# Programme

Arrival is at 09.00, with the welcome from the Principal at 09.40. The timetable gives details of the main talks. There will also be two sessions of small group talks at 11.30 and 13.15. The day will end at 15.45.

# Booking

Exploring Maths 2017 is a free **invitation only** event. If you would like to attend please contact us. Attendance should be organized by a teacher or other authorized representative of your school.

Exploring Maths 2017 is free.

All students must have signed parental permission to attend the event. This is the responsibility of the organizing school, and Royal Holloway accepts no responsibility for unauthorised attendance.

The organizing school is responsible for ensuring that students attend if places are booked on their behalf.

If you need to cancel your attendance at the event after May 31 2017 we reserve the right to levy a cancellation charge of £100 against the school to cover administration costs. If not all places reserved by the school are taken up we reserve to right to levy an administrative charge of £50. Any charges will be submitted to the organizing school by invoice, payable within 30 business days of receipt.

# Travel

## By Car

The College is on the A30, 19 miles from central London and about a mile south-west of the town of Egham. It is two miles from Junction 13 of the M25.

We have reserved parking for cars and minibuses in Car Park 12. All vehicles must display the official ‘Exploring Maths’ parking permit on the dashboard!

#### Directions from the M25

- After leaving the M25, follow the A30 west (signposted Bagshot and Camberley); this is the Egham bypass.
- At the end of the Egham bypass, continue on the A30 up Egham Hill (a petrol station is on your left).
- The main College entrance is on the left immediately after the second footbridge.

#### Directions from the West

- Leave the M3 at junction 3, at the roundabout take the 1st exit onto the A322.
- Proceed under the road bridge and turn left, then merge onto the A30.
- At the roundabout, take the second exit onto the A30.
- Continue on the A30, through Sunningdale towards Staines-upon-Thames.
- The main College entrance is on the right immediately before the footbridge.

#### Avoiding the M25

- To avoid the M25 when travelling from the South or South West on the M4 or the M3, take the A329(M) from the M4 at junction 10, or the A322 from the M3 at junction 3.
- The A329(M) becomes the A329 at Bracknell, and then the A322, and connects with the A30 at Bagshot. Continue on the A30 through Sunningdale and Virginia Water towards Staines-upon-Thames and you will pass the College campus.
- To avoid the M25 from the M40 coming from the Midlands, leave the M40 at junction 4 on to the A404 towards Maidenhead. Join the A308 at Maidenhead which will take you through Windsor and Old Windsor. Turn right at the roundabout after a Harvester Pub called the Bells of Ousley for a shortcut to the College through Englefield Green village on the A328.

## By Rail

The College is connected by fast rail links to the rest of the country, with connections to London and Reading stations.

There are four trains an hour from London Waterloo to Egham (40 minutes), Reading to Egham (40 minutes) and Woking to Egham (35 minutes: change at Weybridge). You can also reach Twickenham (17 minutes) and Richmond (20 minutes) as well as Clapham Junction on the Egham to London Waterloo train.

For local train timetables, please visit South West trains and for information about connections, please see National Rail Enquiries.

**Please be aware that Royal Holloway is located outside of the Oyster Card travel zone.**

#### Walking Directions from Egham Station

The College is about a mile from the station, which takes about 20 minutes to walk.

- Turn right out of the station along Station Road and walk about 100 yards to the junction and traffic lights.
- Turn left at the junction and follow the road around to the large roundabout with a petrol station on the left. Walk up Egham Hill (A30) and follow the footpath.
- The main College entrance is on the left immediately after the second footbridge.

#### Taxi from Egham Station

There is a taxi rank just outside Egham station. A taxi will cost £7.

#### By Bus from Egham Station

Get the 8 or 441 bus from the bus stop near the station: follow walking directions for about 200 yards and it is on your left. There is a bus stop outside Royal Holloway.

# Arrival

On arrival at Royal Holloway you will be directed to the upstairs of the Windsor Building by our friendly student helpers. You will be registered and given an information pack.

After registration a light breakfast will be provided. There will be a chance to relax and meet our student volunteers and other students before the first talk.

For information on getting to Royal Holloway, please see our Travel page.

# Food

Coffee, tea, soft drinks and a light breakfast will be available (free of charge) in the Windsor Building from 09.00. We recommend you bring your own packed lunch. Failing that you may use one of the food outlets open on campus. These are:

**Crossland Café (Founders Building):**Sandwiches, snacks including baked potatoes, and drinks.**Starbucks (Boiler House):**Sandwiches, cakes, drinks.**Union Shop**: Take-away sandwiches, snacks, drinks, confectionery.

Please be aware that these outlets will only accept debit cards for payment for amounts of £10 or more: cash only for amounts less than this.

# Timetable and Main Talks

## Timetable

**09.00**Registration and refreshments**09.40**Welcome from the Principal, Professor Paul Layzell**09.45**Quick quiz with prizes, Mr Laurence O’Toole**10.20**The Music in a 2000 year old proof, Professor Rüdiger Schack**11.30**Small Group Talks**12.00**Lunch**12.30**Tours of campus from outside the Windsor Building**12.30—13.00**Further Mathematics (for teachers only), Mr Steve Collins**13.15—13.45**Small Group Talks**14.00**Mathematics at University, Professor James McKee**14.45**Juggling—Theory and Practice, Dr Colin Wright**15.45**Close and questionnaires

## Main Talks

### Professor Rüdiger Schack: *The* *Music in a 2000 year old proof*

Are all quantities in mathematics ratios of integers, such as 5/7? The ancient Pythagoreans thought so. For them, mathematics, cosmology, and music were firmly based on the integers. The Pythagorean world view was thrown into crisis in the 5th century BC when it was discovered that some distances, and by implication some musical intervals, cannot be expressed as ratios of integers, that is, they are “irrational”. It took more than 2000 years for mathematicians and musicians to fully come to terms with the discovery of irrational numbers. This talk will present a simple and beautiful proof that the square root of 2 is irrational, and through it explore connections between music and mathematics.

*Professor Rüdiger Schack has been teaching at Royal Holloway’s Mathematics Department for more than 20 years, including 5 years as Head of Department. He has made numerous research contributions in the field of quantum theory ranging from foundations to optics and cryptography. Recently he was a panelist at the World Science Festival in New York. His musical interests include singing in a choir and playing piano and harpsichord.*

### 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.*

# Small Group Sessions

There will be two sessions of short 30 minute talks at **11.30** and **13.15**. Each talk will be given twice

*Mathematics and the Laws of Nature: A Variation on the Theme of Wigner*, Professor Jens Bolte*Further Mathematics for Teachers*, Mr Steve Collins. For teachers only, from**12.30**to**13.00***Big Numbers and Securing the Internet,*Mr Benjamin Curtis*Prime Numbers, Perfect Numbers and Amicable Numbers*, Professor Rainer Dietmann*Beyond the Third Dimension*, Further Mathematics Support Programme, Mr Mark Hughes*Exploring Mathematics with MATLAB*, Dr Alexey Koloydenko*Infinity and Computability*, Professor Chris Mitchell*Fun Maths Roadshow and Problem Solving*, Further Mathematics Support Programme, Ms Cath Moore*The MU Puzzle*, Professor Rüdiger Schack*Probabilistic Models: Examples*, Dr Vadim Shcherbakov*The Liar Game*, Dr Mark Wildon

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

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

At least since Galileo Galilei, the laws of nature have been 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 the 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 developed in 1928.

#### *Further Mathematics for Teachers*, Mr Steve Collins

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.

*Big Numbers and Securing the Internet*, Mr Benjamin Curtis

*Big Numbers and Securing the Internet*, Mr Benjamin Curtis

Big numbers keep the internet safe. In this talk we will begin with a discussion about large numbers, and how we can put them into context. For example we can understand how large the number 10 is, but it is much more difficult to understand how large 13,700,000,000 is. We will then discuss exactly how large numbers need to be in order to secure the internet, as well as how these numbers are used. We will consider a simple example of how some mathematical problems get ‘harder’ as the numbers get bigger. Finally, we will look at how future developments in computers will mean that we need a different way to secure the internet in the future.

*Prime Numbers, Perfect Numbers and Amicable Numbers*, Professor 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.

*Beyond the Third Dimension*, Further Mathematics Support Programme, Mr Mark Hughes

Have you ever wondered what lies beyond the third dimension? In this journey through the dimensions, we will be exploring a branch of mathematics called topology (sometimes playfully referred to as ‘geometry on a rubber sheet’). We will explore some shapes that can’t exist unless you add more dimensions. We will discover that transitioning from the third to the fourth dimension is not much harder than transitioning from the second to the third dimension. We will also find that time isn’t necessarily the fourth dimension, despite physicists’ claims.

*E xploring Mathematics with MATLAB*

**, 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.

*Infinity and Computability*, Professor Chris Mitchell

*Infinity and Computability*, Professor Chris Mitchell

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

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 teamwork 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.

*The MU Puzzle*, Professor Rüdiger Schack

*The MU Puzzle*, Professor Rüdiger Schack

Starting from a given sequence of letters and four simple rules, can one arrive at the word MU? We will show that this simple mathematical puzzle leads to surprising insights into the nature of mathematical proof and the limitations of computers. And, of course, we will also solve the puzzle.

*Probability 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.