# Mathematics

## What can we learn from students' reflections on their difficulties and misconceptions in math?

### Supervisor

#### Project code: SCI004

Research in Mathematics Education knows quite a lot about misconceptions and difficulties that students can experience. Consequently, teachers and researchers invest a considerable effort in supporting students in these situations. However, it is clear that students overcome many difficulties and misconceptions all the time without us being aware. How and what can we learn from their successful experiences?

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

### Supervisor

#### Project code: SCI005

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-equipped students. How will they struggle and cope?

## Finding equiangular lines and their (projective) symmetries

### Supervisor

#### Project code: SCI010

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 ind-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 sphericalt-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

#### Project code: SCI011

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.

## Predicting lung function from incomplete data: graph dynamical systems under uncertainty

### Supervisor

#### Project code: SCI012

Dynamical systems can be used to help predict lung function from structure. However, experimental limitations mean that structure is only partially known. This project will explore the ways in which we can account for this uncertainty mathematically.

## Numerical methods for finding homeostasis in biological models

### Supervisor

#### Project code: SCI013

Homeostasis is an important biological concept, in which system output is robust over a range of the input (e.g. parameters). However, locating homeostasis in mathematical models of biological systems is not easy to do; this project will explore newly developed numerical methods for locating and identifying homeostasis in a broad class of mathematical models of biological systems.

## Evolutionary robotics and heteroclinic networks

### Supervisor

#### Project code: SCI014

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

#### Project code: SCI015

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

#### Project code: SCI016

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 turn a pea into the sun by playing ping-pong

### Supervisor

#### Project code: SCI017

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.

## K-theory and index theory

### Supervisor

#### Project code: SCI018

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

#### Project code: SCI019

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.

## Polynomial convexity

### Supervisor

#### Project code: SCI020

What is a convex set?  One way to characterize convexity uses linear functions. K is convex if it has the following property: if the value of any linear function at a point z is smaller than the maximum value of the linear function on K, then z is in K. We can use this to develop various notions of convexity using different classes of functions. For example, replacing 'linear functions' with 'polynomials' gives the notion of polynomial convexity. Such sets have interesting geometric and analytic properties.

## Orthogonal polynomials

### Supervisor

#### Project code: SCI021

A system of orthogonal polynomials is a (suitably normalised) infinite collection of polynomials that is an orthogonal basis for the vector space of all polynomials with respect to some inner product, usually given by an integral. For integrals over the unit circle in the complex plane or a real interval, these polynomials have been well-studied (orthogonal polynomials on the unit circle/real line: OPUC/OPRL). In particular, 1) these polynomials obey recurrence relations; 2) the sequence of orthogonal polynomials has interesting asymptotic behaviour (as the degrees increase). In this project we will look at these polynomials, and also look more generally at orthogonal polynomials on polynomial lemniscates (of which OPUC are an example).