## Timetable

**9.00**Registration and refreshments**9.40**Welcome from the Vice-Principal, Professor Bob O’Keefe**9.45**Quick quiz with prizes, Mr Laurence O’Toole**10.20**Jugging: Theory and Practice, Dr Colin Wright**11.30**Small Group Talks**12.00**Lunch**12.15**and**12.45**Tours of campus from outside the Windsor Building**12.45—13.15**Further Mathematics (for teachers only), Ms Gill Buque**13.30—14.00**Small Group Talks**14.20**Mathematics at University, Professor~James McKee**15.00**Cryptography—Everywhere!, Professor Kenny Paterson**15.55**Close and questionnaires

## Main Talks

### Dr Colin Wright: *Juggling: Theory and Practice*

Juggling has fascinated people for centuries. The skilled practitioner will keep several objects in the air at one time, and weave complex patterns that seem to defy both gravity and analysis. In this talk we develop a simple method to describe and annotate juggling patterns. We see how simple mathematics can be used to classify them, to describe those patterns that are known already, and discover a technique for creating new ones.

*While earning his Ph.D from Cambridge University in the 1980s, Colin also learned how to fire-breathe, unicycle, juggle and ballroom dance. He has worked as a research mathematician, a computer programmer, and an electronics hardware designer, and takes time to give talks all over the world on various mathematical topics.*

### Professor James McKee: *Mathematics at University*

Are you interested in studying Mathematics at University? This session will deal with the types of course available and the qualifications required, the ways in which university mathematics is different from or similar to mathematics at A-level, and the careers available.

*Professor James McKee is the Head of the Department of Mathematics. He works at the interface of number theory and combinatorics, with computational leanings. His recent work has mostly been connected with associating algebraic numbers with certain combinatorial objects.*

### Professor Kenny Paterson: *Cryptography—Everywhere!*

Cryptography was once the preserve of emperors and generals but now we all use it (without knowing it) every time we make a mobile phone call, log in to Facebook or buy groceries using a debit card. This talk will try to explain how cryptography uses many different kinds of mathematics to make our world more secure.

*Kenny is a Professor of Information Security at Royal Holloway. His Ph.D is in Mathematics, but he now works at the interface between Mathematics and Computer Science, using any tools that come to hand to solve problems in cryptography. His research has directly led to improvements in the security of communications on the internet.*

## Small Group Sessions

There will be two sessions of short 30 minute talks at 11.30 and 13.30. Each talk will be repeated twice. Some talks will only be given on one day. Here is the timetable.

*Various Proofs of the Inequality of Means*, Dr Yiftach Barnea*The Mathematics of Matches*, Professor Simon Blackburn*Mathematics and the Laws of Nature: A Variation on a Theme of Wigner*, Dr Jens Bolte*Further Mathematics for Teachers*, Ms Gill Buque. For teachers only, from**12.45**to**13.15***Primes, Perfect Numbers and Amicable Numbers*, Dr Rainer Dietmann*Fun Maths Roadshow and Problem Solving*, Further Mathematics Support Programme, Ms Cath Moore and Mr Mark Hughes*Traffic Jams*, Dr Alastair Kay*How is Doing Mathematics in the 21st Century Different from that in “The good old days”?*, Dr Alexey Koloydenko*How to Really Share a Secret*, Professor Keith Martin*Graphs and Friendship*, Dr Iain Moffat*The Mathematics of Sudoku-Like Problems*, Professor Sean Murphy*The Liar Game*, Dr Mark Wildon*Pi: A Fundamental Constant of the Mathematical Universe*, Mr Pavlo Yatsyna

The abstracts (i.e. short summaries of the talks) are below.

*Various Proofs of the Inequality of Means*, Dr Yiftach Barnea

One of the most fun things in mathematics is to explore different proofs of the same result. In this talk I will present various proofs of the inequality of means, that is, if are positive real numbers, then

and equality holds if and only if . While the inequality of means is one of the most basic inequalities in mathematics, the proofs are non-trivial and each requires some ingenuity. Indeed, we will even see a slight generalization of the inequality that is still open for .

*The Mathematics of Matches*, Professor Simon Blackburn

Who wins when two good players play a game? What is the winning tactic? There is often some beautiful and surprising mathematics behind these questions. This session explores one particular game (often played with piles of matches) to illustrate some of the mathematics involved.

*Mathematics and the Laws of Nature: A Variation on a Theme of Wigner*, Dr Jens Bolte

At least since Galileo Galilei the laws of nature are formulated in mathematical language. The Mathematical Physicist E.P. Wigner once gave a talk on this subject, under the title “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, in which he elaborates on “why the success of mathematics in its role in physics appears so baffling”. In this talk I shall explain Wigner’s ideas in examples, from very simple ones to some of the more “baffling” ones. Among the latter is P.A.M. Dirac’s prediction of anti-matter, solely based on the mathematical consistency of the (Dirac-) equation that he had developed in 1928.

*Further Mathematics for Teachers*, Ms Gill Buque

This is an informal opportunity for current or potential teachers of Further Maths to share ideas and to find out how the Further Maths Network can support them. It will also provide an opportunity for teachers to get together and discuss different aspects of Further Maths teaching.

*Primes, Perfect Numbers and Amicable Numbers*, Dr Rainer Dietmann

Primes are amongst the most fascinating objects in mathematics. In this session we want to discuss some of their basic properties such as the fact that there are infinitely many prime numbers. This can for example be demonstrated by using so called Fermat numbers, which were conjectured to be all prime until Euler found a counterexample in 1732. Another interesting class of primes, Mersenne primes, are closely connected to so-called perfect numbers which are subject to many interesting unresolved conjectures and are related to so-called amicable numbers.

*Fun Maths Roadshow and Problem Solving*, Further Mathematics Support Programme, Ms Cath Moore and Mr Mark Hughes

This is a hands on workshop where students can experience a variety of problems in small groups. There will be the opportunity to demonstrate maths, logic, communication and team work skills as different tasks covering a variety of topics are tackled. This workshop is ideal for any student in the first year of A level Maths or Further Maths.

*Traffic Jams*, Dr Alastair Kay

Have you ever been sat in a really long traffic jam and then, when you get to the front of the queue, there’s no apparent reason for why it took so long? This talk will attempt to explain how that happens.

*How is Doing Mathematics in 21st Century Different from that in “The good old days”?*, Dr Alexey Koloydenko

MATLAB is a powerful package for scientific computing, typical of the facilities available in mathematical laboratories. We make considerable use of such packages, both in teaching and in research. Algebra and calculus can nearly all be done ‘automatically’ on the computer rather than by hand, thereby avoiding ‘getting the sign wrong’ or ‘forgetting the factor of 2’ that plague all of us at times. This is particularly important in applications where the equations can spread over several pages at a time. In this introduction, you will be guided through some basic algebra and calculus examples, including 2D and 3D graphs and a demonstration of solving a real life problem.

*How to Really Share a Secret*, Professor Keith Martin

Let’s face it, we all have secrets and just occasionally we need to share them. However, the security of our modern digital world relies on crucially important secrets that really do need to be shared. We will discuss some mathematical techniques for sharing secrets, some of which will require you to draw elegant pictures. Along the way we will learn how to look after the keys to a bank vault.

*Graphs and Friendship*, Dr Iain Moffat

In any group of people, there will always be two people who have the same number of friends in that group. Why is this? Have we discovered some strange social phenomenon? In this talk I will use an area of maths called graph theory to show that this, and other “social phenomenon”, result from mathematics rather than sociology.

*The Mathematics of Sudoku-Like Problems*, Professor Sean Murphy

Many sudoku-like logic problems can be described as problems in algebra and geometry. We examine this relationship and explore connections with other branches of mathematics, such as cryptography, statistics and possibly complex numbers.

*Probabilistic Models: Examples*, Dr Vadim Shcherbakov

Probability has been developed as a mathematical theory since the 17th century and was originally motivated by the study of gambling games. Nowadays probabilistic models are widely used in many applications. In this talk we will try to demonstrate by example that some probabilistic models and related problems of interest can be easily formulated and understood intuitively but it is not obvious how to solve them formally.

*The Liar Game*, Dr Mark Wildon

Ask a friend to think of a secret number between 1 and 15. How many questions with yes/no answers do you need to discover your friend’s number? How many questions would you need if your friend is permitted to lie in one answer? We will answer these questions and learn how to play these games optimally, using the mathematics of coding theory to detect lies.

*π: A Fundamental Constant of the Mathematical Universe*, Mr Pavlo Yatsyna

When one divides the length of the perimeter of a circle by the length of its diameter one gets the number 3.141592653589793… , independent of the size of the circle. Somewhat surprisingly the same number appears when one divides the area of a circle by the square of its radius as Archimedes has shown about 2300 years ago. This “circle number” is called π Amazingly enough, π is somehow omnipresent, even if we don’t see any circles. For instance, the infinite alternating sum of the reciprocals of all odd positive integers 1 + 1/3 + 1/5 + 1/7 + 1/9 + … yields π/4 , or the infinite sum of the reciprocals of all squares 1 + 1/4 + 1/16 + … is π^{2}/6, and in turn can be interpreted as the probability that two positive integers are relatively prime. We will discuss a few interesting facts about π and ￼￼￼￼￼￼how little we actually know about one of the most fundamental constants in mathematics.