Project Supervision (Wolfram Just)
Topic outline
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The term chaos refers to dynamical behaviour which looks like a random process, while the underlying system does not contain any random component. A classical example of chaos is the turbulent motion in fluids which is at the heart of the limitations of weather forecasting. A main feature of chaos is the so called sensitivity on initial conditions ("butterfly effect"), i.e., that small changes may trigger dramatic changes. Chaos can be found in simple systems of differential equations and the project aims at studying some of the classical examples, predominately either by summarising results available in the literature or by illustrating chaotic behaviour in numerical simulations.
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The term chaos refers to dynamical behaviour which looks like a random process, while the underlying system does not contain any random component. A main feature of chaos is the so called sensitivity on initial conditions ("butterfly effect"), i.e., that small changes may trigger dramatic changes. Chaos can be found in systems which are as basic as one-dimensional maps, where one can on the one hand uncover universal behaviours, and on the other hand may even apply analytical tools such as symbolic dynamics to study chaotic behaviour from a more rigorous perspective. The project aims at studying the chaotic dynamics of one-dimensional maps either by analytical or by computational means.
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Predicting and forecasting market evolution is one of the holy grails in economics and financial mathematics. Within this project we look at time series data of the electricity system price of the Nord Pool spot market. A particular feature of electricity prices is an their inherent periodic component which reflects the variation of electricity consumption during the day. To cope with such trends we employ an idea developed in the context of signal processing where one uses concepts from complex variables and Fourier transforms to discount for such periodicities by computing a time dependent amplitude and a corresponding phase. The project then will look at a statistical analysis of these two quantities (amplitude and phase) to uncover hidden structures of electricity price fluctuations.
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Predicting and forecasting market evolution is one of the holy grails in economics and financial mathematics. Of particular interest is the prediction and explanation of extreme events, i.e., massive price fluctuations. For that purpose we look at the time intervals between such extreme events which display an intermittent pattern. Dynamical systems theory predict a certain distribution of such time intervals (so called on-off intermittency) and the projects aims at testing such an hypothesis by computing the histogram of time intervals between extreme events. The main aim of the project is the detection of scaling behaviours known from so called on-off intermittent systems.
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Predicting and forecasting market evolution is one of the holy grails in economics and financial mathematics. Data analysis tools are used in this context to provide useful information for traders and market participants. Within this project, time series data of the electricity system price of the Nord Pool spot market are used to benchmark mathematical models of economic markets. One aim of the project is the comparison of the actual spot price with simple stochastic models such as geometric Brownian motion. In addition the project may focus on more advanced models which are able to take the fluctuations of real market data into account.
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Predicting and forecasting market evolution is one of the holy grails in economics and financial mathematics. Data analysis tools are used in this context to provide useful information for traders and market participants. Within this project, time series data of the electricity system price of the Nord Pool spot market are used to benchmark data analysis tools.
Of particular interest is to understand whether extreme price fluctuations are related with particular time instances, such as rush hours or busy days within the week, and to uncover correlations in the time series as for instance those induced by electricity consumption as well as to uncover the relevance of non-stationary features. The analysis will be based on a spot price data set and will use tools from statistical data analysis. -
Mathematical tools, such as those introduced in Statistics, Graph Theory, and Network Analysis can be used to understand the performance of football teams and to uncover the strength and weaknesses of players and team managers' strategies. Nowadays data sets are publicly available which can serve as an input for the mathematical analysis of football matches. The project aims at applying various mathematical data analysis tools to study the performance of teams and players for different leagues and competitions.
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Which kind of mathematics can be used to describe e.g. the shape of a tree or a leave, and how is that related to computer generated images used e.g. in contemporary video games. The answer can be found to some extent in modules such as Chaos and Fractals where a mathematical concept, so called iterated function systems, are introduced to generate fractal sets which share some properties with natural shapes. The theoretical aspect of the project aims at a deeper understanding and classification of fractal sets and iterated function systems, while the applied aspect aims at using iterated function systems to obtain computer generated images which resemble natural shapes.
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Classification of objects and data is one of the main aims of what is called machine learning. Various of the mathematical ideas have been inspired by neurobiology and the project aims at understanding some of the basic concepts such as the perceptron, while more advanced ideas, artificial neural networks, can be applied to perform fairly subtle classification tasks. Depending on the particular flavour the project can centre around a mathematical study of machine learning algorithms, or on applications of machine learning for pattern recognition tasks.
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Square matrices are one of the key subjects in Linear Algebra. They are also a key concept in various other disciplines, even beyond mathematics. Square matrices can be fully characterised by their eigenvalue problem. In fact eigenvalues and eigenvectors can be used to define functions of matrices. The exponential function of a square matrix is a fascinating object from the theoretical and the applied perspective. Within this project we study how the exponential of a square matrix can be defined and why this concept is so powerful in mathematics and beyond.
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The simplest example of an oscillator is a pendulum which can be described by a linear second order differential equation. Such linear systems are fairly well understood and can be studied using ideas from linear algebra. If one considers oscillators where nonlinear behaviour becomes important, and which are described by nonlinear systems of differential equations the motion becomes more complex. Depending on the parameters of the system the motion may change considerably and instabilities may occur. The project aims at investigating such changes in dynamical behaviour, a subject which is known in mathematics under the notion of bifurcation theory. The project can be given either theoretical and rigorous flavour by summarising results on various types of bifurcations, or a more applied flavour by studying systems of differential equations by numerical means.
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Partial differential equations occur in various branches of science. Their mathematical study poses substantial challenges and the computation of solutions to partial differential equations is far from trivial. Within this project we focus on various types of partial differential equations which occur in the context of hydrodynamics, such as the Korteweg De Vries equation (for the study of stable localised wave patterns, so called solitons), the Burgers' equation (a simple model which explains shock wave formation, that means the formation of waves used by surfers in Hawaii), or the Kuramoto Sivashinsky equation (a simple model for one-dimensional turbulence). The project can be given a more rigorous analytic flavour or can focus on computational aspects of solving partial differential equations.
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Classification of short texts, say identifying whether email is spam or classifying the views of people about political candidates during an election campaign, is an important aspect in modern day life. Mathematical statistics has developed techniques to retrieve the attitude of a writer with respect to a certain topic from a written document. These techniques are summarised under the notion sentiment analysis. Sentiment analysis uses methods which have emerged in the context of machine learning. The project will explore the use of simple schemes, such as the Naıve Bayes classifier to predict the polarity of texts (i.e positive or negative) and the associated emotion (happiness, sadness, anger, etc). The implementation can be done using standard statistics software such as R.
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Data collection is one of the key issue in any Applied Statistics task. The project aims at addressing a question of your choice by collecting relevant data, e.g., within a survey, and to perform a statistical data analysis to test various research hypothesis. Examples are for instance career opportunities of graduates from different higher education institutions, or the impact of Covid-19 on travel behaviour.
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The motion of flock of birds, a school of fish, or a herd of horses are examples of swarm behaviour, a peculiar example of collective motion. Surprisingly there are general mathematical mechanisms which support the formation of swarms. Their study is the main interest of this topic, and the results can then be used as well in a technological setting, for instance, by optimising the motion pattern of robots.
The most fundamental model of swarm formation is the so called Vicsek model which is a simple illustration of agent based modelling. Each agent obeys simple realistic movement rules which finally result by self-organisation in swarm formation. The project studies the theoretical background of this model and provides the opportunity to illustrate swarm formation with numerical simulations. -
Synchronisation is a ubiquitous phenomenon in science and engineering. Without synchronisation your mobile phone would not work, and in fact, you would be dead as your heartbeat needs to synchronised with your breathing, the so called, cardio-respiratory synchronisation. In a simple form synchronisation can be nicely illustrated by the coherent flashing of fireflies. Such counter-intuitive behaviour can be explained and modelled with the help of some basic and fundamental mathematical concepts. Within the project synchronisation phenomena are analytically studied using a simple model of phase coupled oscillators developed by Kuramoto. Conditions for the existence and the stability of the synchronised state will be obtained by methods developed, e.g., in modules on dynamical systems.
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While computing the roots of a quadratic equation is pretty straightforward, computing the roots of a polynomial of higher degree is already a fairly difficult task. While for polynomials one can still determine the number of roots, the matter becomes even more challenging when equations containing transcendental functions (i.e. equations containing expressions such as exponentials or trigonometric functions) are at stake. One may locate solutions of such equations in the complex plane by making use of techniques from complex variables, in particular by exploiting the concept of analyticity. The project aims at exploring this so called argument principle which gives a simple criterion how to find complex valued solutions of transcendental equations.
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Estimating the size of a population can be a matter of life or death. The tanks produced by Germany in WWII had equipment with a consecutive serial number, i.e., tanks were essentially labelled by integers. The Allies knew about this practise and they were interested to get a good estimate about the number of tanks produced. They had serial numbers from captured or destroyed tanks, i.e., they had a random sample of serial numbers. Given such an information what would be a suitable estimate for the number of tanks produced?
Problems of this type, which appear frequently and which are summarised under the notion of population size estimates, are at the heart of Statistics, when the setup is cast into a proper mathematical language. Within this project we are going to illustrate and apply various basic and advanced Statistics concepts, such as unbiased estimation, hypothesis testing, maximum likelihood estimation, to determine the size of a population from a random sample. In addition, within this setup one can illustrate as well the fundamentals of Statistics, i.e., the distinction between Bayesians and frequentists. The project can be given a theoretical flavour which does not involve any data analysis or computing, or it can be cast into an applied data analysis project using, e.g., the statistics software R. -
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One may wonder why mathematics is so obsessed with rigour and proofs. One can use some simple concepts from Calculus, such as integration, to illustrate why there is need to exercise rigour in mathematics, as otherwise one may end up with false statements. The project aims at addressing the issue for the need of proper proofs using examples from Calculus, such as convergence, continuity, or the relation between integration and differentiation.
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A large class of dynamical systems can be mathematically modelled by differential equations, for instance the equations of motion in mechanics. Such an approach fails when the propagation time of signals becomes important, e.g., in contemporary communication technology, where the speed of light becomes a relevant quantity in fast signal processing networks. Such propagation effects result inequations of motions where the right hand side contains a time delay. Historically the first example of such delay equations occurred in the context of the balancing problem, i.e., when studying the impact of the physiological delay on the ability of humans to balance a stick (or even their own body). Within this project we study the simplest instalment of delay equations, i.e., linear equations where the analysis can be done to a good deal by analytical methods. Among others this analysis helps to understand how time delay impacts on the stability of motion.
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Traffic jams are not just annoying, they do cost the economy an enormous amount of money. jams occur of course, e.g., during rush hour but they are not limited to car traffic. jams may occur as well, e.g., on the internet, in production lines, or even on your DNA (affecting your life at a very fundamental level). Hence understanding and modelling traffic jams from a mathematical perspective has far reaching consequences. We will analyse a simple car traffic model, the so called Nagel-Schreckenberg model, to obtain some insight into the formation of traffic jams. The project uses tools you have learnt in Calculus and Probability modules. Depending on your interests it is possible to base the project on numerical simulations and programming.
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w.just@qmul.ac.uk