy.

The project will investigate an aspect of finite tight frames appropriate for the student's background.

## Condorcet domains

### Supervisor

#### Faculty of Science

#### Project code: SCI019

Condorcet domains are sets of linear orders such that, if a society restricts themselves to choosing from these, they will never run in the paradox of intransitivity of the majority voting. Condorcet domains appear to be interesting combinatorial objects. In particular, they are closely related to median graphs, Coxeter groups and Bruhat orders.

## Numerical methods for finding homeostasis in biological models

### Supervisor

#### Faculty of Science

#### Project code: SCI020

Homeostasis is an important biological phenomenon, in which the output of a biological system is relatively insensitive to changes in input over a certain range; e.g. body temperature (output) as a function of air temperature (input). One key question is: how can we find homeostasis within (often large) parameter spaces? This project will explore numerical methods for finding homeostasis, using as an example system models of gene regulatory networks (GRN).

In GRNs, the relatively low dimensionality allows one to use a geometric approach in which homeostasis is expected to emerge generically only via certain geometric structures. These geometric insights allow direct methods of finding such structures via construction of augmented systems in which the desired structures are attracting equilibria. Interested students should have at least some background in (ordinary) differential equations and numerical methods, and an interest in biological applications.

## Can undergraduates convert $100 to a foreign currency?

### Supervisor

#### Faculty of Science

#### Project code: SCI021

Numeracy is often associated with the capacity to use mathematics sensibly in everyday situations. Conversion of currency is an instance of such a situation and, in the first glance, it requires basic arithmetic reasoning that any undergraduate should have. However, the situation becomes more complicated when one realizes that currency exchange works differently in different countries. The conceptual jumps that are needed for interpreting and operating with differently represented information may be complicated even for mathematically versed students. How will they cope and succeed?

## What can we learn from students' difficulties and mistakes on their thinking?

### Supervisor

#### Faculty of Science

#### Project code: SCI022

Mistakes are often treated by students and teachers as something to be avoided (marks, in some sense, are an institutionalized punishment for mistaking). For mathematics education research, in turn, students’ difficulties and systematic errors is a rich source for learning about possible ways of thinking. The person working on this project will engage with a variety of written data collected from different populations of students and derive common of ways of their mathematical thinking.

## And what if the students were teaching themselves?

### Supervisor

#### Faculty of Science

#### Project code: SCI023

If asked to think about a typical university teaching, many of us would probably imagine a large lecture theater, a mathematician telling stuff to students, and them taking notes. Now imagine an environment where the students teach their classmates, collaborate on problem solving, and work as a class to improve their proofs. The person working on this project will get access to video-recordings of such an environment, and will explore the potential and limitation of learning that can take place there.

## Evolutionary robotics and heteroclinic networks

### Supervisor

#### Faculty of Science

#### Project code: SCI024

People often compare our brains to computers, but there are many things that we can do that computers can’t and vice versa. If the brain isn’t a computer, then what is it? Evolutionary robotics allows us to investigate non-computational forms of artificial intelligence. A genetic algorithm is used to tune the parameters of a set of equations so that when those equations are used as a robot controller, a desired intelligent behaviour is produced. Most work in evolutionary robotics has focussed on a particular form of coupled differential equations called continuous time recurrent neural networks. In this project we will investigate using a different dynamical structure called a heteroclinic to see how these systems can produce intelligent behaviours such as categorical perception or associative learning.

## The mathematics of buildings on shaky ground

### Supervisor

#### Faculty of Science

#### Project code: SCI025

Structural integrity of buildings during and after an earthquake is an important and difficult problem in civil engineering. Mathematical modelling and analysis is used to gain a better understanding of the way different forces are interacting. Classical textbook models assume that the force experienced at the top of the structure is identical to the force experienced at the bottom, where footing of the structure interacts with the soil. Experimental results indicate that this assumption is false. In this project, you will investigate better modelling assumptions with the aim of obtaining a model with predicting power that is tested against experimental data.

## Higher-dimensional expanders

### Supervisor

#### Faculty of Science

#### Project code: SCI026

Expander graphs are highly connected finite sparse graphs which are of importance in number theory, group theory, geometry and combinatorics, with significant applications in computer science as basic building blocks for network constructions, error correcting codes and algorithms. In recent years there has been strong interest in defining and constructing higher dimensional expander graphs, one such construction arises as a finite quotient of a Bruhat-Tits building in analogy to the spectrally optimal Ramunujan expander graph of Margulis and Lubotzky, Phillips and Sarnak.

## How to turn a pea into the sun by playing ping-pong

### Supervisor

#### Faculty of Science

#### Project code: SCI027

In 1924 Banach and Tarski showed that given a solid ball in 3 dimensions there exists a decomposition of the ball into a finite number of disjoint subsets which can be reassembled in a different way to yield two identical copies of the original ball. A key component in the proof is the ping-pong lemma, a technique first used by Felix Klein in the 19th century and used by Jacques Tits in the 1970s to prove the celebrated Tits alternative.

## Buildings

### Supervisor

#### Faculty of Science

#### Project code: SCI028

In the 1950s Jacques Tits reversed the celebrated Klein’s Erlangen programme by studying groups through their actions on associated geometric objects, called buildings. Two main classes of buildings are the spherical ones, coming algebraic groups and Euclidean buildings, which come from algebraic groups over a local field. Already the theory of one-dimensional affine buildings, better known as trees, is very interesting.

## K-theory and index theory

### Supervisor

#### Faculty of Science

#### Project code: SCI029

K-theory is an invariant that was introduced by Sir Michael Atiyah and Friedrich Hirzebruch in 1959 and bears all the hallmarks of a great mathematical structure. It draws upon and relates several areas of mathematics, and allows for various district realisations. At the time of its inception, it furnished one of the first examples of a generalised cohomology theory. It is further the natural habitat for the Atiyah-Singer index theory, one of the greatest mathematical achievements of the 20th century. Over the past two decades, K-theory has also found a prominent place in physics, particularly in the classification of topological insulators in condensed matter physics and in the study of chiral anomalies in particle physics and D-brane charges in string theory. The aim of this project is to learn about K-theory and explore some of its applications.

## Loop groups and their applications

### Supervisor

#### Faculty of Science

#### Project code: SCI030

Loop groups are an important class of infinite dimensional Lie groups that are ubiquitous in mathematics. They have a beautiful structure theory and representation theory. They have further vast applications to theoretical physics (gauge theory and string theory), number theory and affine Kac-Moody algebras. This project is focused on learning about the basic properties of loop groups and some of their applications.

## Hamiltonian cycles in vertex-transitive graphs

### Supervisor

#### Faculty of Science

#### Project code: SCI031

A vertex-transitive graph is, informally, a graph in which all the vertices are "identical" with respect to the structure of the graph. A Hamiltonian cycle in a graph is a cycle going through every vertex exactly once. There are only five known connected vertex-transitive graphs without Hamiltonian cycles and some conjecture that there are no others. The general problem is probably hard, but various cases can be investigated, including graphs of small order, where a computer can be very helpful.