# Summer Project Examples

The following link will give you an idea of what to expect from an undergraduate summer project.

http://www.sussex.ac.uk/maths/ugstudy/summerprojects

## Optimal alignment of random sequences drawn from a finite alphabet

Faculty Supervisor: Dr Nicos Georgiou

Consider three words in different languages, for example Henry, Enrico and Heinrich. The alphabet used to construct these words is the Latin alphabet and just by looking at them we are convinced that the words "look the same". One way to make this rigorous is to "align" the words so that similar letters line up. For example:

H e _n r y _ _

H e I n r I c h

_ E _ n r I c o

and then introduce a score function for which aligned letters gain a score +1, mismatched letters (like h and o in the last two words) get penalised by a -a and gaps (denoted by underscores) are penalised by -b.

An alignment is called optimal if it achieves the highest possible score from all possible aligments between the words and a high score indicates a higher probability that the words are "similar". This can be used for example when comparing DNA sequences of two species.

While the example above is deterministic, once can add randomness to this by creating two words over any finite alphabet with random letters, and then trying to find the optimal score and alignment, as well as the behaviour of these scores when the alphabet size k, costs a, b and size n change.

## Nonlinear Differential Equations in Finance

Faculty Supervisor: Dr Max Jensen

Many problems in finance lead to interesting nonlinear differential equations. The aim of this project is to develop rigorous tools to analyse these problems, for example to study the existence and uniqueness of solutions. (The financial modelling leading to the differential equations is not part of the project.)

The project could focus on one or two more advanced aspects. Examples are:

a) Analysis of the Black-Scholes equation with transaction costs

b) Option pricing with unknown volatility

c) Theory of fully nonlinear differential equations and viscosity solutions

## The Monge--Ampère equation on Riemannian manifolds: analysis and computations

Faculty Supervisor: Dr Omar Lakkis

The Monge--Ampère equation is at the same time one of the hardest yet ubiquitous partial differential equations. It arises in fields as diverse as optimal mass transportation, geometric optics and computational meteorology. In this project we are interested in understanding the analysis of the Monge--Ampère equation on Riemannian manifolds following the breakthrough papers of McCann (2001) and Brenier (1991). We may also look at the possibility of extending some of the known numerical methods, especially that of Lakkis and Pryer (2013) to the case of surfaces of positive curvature in \$R^3\$.

## Dynamical Systems: Lyapunov functions and their computation

Faculty Supervisor: Dr Peter Giesl

Dynamical systems study systems that evolve in time. Differential equations naturally lead to dynamical systems: given an initial condition, the system evolves over time. Examples of dynamical systems from differential equations come from a variety of applications such as Physics, Engineering, Finance, Biology, Sport Sciences and many others.

An equilibrium is a constant solution. The stability of an equilibrium describes the behaviour of nearby solutions, and all solutions tyhat approach the equilibrium in the long run are called its basin of attraction. The basin of attraction can be determined by a Lyapunov function, i.e. a function that decreases along solutions of the differential equation. Level sets of the function will then determine the basin of attraction.

We are interested in construction methods for such a Lyapunov function. There exist several methods, requiring different mathematical techniques such as numerical approximation or linear optimization. For the latter method, the Lyapunov function will be a piecewise linear function.

## Foundations of Fourier analysis

Faculty Supervisor: Dr Masoumeh Dashti

Consider the most natural shape of a vibrating string, that is, the triangle of a plucked string as it is released. Around the middle of the 18th century, Daniel Bernoulli claimed, on physical grounds, that the motion of the vibrating string could be expressed by an infinite trigonometric  series. This was based on the intuition that any mode of vibration results from the superposition of simple modes. We now know that Bernoulli's intuition was correct, but it was well into the 19th century before a more clear understanding of trigonometric series was obtained. The fact that the triangular shape of a plucked string can be represented by a series showed that a series representation does not guarantee differentiability. Later, questions regarding continuity and convergence of trigonometric series led to the development of the theory sets.

After Jean-Baptiste Joseph Fourier applied the trigonometric series very successfully to the theory of heat, they are called Fourier series. They have however, as briefly mentioned above, proven valuable all over mathematics, from partial differential equations to the theory of sets and numbers.