# Small Group Sessions

There will be two sessions of short 30 minute talks at 11.30, 13.15 and 14.00. Each talk will be given three times

• The mathematics of matches, Professor Simon Blackburn
• Mathematics and the Laws of Nature: A Variation on the Theme of Wigner, Professor Jens Bolte
• Pi, Mr Joshua Coyston
• Mathematics at University, Professor Stefanie Gerke
• The Birthday Paradox and its applications, Haibat Khan
• Exploring Mathematics with MATLAB, Dr Alexey Koloydenko
• The shape of space, Professor Brita Nucinkis
• Puzzles and Problem solving
• Probabilistic Models, Dr Vadim Shcherbakov
• The Liar Game, Professor Mark Wildon

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

#### 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 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.

#### PI, Mr Joshua Coyston

When one divides the circumference of a circle by its diameter one gets the number 3.141592653589793 . . ., regardless 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 showed about 2300 years ago. This “circle number” is called Pi. This number Pi keeps cropping up in mathematics, 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 Pi/4; or the infinite sum of the reciprocals of all squares 1+1/4+1/9+1/16+. . . is PI^2/6; the probability that two positive integers are relatively prime is 6/PI^2. We’ll ex- plore a few interesting facts about PI and how little we actually know about one of the most fundamental constants in mathematics.

#### Mathematics at University, Stefanie Gerke

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.

#### The Birthday Paradox and its applications, Mr Haibat Khan

How many people do you have to invite to a party so that there is a 80% chance that there are at least 2 that have the same birthday? The number is surprisingly small and this is known and the “birthday paradox”. Today the birthday paradox is used to compromise passwords and we will discuss how these attacks work and how to prevent them.

#### Exploring 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.

#### The shape of space, Professor Brita Nucinkis

How big is the universe? Is it finite or infinite? Does it have a boundary? These and other questions lead us to an area of mathematics called Topology. We will explore these topics, first in dimension 2, and will then see how to extend this to higher dimensions.

#### Puzzles and Problem Solving,

This is a hands-on workshop where students can experience a variety of problems individually or 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. Some of the problems are from the NRICH roadshow.

#### Probabilistic Models, 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 that 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.