Geometric Diagrams for Gaining Insights Into the Microstructure of Metals
Voronoi diagrams, considered as early as 1644 by philosopher Rene Descartes, divide the space into regions close to each of a given set of points. These diagrams are not only interesting from a computational geometry point of view (how can one compute such diagrams efficiently?), but they are also related to data science (how can one “cluster” data points?) and, most recently, material science (how can one model and simulate microstructures in metals?). In this project we will explore some of these aspects from a mathematical perspective. Depending on interest, there will be opportunities for analysing, computing, and experimenting with these diagrams.
https://alexbeutel.com/webgl/voronoi.html
http://www.ams.org/publicoutreach/feature-column/fcarc-voronoi
The geometry of complex functions
It is well-known that the square of any real number is non-negative, and that negative numbers therefore have no real square roots. Complex numbers - which you may have already encountered in school - are an extension of the real numbers in which every number has a square root.
Unlike integers, fractions and real numbers, which are arranged along a number line, complex numbers form a two-dimensional plane. While they may at first appear exotic - are numbers not meant to measure quantities? - it turns out that complex numbers have natural and beautiful geometric properties. The goal of this project is to explore these properties, and in particular understand the geometric meaning of operations such as addition, multiplication and exponentiation. It turns out that such complex functions are closely related to "conformal" - angle-preserving - transformations, which play an important role in cartography.
For students with some programming experience, there will be an opportunity to develop computer programs that visualise the way that complex functions transform the plane.
Computer-assisted proofs
Recently, there has been a substantial interest in computer-assisted proofs: The formal verification of mathematical arguments by computer. In particular, there is a considerable community of mathematicians who are using the proof assistant Lean (https://leanprover-community.github.io/). In this project, students will first work through the "Natural Number Game" (https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_game/), and develop experience both in formal mathematical reasoning and in working with Lean from there. The goal of the project is to be able to produce formalisations of some simple mathematical proofs - perhaps even some that have not previously been formalised.
Calculating Orbits
Chaos and fractals
Dark matter phase transition
This project is designed for students with an interest in computational data analysis and machine learning. The physics context is the possibility that dark matter may have undergone a phase transition --- analogous to boiling water --- in the early universe, leaving an observable imprint in the form of gravitational waves. While you're welcome to learn about the physics details if interested, they don't need to be understood to work on the computational task: using machine learning to classify data between the two phases involved in the transition. You will begin by working through a warm-up tutorial using machine learning to classify a standard set of images. The challenge is then to adapt the tutorial into a program that operates instead on the physics data. For more information see https://www.tinyurl.com/ULMaS-DMPT.
Percolation, fractals and critical phenomena
Interactions between a large number simple physical systems (such as e.g. atoms in magnetic field) can lead to emergent phenomena ("criticality") with complexity that goes far beyond the simplicity of each individual system. In this project, we'll look at "percolation" of fluid (or, alternatively, electric current) in disordered landscapes consisting of elementary blocks/cells that are either "passable" or "impassable". Despite their simplicity, such disordered system exhibit the emergence of percolating clusters with beautiful and nontrivial fractal geometry. Working on this project, you'll learn about practical approaches to fractals such as fractional (Hausdorff) dimensionality.
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