Tomorrow's Mathematicians Today

Abstracts

Abstracts for Keynote Speech - Morning Session - Airy Stream - Burnside Stream - Comrie Stream - Eddington Stream - Halley Stream - Afternoon Session.

Important note: abstracts involving mathematical symbols have been edited to try to make them readable on the website. Our apologies for any errors or ambiguities in this editing.

Keynote Speaker

The keynote speaker is Professor Ian Stewart FRS, who will talk about "Mathematical Curiosities and Treasures from Professor Stewart's Cabinet". (Ian Stewart's website - Ian Stewart's Wikipedia entry) The photo is by Avril Stewart from Wikipedia Commons.

Abstract: Professor Stewart’s Cabinet of Mathematical Curiosities, a surprise 2008 Christmas bestseller, is now joined by his Hoard of Mathematical Treasures. Both books are mathematical miscellanies for the general public, ranging from one-liners to mini-essays on the great problems and applications of mathematics. The lecture will present a selection of their contents, in an accessible and highly illustrated way, to demonstrate that maths is enjoyable, interesting, and applicable to the real world.

Morning Session

Nic Mortimer, University of Greenwich, How to Play the ‘Weakest Link’ Optimally: an exploration into the mathematics behind the popular game show
Abstract: I intend to present my research on devising an optimal strategy for playing “The Weakest Link”. This will include when to “bank” money depending on the varying skills of your opponents, how to vote off your fellow contestants, some unconventional thoughts on answering questions incorrectly on purpose, and how to give yourself the best chance of surviving if you undergo that greatest sin of answering a question incorrectly in the very first round. I will begin the presentation with a short introduction/reminder of the rules, for the audience to familiarise themselves (I am assuming a large percentage of the audience will have seen the show at least once).
My inspiration comes from the book “How long is a piece of string?” by Rob Eastaway and Jeremy Wyndham and also from watching many hundreds of episodes of the programme myself.
I hope that my short lecture will make people think a little deeper about game shows and games in general, especially with regards to the mathematics behind them.

David Sims, University of Oxford, Cayley and the Abstract Group
Abstract: Arthur Cayley is often regarded as the grandfather of abstract group theory for his work in the middle of the 19th Century. That he has three fundamental concepts in group theory named after him (Cayley Tables, Cayley Graphs and Cayley's theorem) only seems to re-enforce this claim. However, when one takes a closer look at Cayley's group theory questions begin to arise: was it really as pioneering as it is nowadays claimed to be? Was Cayley's definition of a group really that abstract? How deep were Cayley's insights? And did he really come up with the theorem named after him? In this talk I will attempt to address some of these issues (though probably not all, depending on the time available), as well as Cayley's influence on later generations of group theorists.

Anupam Das, University of Oxford, Talking about this Sentence
Abstract: What are paradoxes? Is "This sentence is a lie" true? Mathematicians and philosophers have discovered a massive variety of paradoxes; some of these peculiarities pose a real problem in how we formalise mathematics in the language of logic, although they have also been used to yield important fundamental results, for example in Godel's famous Incompleteness Theorems. In this talk the speaker will look at the common themes in certain paradoxes (in particular self-reference) and how the paradoxes are avoided when formalising mathematics.

Yasmeen Chaudhry, University of St Andrews, Mathematics in the Development of Art and Architecture in Renaissance Italy
Abstract: The Renaissance was a period of cultural and intellectual reform throughout Europe which bridged the transition from the Middle Ages to early modern day. Beginning in Italy, this era generated several polymaths who amalgamated their wisdom of various subjects, bringing together the previously disjointed studies of mathematics and art.
Giotto was the first to bring naturalism into art and other artists also began attempting to replicate their perception of the world around them, adding a sense of reality into art. This stimulated, in particular, the revolutionary development of linear perspective. The slow but steady mastery of this enhancement can be traced in the work of various artists, of whom some were particularly adept.
Geometry, proportion, the golden ratio and Archimedes’ On the Sphere and Cylinder also played an important part in the development of art and architecture. Domes such as the Pantheon’s required advanced mathematics for their construction, and Leonardo da Vinci‘s Vitruvian Man epitomises his passion for replicating the proportions of the human body in architecture. Brunelleschi and Alberti were such firm believers in the beauty of mathematics that they did not believe any building could be beautiful without being perfectly proportional. Modern art and architecture is indebted to the pioneering masters of the Italian Renaissance.

Steve Cockram, University of Exeter, How can we Measure the Radius of the Earth?
Abstract: The presentation gives an explanation of multiple ways of measuring the curvature and by extension radius of a sphere, such as the Earth.
Measuring such a quantity is very easy when you are on the outside of a surface looking onto it, but to accomplish while you are on the surface is significantly more difficult. There are a few clever intrinsic methods to accomplish this. These will involve thinking about the limitations of Euclidean geometry, as well as using geometric and topological ideas to come up with a method that will actually work.
Additionally a small history of the usage of these techniques will be given, including the technique used by the Alexandrians and also Egyptian slaves.

Erika Nilsson, University of Aberdeen, The Banach-Tarski Paradox
Abstract: By a theorem proved by S. Banach and A. Tarski in 1924, it is possible to divide a solid ball in 3-space into a finite number of non-intersecting pieces, which can then be reassembled, using only rotations and translations, to form two identical copies of the original ball. The result holds for any dimension greater than 3 and may, in dimension 3, be achieved with as few as five pieces. Besides being highly counterintuitive, this theorem is an illustration of the fact that the concept of 'volume', or 'measure', in R^n is not as straight-forward as one might expect. This talk will outline a group theoretic proof of the Banach-Tarski theorem, using 24 pieces, and will also indicate why the result relies in a substantial way upon the Axiom of Choice.

William M.R. Simpson, University of St Andrews, The Sun under Stress
Abstract: Solar flares involve the sudden, fast release of magnetic energy from the sun. These explosive events can significantly affect terrestrial space and life on Earth, and cannot (at present) be predicted. The mechanism generally believed to be responsible for this energy release is magnetic reconnection. But just where and how the energy is stored is less certain. It is hypothesised that, in the time leading up to a flare, the magnetic field in the vicinity of the solar flare site becomes increasingly ‘stressed’ until some critical point is reached, and reconnection is initiated. By mathematically modelling a flaring active region with a time-indexed, point-source topology, we were able to relate groups of separator field lines (tracked through time) to probable reconnection locations (deduced from hard x-ray observations) by studying the stresses they developed close to the time of the solar flare. Many of the connectivity groups that evidenced significant stress were found to be associated with reconnection sites, suggesting a strong correlation between the two phenomena. This may be a small but significant step towards predicting solar flares.

Kathryn Garcin, University of St Andrews, Bemusing the Public with the help of Combinatorics
Abstract: The advanced study of Mathematics can have an isolating effect in social settings. You are much more likely to face blank stares when describing your most recent proof than the history major discussing his research. Because of this information gap, I propose we use our knowledge for entertainment and inspiration when meeting the general public.
There are many mathematics-based number guessing ‘tricks’. These involve a contestant choosing a number and the host correctly identifying the choice based on truthful yes or no answers to given questions. Many of these games are accessible to the general public and require only simple sums or inverse operations. This paper instead explores perfect codes and projective planes and how they can be applied in developing a more advanced game. The error correcting nature of a (7,4) Hamming code even encourages a contestant to lie to a question of their choosing with the host able to correctly identify both the lie and the original number, thus making the process more interesting for both parties.
This is a single example of a bridge for mathematics majors to exploring the use of their subject for entertainment and a more general audience application.
When explanation of our subject is futile, use it to bemuse.

Airy Stream - Applications

Alex Cole, University of Greenwich, The Mathematics of Man
Abstract: In 1954 the eminent anthropologist Claude Lévi-Strauss introduced the methods from pure maths into the study of social relations in anthropology, the Mathematics of Man. He was interested in the study and comparison of populations at the level of the family, the atom of kinship. This structuralism, as it became known, used theory and techniques usually associated with pure mathematics; graph theory, group theory and permutations.
In this presentation I will introduce the discipline of anthropology to the audience by identifying the main fields and introducing some of the important names. I will then focus on cultural anthropology and how graph theory has been applied to prescriptive kinship systems.

James Anderson, University of Oxford, Phase-Type Distributions in Insurance and Finance
Abstract: Phase-type distributions are a flexible class of distributions for which several problems in insurance and finance have found explicit solutions. They are the distribution of time until absorption of a continuous time Markov chain with one or more absorbing states.
The presentation will follow a similar course to my fourth year dissertation. It will begin by establishing basic properties about phase-type distributions and consider the application of these properties to a compound poisson risk process with phase-type claim amounts, including the distribution of surplus at ruin calculated explicitly. It will then generalise to models with non-exponential waiting times, phase-type renewal processes, and look at the possibilities brought by this. Furthermore, phase-type jump sizes in more general Levy processes have found applications in pricing of American and Russian options, which may or may not be discussed dependent on the development of the dissertation at this stage.

James Howe, University of Greenwich, An Introduction to Modern Cryptography
Abstract: My topic choice for the presentation is Cryptography. It is one of the first subjects that got me into mathematics and I enjoy everything about it.
The structure will be a brief historical timeline, but with some of my favourites in greater detail, probably my favourites being the Enigma code and Public Key cryptography, especially prime number public key which they use for online banking security. Finally I will conclude with the future of cryptography.

Adam Sebestyn, University of Greenwich, Comparmental Models in Epidemiology
Abstract: My presentation will focus on different approaches to the modelling of epidemics and diseases. It will show how these can be used to successfully analyse the seriousness of a disease and how it can help to make the proper actions to evade an epidemic. It will show how vaccination can help even if not every person is treated to counter the spread of an illness and even eliminate it from a population. The presentation will focus on compartmental models.

Burnside Stream - Group Theory and Number Theory

Magdalena Jasicova, Royal Holloway, University of London, Multidimensional problems in additive combinatorics
Abstract: My presentation will include overviews and analysis of recent papers written by: Paul Erdos & Paul Turan, Raphael Salem & D. Clayton Spencer, Felix A. Behrend, Leo Moser and Michael Elkin. I am working on a problem very closed to the Moser cube problem: What is the cardinality of the subset of the n-dimensional cube of the side-length 4, containing no three distinct elements x,x+y,x+2y lying on a geometric line of length 3.

Zhu Gong, University of Cambridge, On disjoint systems of residue classes
Abstract: For integer a and n > 0, we call a(n) = a + nZ = {x in Z : x is congruent to a (mod n)} a residue class with modulus n. A finite system A = {a1(mod n1), ... ak(mod nk)} of residue classes is said to be disjoint if the k residue classes in it are pairwise disjoint, i.e. ai(ni) interssect aj(nj) = the empty set for all 1 <= i <= j <= k: A natural extension is a
disjoint system of finitely many left cosets a1G1 ... akGk: In this talk we introduce some fascinating results concerning the disjoint systems. This field is connected with number theory, combinatorics, algebra and analysis. We will also talk about recent progresses on the challenging conjecture concerning the moduli of a disjoint system.

Jonathan Constable, University of St Andrews, Free Groups, the Nielsen-Schreier Theorem and the Andrew-Curtis Conjecture
Abstract: In this talk I provide an overview of free groups and show that every group is isomorphic to a factor group of some free group.
We then consider one of the most well-known and important theorems for free groups: The Nielsen-Schreier theorem for free groups. I will outline this theorem and Nielsen's approach to its proof via Nielsen transformations based on a version of Nielsen's proof presented in D. L. Johnson's book "Presentations of Groups".
Finally I will provide motivation for the study of free groups by discussing the Andrew-Curtis conjecture, one of the leading open problems in group theory as listed by the New York Group Theory Cooperative on their website.

Andrew Bestel, University of Glasgow, Visualisations and calculations of Nil Geometry
Abstract: The 3-space endowed with the structure of the Heisenberg matrix group is known as Nil geometry. The group action is defined as the usual matrix multiplication so, maybe as one may expect, translations are non-commutative. A number of models of Nil geometry can be developed each offering different benefits for studying the space. As such some of these models will be utilised to prove some fundamental results and also to visualise the space. This will give an exciting opportunity to see objects of Nil geometry sitting in Euclidean 3-space through writing a computer graphics program. With this, simple geodesics will be defined. From these more elaborate constructions can be made. The unit sphere, 3 points joined by geodesics to make a triangle, then a pyramid, and so on. With these objects defined, the user of the program will be able to apply translations to see how the objects distort as they move from the origin. The result of this project will be to give an exciting glimpse at the unusual and interesting world of Nil geometry, while proving some of the underlying mathematics.

Comrie Stream – Probability, Chaos Theory and Set Theory

Rizwanur Rahman, University of Exeter, Title to be announced
Abstract: I would like to talk about statistics. In particular, I would like to summarise the main concepts that I have learnt along with the key background information I have gained through undertaking modules related to probability and statistical theory. I would then like to provide some brief information on how statistics is involved in many real world situations.
Statistical modelling overall is hard and complex. At best they are not wrong rather than right.
Areas involving the applications of statistics include Genetics, Agriculture and Epidemiology.
In general, statistics related study could involve plenty of algebraic manipulation and number crunching but also statistics is a very powerful tool in tackling problems involving uncertainty and/or predicting future outcomes.

Colin Price, Birkbeck, University of London, An Investigation into the stabilisation of Chaotic functions
Abstract: We investigate a chaotic function

X_T+1= 2X_T²-1

As the function is symmetric only positive values of X need to be considered as a negative value of X will give the same function as the corresponding positive value apart from the first tem.
If X is greater than 1 the function will diverge. If X=1 or X= -1/2 the function will be constant. However for other values of X between 1 and -1 the function appears to be chaotic.
In a effort to find how the function will stabilise I inverted it to find the values which generate the stable values of 1 and -1/2 and then continued the process the find the values that generated those values and so on. This produces a possibly infinite string of values which generate the stabilisation of the function.

Izabela Petrykiewicz, University of St Andrews, Iterated Cesaro Averages, Frequencies of Digits and Baire Category
Abstract: In Summer 2009, I took part in Summer Research Project organized by University of St Andrews. Together with three other students (James Hyde, Vaios Laschos and Alison Shaw) under supervision of professor Lars Olsen I worked on frequencies of digits in the expansion of typical number. We wrote a paper titled Iterated Cesaro Averages, Frequencies of Digits, and Baire Category and submitted it to Acta Arithmetica on the 9th of July 2009. I would like to present our findings. Namely, if we fix a positive integer N >= 2, for a real number x in [0, 1] and a digit i in {0, 1, ...,N − 1}, let Pi(x; n) denote the frequency of the digit i among the first n N-adic digits of x. It is well-known that for a typical (in the sense of Baire) x in [0, 1], the frequencies Pi(x; n) diverge as n tends to infinity. In summer we found a substantial strengthening of this result. We proved that for a typical x in[0, 1] all higher order Cesaro averages of the sequences (Pi(x; n))n of frequencies also diverge.

Alexander Collins, University of Leeds, Derived Categories and Equivalences
Abstract: Derived categories are (not uncontroversially) ubiquitous in modern representation theory and their philosophy permeates much of algebraic topology and geometry, even finding its way into parts of physics (well, string theory anyway).
I will try to give some explanation of what these objects are, how they arise and why we should care about them. I will also discuss the various ways in which one can construct equivalences between derived categories: how seemingly unrelated things can turn out to be derived equivalent. As an example of this phenomenon I'll wave my hands at Beilinson's celebrated theorem on the existence of derived equivalences between projective n-space and certain associative algebras.

Eddington Stream – Popular mathematics and history

Lindsay Munroe, University of St Andrews, An Introductory Problem in Combinatorial Game Theory
Abstract: We will be considering the mathematical two-player game nim. The rules are are as follows: the players take turns removing objects from a set of heaps, and on each turn the player can remove any number of objects (at least one) such that they are from the same heap. The player who takes the last object loses. We will analyse the game and deduce the winning strategy.

Khadija Khairoun, University of Greenwich, History of Mathematics
Abstract: My presentation will be about the History of Mathematics, starting from Egyptian Mathematics all the way to present-day mathematics.
Egyptian mathematics can be traced back to as early as 3000BC. Hieroglyphs (a pictorial character) were used on stones to represent numbers so that the Egyptians were able to calculate in their day-to-day lives. Most of what we know of Egyptian mathematics comes from two papyri, The Rhind Papyrus and The Moscow Papyrus. Both were from around 1850BC, and both had many problems. On the papyri, instead of using hieroglyphs, Egyptians used hieratic symbols. Problems in the Rhind Papyrus included simple linear equations and the use of geometry and proportions. Problem 26 in the Rhind Papyrus is: a quantity added to a quarter of that quantity becomes 15. This translates simply to x (a quantity) + (1/4)x (a quarter of that quantity) equals (becomes) 15. x + (1/4)x = 15. This solves to become x=12. Similar problems to the one above occur a lot throughout both papyri.
I will move on from there to Babylonian Mathematics, Greek Mathematics and so on.

Kris Hristakev and Charlotte Vickers, University of Exeter, An Introduction to the Fibonacci Sequence
Abstract: The Fibonacci sequence was popularised in Western Europe by Leonardo Pisano in the 13th century and has since been linked to many real-world phenomena, ranging from the growth of natural structures such as flower heads and shells, to popular and high culture. The Fibonacci sequence is also a fertile ground for studying proof as many fascinating but easily proved results can be derived from its simple defining relation. We will survey some of the results, proofs and applications associated with the Fibonacci sequence. These will include Binet's formula, the connection with the golden ratio and Pythagorean triples. When looking in more detail we can see there are many applications that are perceived to be the same but have significant, underlying differences such as the golden rectangle and the Fibonacci rectangle.

Philipp Legner, University of Cambridge, Analysis of the Game of Nim and some interesting Variants
Abstract: Combinatorial game theory is very different from classical game theory, since it doesn’t involve chance, cooperation or conflict. Therefore combinatorial games can be analysed completely using mathematical theory. The most famous combinatorial game is Nim: the opponents alternately remove some counters from distinct heaps and the player to remove the last counter wins. During this presentation I want to show how to derive and prove an optimal strategy for either player, which involves graph theory, the Sprague-Grundy function and the binary digital sum of the heap sizes (called the Nim-sum). I will then prove the Sprague-Grundy theorem, that every impartial game is equivalent to a
certain game of Nim. This is fundamental to combinatorial game theory and can be used to analyse many variants of Nim, such as Grundy’s Nim.
In the second section of the talk, I want to present “Lucky Nim” – a combination of an impartial game that can be analysed mathematically, and coin tossing. This noncombinatorial game seems to depend on pure chance. However analysing the game using the Sprague-Gundy function gives a very surprising result.

Halley Stream - Applied Mathematics

Daniel Colquitt, University of Liverpool, Frequency Related Localisation of Harmonic Elastic Waves in Stratified Welds
Abstract: Over the past two decades there has been much interest in the propagation of elastic waves in inhomogeneous and anisotropic materials, motivated by the requirement for effective ultrasonic non-destructive examination (NDE) of strongly anisotropic welds. I shall present a novel computational model for elastic waves in a structured weld adjacent to the free surface of an elastic solid. The emphasis is on the interaction of waves with the microstructure of the weld. Effects of localisation will be addressed. A model of a grain structure within the weld is also considered.
I shall briefly outline the governing equations for the propagation of time-harmonic waves in a linearly elastic medium. A simplified model of a typical weld will be discussed and the spectral problem will be formulated. Dispersion diagrams will be presented and discussed together with a series of illustrative finite element computations, to demonstrate the localisation. Finally, I will outline a more realistic model for the weld and present a series of computations which use numerical values taken from physical measurements of a typical weld.
The methods discussed are novel and while at an early stage of development, potentially present alternatives to those methods currently used to design NDE inspec tions.

Sandy Black, University of Glasgow, Self-excited oscillations in collapsible tubes
Abstract: A naturally collapsible tube such as the vein can collapse or oscillate at random. This poses a problem in understanding when and how these events occur. A simple analytical one-dimensional model can be developed to describe the steady and unsteady flow through the collapsible tube, held open at two ends inside a pressurised chamber. By changing certain values of the tube such as the tube length, circumferential bending, inlet velocity and external pressure, then aspects can be measured such as the cross sectional area, average velocity and internal pressure of the tube. Steady flow results in particular show that collapse occurs at half the length of the collapsible tube segment whereas the unsteady flow model presents self excited oscillations for particular values of transmural pressure and tension. These results largely agree in comparison with other experiments and theoretical studies.

Benjamin Harrop-Griffiths, University of Oxford, Helmholtz decomposition of L^p-vector fields on Euclidean n-space
Abstract: The Helmholtz decomposition of a vector field into a gradient field and a solenoidal (divergence-free) field has a wide range of applications in anal ysis, in particular in connection with the Navier-Stokes equations. Origi nally studied as a decomposition of sufficiently smooth, rapidly decaying vector fields on R^3, the decomposition may be generalised to L^p-vector fields on R^n. This talk would aim to give a brief overview of how the decomposition can be formulated, using weak derivatives, as an orthogonal decomposition of L^2-vector fields and extended to L^p-vector fields using Fourier multipliers.

Luke Miller, University of Leeds, Wavelets Models for Diagnosing Fluid Flow in Pipes
Abstract: We consider using wavelets on the inverse problem of determining the nature of fluid flow in a pipe using electrical tomography; an important problem in many industrial processes, and an ongoing research theme at the University of Leeds. Wavelets are powerful tools, only recently developed, and an active research area in statistics. We compute the Evolutionary Wavelet Spectrum 1 and Wavelet Variance 2 of simulated data in an attempt to categorize flow as 'bubble' (desirable) or 'churn' (problematic), using logistic regression to model flow type. To test these models, they were used to predict the flow type of independently generated data sets. By testing over 3000 parameter settings, it was found that the predictor could produce excellent results, the best being a success percentage of 91.3%.
Many exciting areas of mathematics were combined in this project: the use of transformations that have only been discovered in recent times; the application of powerful statistical techniques; and tying these together with the aid of computer programming to develop efficient algorithms.

Afternoon Session

Isabel Peters, University of St Andrews, Mathematics behind Google
Abstract: Searching machines have become indispensable in our society in order to find information through the Internet and their popularity rise every day. With about 40% market share Larry Page and Sergey Brin have created the most famous searching machine in the world - Google. Looking closer at Google’s searching algorithm one can figure out that this is not only a computational algorithm but also involves a lot of mathematics, including matrix calculations and vectors. Google’s Page Rank is based on the idea to examine the links a web page has. The
more links a web page is pointed to the higher the page rank is for this page. However, other factors like the importance of a site (BBC has more importance than a small personal website), the immense structure of the web and the total amount of links pointing to a page make the calculation of a web page quite complex. The question behind the page rank is which properties a link matrix needs to have to get the best and most accurate result. In my presentation I will give an overview of how Google works and what kind of mathematics is needed to calculate an adequate page rank.

Ameli Gottstein, University of Greenwich, The Heat Equation in 1D – derivation and presentation of the solution Green’s function
Abstract: The heat equation (HE) is of significant interest in undergraduate analysis. Solutions such as Green’s function have applications in physics and engineering where they help to model systems. I will in particular use the boundary value problem of an insulated metal rod to derive the HE in one, two and three dimensions (here fore I will assume the audience to be familiar with vector calculus and the basic concepts of PDEs). Further I want to prove Green’s function as a solution, connect to the normal distribution and aim to explain the meaning of it being a Gaussian. For the last part I will talk about conservation of substance, again using vector calculus. My talk will be in two sections, the first one will be the derivation of the HE and the second one will focus on Green’s function.
When I studied this subject first, I felt various areas of mathematics such as differential equations, proofs, vector calculus and things I had learned about in statistics perfectly falling into place. I hope with my talk I will provide an appropriate framework in which the audience can feel the same.

Alexander Kuznetsov, Birkbeck University of London, Galois Theory
Abstract: If asked what was my most inspiring and exciting experience of studying mathematics during these two years I would clearly cite abstract algebra in general and Galois Theory in particular.
Being at first fascinated by the dramatic and controversial life story of the young genius Évariste Galois, who died before his 21st birthday and has since been named the founder of all abstract algebra, I tried then to familiarize myself with this beautiful and elegant mathematical theory.
During the talk I would like to demonstrate the main concepts of the theory by considering two different polynomials: one with a solvable galois group and one, whose galois group is not solvable. Having a quick look then at general quintic I would like to show where this theory came from in the first place.
I will also say a few words about the modern treatment of this theory and its value for the Fields and Algebraic Number Theories and will conclude the talk with some open questions as well as possible further theory extensions which may inspire other young fellows in future
research.