The Oxford University Maths Society

# Termcard for Michaelmas 2012

Unless otherwise stated, events take place on a **Tuesday** in the **Maths Institute** at around **8:15pm**

### Tuesday Week 2, 16th October - Desert Island Maths

#### David Acheson

If you were marooned on a desert island, which 8 pieces of mathematics would you want with you, to help keep your spirits up? David presents his own personal choice, drawn from both pure and applied mathematics, with reasons.

### Tuesday Week 3, 23th October - Poker Night

#### N/A

Quite self-explanatory; come along to the Mathematical Institute for a night of poker and generally good fun!

### Tuesday Week 4, 30th October - Did Galois deserve to be shot?

#### Peter Neumann

Évariste Galois died aged 20 in 1832, shot in a mysterious early-morning duel. His ideas, after they were published fourteen years later, changed the direction of algebra and have had a huge influence on mathematics. In this talk I propose to explain something of his mathematical insights and legacy in non-technical terms, and why it seemed worthwhile to produce a meticulous bilingual edition of his writings.

### Tuesday Week 5, 6th November - Members' Papers

#### N/A

Phil Tootil:

Transcendental Number Theory

It is commonly known that pi and e are irrational numbers, but there is more that can be said. In fact, they are transcendental, meaning that they satisfy no rational polynomial.

There are many open problems about transcendental numbers, such as whether the number pi+e is transcendental or not.

We shall investigate Schanuel's Conjecture, an unproved result which provides answers to many of these.

Sam Evington:

The “Paradox” of conditional convergence

One of the most surprising results in elementary analysis is that re-arranging the order of the terms in an infinite series can change the sum.

Indeed a theorem of Riemann states that for certain real series you can, by rearranging the terms, make it sum to any number you like! We shall investigate this phenomenon in the simplest cases and then study its generalisation in the setting of Banach Spaces.

### Tuesday Week 6, 13th November - Mathematical Potpourri on Airplane Boarding

#### Anne Henke

When passengers board an airplane, they queue in arbitrary order. For simplicity, we assume that passengers are arbitrarily fast to get to their seat row, they are arbitrarily broad and arbitrarily thin. Each passenger will need precisely one minute to store his luagge away and take his seat. During this time, he blocks the way for passengers in rows further back in the plane.

What is a good scenario to minimise the expected boarding times? Should, for example, passengers in back rows board first? Or should passengers on a window seat board first? The solution to these questions is deep and it connects different areas of mathematics: probability, algebra, differential equations.

My talk will be entirely elementary and no particular mathematical background is needed. We will encounter elementary combinatorics, a proof using the pigeonhole principle, and we will learn what the above question has to do with mathematicians going to the cinema.

### Tuesday Week 7, 20th November - Attempting to Model the Mathematical Mind

#### Roger Penrose

Alan Turing’s ground-breaking 1937 paper introduced his concept of Universal Turing machine, which underlies the modern general-purpose computer. In 1939, he proposed generalizations based on ordinal logic and oracle machines, these being apparently motivated by attempts to model the mathematical mind in ways that evade the apparent limitations presented by Gödel’s incompleteness theorems. In this talk, I introduce the idea of a “cautious oracle” as a more human version of Turing’s oracles. Nevertheless, I show that even this fails to capture the essence of the full capabilities of our understanding. Are there possible physical processes that might circumvent these Gödel-type restrictions? I shall briefly report on some startling new experiments, which may shed light on possible physical processes underlying brain activity, and I speculate on its role relating to the power of human understanding.