Undergraduate Projects

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[5]  Div, Curl, and the Weather (2019)

Institution: Heriot-Watt University

Supervisor: Mark Wilkinson

Second Supervisor: Simon Malham

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Project Description

It is likely that you have met the notion of a {\em conservative vector field} at some point during your degree. It's just a map G from R3 (or a subset thereof) into R3 which is realised as the gradient of a scalar function, i.e. G=Dg for some smooth scalar-valued function g. It might sound like a strange question, but in what ways is it possible to “push this vector field around” in such a way it keeps its conservative nature? It might not look like it, but this question is closely connected to the equations of meteorology which are used to predict the weather!

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In this project, we shall analyse div-curl systems which are coupled systems of elliptic-type partial differential equations. We shall try to understand when solutions of these systems exist and are unique. We shall also explore the connection these systems have with important equations of mathematical meteorology.

 

Prerequisites

Any student who is interested in this project should have an aptitude for real analysis, and should be comfortable with partial differential equations.

 

References

1. Mike Cullen, A Mathematical Theory of Large-scale Atmosphere/ocean Flow, World Scientific (2006).

2. Lawrence C. Evans, Partial Differential Equations (2nd Edition), AMS (2010). 

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[4]  Statistical Mechanics of Hard Sphere Systems (2019)

Institution: Heriot-Watt University

Supervisor: Mark Wilkinson

Second Supervisor: Heiko Gimperlein

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Project Description

We were all taught at school that matter is made up of a very large number of atoms and molecules. For instance, in a typical room, there are approximately 1029 atoms that make up the air. Mathematicians like to build and analyse models of matter, but when a given matter system is made up of 1029 particles with each particle being governed by 2 ordinary differential equations, the analysis of these models can prove a significant (and, sometimes, impossible) challenge! Instead of analysing such a huge number of equations directly, mathematicians often turn to a statistical analysis of the system's behaviour instead.

In this project, we shall think of a gas as being made up of lots of perfect spheres. We shall build and analyse a (statistical) system of partial differential equations known as the BBGKY hierarchy, named after those 5 mathematicians responsible for its original study. We shall prove that solutions to this system exist and are unique, and we shall further analyse these solutions to deduce some physical properties of large systems of hard spheres.

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Prerequisites

Any student who is interested in this project should be comfortable with ordinary and partial differential equations. No knowledge of physics or even statistics is required to embark on this project.

 

References

1. Carlo Cercignani, Reinhard Illner and Mario Pulvirenti. The Mathematical Theory of Dilute Gases.} Applied Mathematical Sciences 106, Springer-Verlag New York, 1994.

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[3]  The Heston Model: Variable Volatility in Option Pricing Theory (2018)

Institution: Heriot-Watt University

Supervisor: Mark Wilkinson

Second Supervisor: Simon Malham

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Project Description

The Black-Scholes model and the associated pricing PDE is an excellent first model to consider for the pricing and hedging of options.  However, it is well known that the model fails to account for many empirically-observed features of real time series.  The Heston model uses more complicated stock price dynamics, known as stochastic volatility, to be able to better capture these empirically-observed phenomena, and is a standard model used in industry.

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This project will initially focus on the numerical analysis of the Heston model and the implementation of its sample paths to value a 2-year ATMF (at-the-money forward) vanilla European put option. Depending on the progress made, the student might also look at the problem of using historical market data for a particular asset to determine “reasonable” values for the free constants in the model. 

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Prerequisites

A successful student on this project will certainly be interested in using his mathematical skills in the financial sector! The student will also be proficient in MATLAB and be comfortable employing numerical methods for difference and differential equations. Knowledge of measure theory and stochastic differential equations is preferred, but is not necessary.

 

 

[2]  Functional Equations for the Boltzmann Equation (2018)

Institution: Heriot-Watt University

Supervisor: Mark Wilkinson

Second Supervisor: Heiko Gimperlein

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Project Description

The Boltzmann equation is a well-studied PDE which is a model for the evolution of a rarefied gas of spherical particles. Its attention from the mathematical community, in the past couple of decades, has been considerable. Indeed, 2 Fields medallists (P. L. Lions, http://www.mathunion.org/general/prizes/fields/prizewinners/o/General/Prizes/Fields/1994/ lions.lecture.ps and C. Villani http://www.ams.org/notices/201011/rtx101101459p.pdf) were cited for their respective work on the Boltzmann equation when the announcements of their medals were made. Despite decades of intense work on the Boltzmann equation, many interesting and difficult problems remain open.

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This project will focus on one particular area of the theory of the Boltzmann equation, namely {\em functional equations}. You may have already encountered the famous Cauchy functional equation for the unknown f during your undergraduate studies:

f(x+y)=f(x)+f(y), which should hold for all x, y in R. If f is assumed to be measurable, it can be shown that f is necessarily linear. Interestingly, if no assumptions are placed on f, by assuming the Axiom of Choice it was shown by Hamel that the functional equation admits infinitely-many ”wild” functions which are not linear!

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Although you might not be able to appreciate it quite yet, this equation and others like it are very important in the theory of the Boltzmann equation. In this project, you will:

1. Investigate collision invariants, which are solutions of an important functional equation in kinetic theory;

2. Try to prove a new result on collision invariants for rotating spheres;

3. If time permits, perform a spectral analysis of a linearised Boltzmann collision operator.

 

Prerequisites

A successful student on this project will have an affinity for pure mathematics (despite the physical origins of the problem!). It would also be helpful if the student has some knowledge of functional analysis and partial differential equations.

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References

1. Carlo Cercignani, Reinhard Illner and Mario Pulvirenti. The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences 106, Springer-Verlag New York, 1994.

 

 

[1]   Thermodynamics of Hard Particle Systems (2018)

Institution: Heriot-Watt University

Supervisor: Mark Wilkinson

Second Supervisor: Bryan Rynne

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Project Description

It is commonly accepted that matter in the Universe is made up of atoms and molecules. However, from the mathematician's perspective, establishing global-in-time existence theories for the equations of motion of N particles -- and proving interesting qualitative properties of these solutions -- can be extremely problematic! One need only turn one's thoughts to the three-body problem (and Poincaré's efforts thereon) to convince oneself there is work here to be done.

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One model for particles in the realm of classical physics, namely the theory of rigid bodies, goes back to the time of Newton and Euler. However old the subject of rigid body physics might be, it is still providing some mathematical surprises and challenges to the present day. 

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It can be shown that solutions of Newton’s equations of motion subject to the classical constraints of conservation of (i) linear momentum, (ii) angular momentum and (iii) kinetic energy are not guaranteed to be unique. This begs the question: which of these solutions of the equations of motion corresponds to “reality”?

In the hope of making some progress on this question, in the project we shall do the following things:

1. Using methods of functional analysis, construct families of global-in-time weak solutions to Newton's equations of motion;

2. Simulate the dynamics of N hard particles in a closed box (or on a torus, in the case of periodic boundary conditions) in the absence of external forces;

3. Investigate the initial behaviour of thermodynamic functionals associated to the particle system (such as pressure, temperature, or free energy);

4. If time permits, investigate the response of the particle system to applied forces.

 

Prerequisites

A successful student on this project will be willing to learn about (a) rigorous existence theories for ODEs subject to constraints, and (b) classical and statistical physics. Such a student should also be comfortable programming in MATLAB (or any reasonable language).

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References

1. C. A. Truesdell III. A First Course in Rational Continuum Mechanics. Vol. 1, Second, Pure and Applied Mathematics, vol. 71, Academic Press, Inc., Boston, MA, 1991.

2. Patrick Ballard. The Dynamics of Discrete Mechanical Systems with Perfect Unilateral Constraints. Archive for Rational Mechanics and Analysis (2000) 154: 199. doi:10.1007/ s002050000105.