The Oxford University Maths Society

# Termcard for Hilary 2012

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

### Tuesday Week 2, 24th Jan - A model of learning under uncertainty

#### David Leslie

Reinforcement learning is a very popular model of machine, human, and animal learning in which the value of taking actions in different states is estimated online from observations. We will observe that convergence of the learning relies only on the fact that averages converge to expected values. However these simple convergence results only hold when an individual receives unambiguous information as to which state the world is in. In any natural environment the state information is noisy, so that the learner cannot be certain about the current state of nature, and extensions to the basic model are needed. We will discuss how to address this problem, using simple tools from probability, statistics, and dynamical systems theory.

### Wednesday Week 2, 25th Jan - Complex Numbers, Quaternions and Beyond

#### Sam Evington

In 1835 Hamilton gave a rigorous definition of a complex number as simply an order pair of real numbers (a,b). He defined addition coordinatewise and multiplication by (a,b)(c,d) = (ac-bd,ad+bc). He then went on to show that the expected properties follow from these definitions and nothing else. There’s no need to pluck a square root of -1 out of thin air! This leads one naturally to ask if there is a similar rule for multiplying triples of reals (a,b,c). Given the many applications of complex numbers to 2-dimensional geometry and physics a generalisation to the 3 dimensions of space would be immensely useful. Alas we shall see that no such system exists. All is not lost however. If one ventures into a fourth dimension (and gives up the notion that A times B should equal B times A) then a generalisation of complex numbers does exist. These are the Quaternions, and we shall establish their basic theory then look at their applications in 3-dimensional geometry, the invention of vector and modern physics. If time permits we shall look beyond Quaternions and discover why (with perhaps 1 exception) there’s nothing else worthy of being called a number system.

### Wednesday Week 3, 1st Feb - DNA, Cells, Organisms and Populations - A Tour of Mathematical Biology

#### Annekatherin Meiburg

I specialise in Mathematical Biology. – Oh, so you do Genetics and Statistics and stuff? – No actually I am really bad at Statistics. – Then what do you do? I do: Biofilm; Brain; Cancer; Cell cycle; Ecological niches; Chemotaxis; Crypt dynamics; Ecology; Enzyme kinetics; Epidemiology; Gene transcription; Heart beat; High altitude Pine rust; HIV; Insect swarm; Life cycles; Membranes; Mutations; Neurons; Organ growth; Pattern formation; Population dynamics; Predator prey relationship; Protein synthesis; Red squirrels; Seismic lines; Signal amplification in bacteria; Sperm motility; Vascular genesis; Wound healing This talk will be explaining what exactly a mathematical Biologist (really it ought to be biological Mathematician, but that sounds wired) does, what problems they have and where their work can lead us.

### Tuesday Week 4, 7th Feb - Games of Persuit and Evasion

#### Imre Leader

A scorpion wants to catch a beetle, a porter wants to catch a student, and a lion wants to catch a man. The beetle, student and man do not want to be caught. What tactics should they adopt?

### Wednesday Week 4, 8th Feb - Berry Phases

#### Jonathan Sykes

A discussion of the Gauss-Bonnet theorem's application to Berry phases. The lecture will briefly introduce the Gauss-Bonnet theorem for simple closed curves. It will then move to derive the Berry phase for a laser light in a helix and for magnetic field of a monopole. Conclusion will mention a brief discussion of measurement and further applications.

### Tuesday Week 5, 14th Feb - Infinitesimals

#### Kobi Kremnitzer

After the work of Cauchy and Weierstrass on the foundations of analysis it seemed like infinitesimals are unnecessary incoherent entities. Infinitesimals have survived and flourished in algebra, algebraic geometry and model theory, even after the brutal attack on them by analysts. I will discuss synthetic differential geometry. This is an approach to analysis and geometry that does not use epsilon and delta, but infinitesimals instead. Many theorems from first year analysis have one line proofs in this language. This approach is now finding many applications in geometry and mathematical physics.

### Wednesday Week 5, 15th Feb - Beyond the Möbius Strip

#### Will Perry

Consider the phenomenon of a vector space attached to each point of a shape, much as the Möbius strip can be formed by attaching a line to each point of a circle with some global "twisting". We will see how this notion is crucial in differential geometry, is a fascinating construction in itself, and can help us understand the topology of the underlying shapes. Throughout, we will lightly touch on some fairly sophisticated algebraic and geometric ideas such as classifying spaces, representable functors, and generalised cohomology. The talk will remain accessible to those not of geometric or algebraic backgrounds.

### Tuesday Week 6, 21st Feb - Rubik's Magic Cube in Oxford

#### Peter Neumann

Rubik's Cube first came to Oxford in September 1978, brought by Professor Roger Penrose from the International Mathematical Congress that had been held in Budapest that summer. It has been enjoyed, analysed, solved, written about ever since. And last year one of the intriguing problems about it, how many moves does "God's Algorithm" require? was solved. This Invariant Society lecture will focus on some of the interesting mathematics that is involved.

### Wednesday Week 6, 22nd Feb - The Story of Arithmetic Descent pt. 1

#### Simon Myerson

In 1659 Pierre de Fermat claimed to have "astonished the greatest experts" with his methods of ascent and descent. They were the first attempt at a systematic approach to solving polynomial equations in whole numbers (Diophantine equations). Descent later became part of modern number theory after being used to prove the Mordell-Weil theorem in the 1920s. Descents – in a form which would have greatly astonished Fermat – are used today in research on the 'hot topic' of elliptic curves. This is the first of two seminars on this topic. We will see precedents for ascent and descent in the work of Diophantos and Bhascara and discuss Fermat's innovations. We will see how these ideas became part of modern algebraic geometry, using the Riemann-Roch theorem to define group laws on curves.

### Tuesday Week 7, 28th Feb - Trapping a line with a curve

#### David Epstein

I will talk about a problem that I have used in the past to motivate undergraduate courses on topology or on metric spaces. In fact I would usually present 6 different problems at the beginning of each course, but I'm only planning to talk to the Oxford Invariants about one of them. What are the features that my problems are required to have? 1. Problem instantly comprehensible to everyone in the class. In fact, the problems I will talk about will be comprehensible to virtually everyone, including those who gave up mathematics in despair in their early teens. 2. Answer to problem can be guessed (but the intuition may be wrong). For this part, you will NEED A RULER (or be sitting next to someone with a ruler). 3. Correct proof of answer requires some non-trivial mathematics. In this case, some topology or metric space theory.

### Wednesday Week 7, 29th Feb - The Story of Arithmetic Descent pt. 2

#### Simon Myerson

This is the second of two seminars on this topic, but it is completely independent and no prior knowledge is required. In 1659 Pierre de Fermat claimed to have "astonished the greatest experts" with his methods of ascent and descent. They were the first attempt at a systematic approach to solving polynomial equations in whole numbers (Diophantine equations). Descents – in a form which would have greatly astonished Fermat – are used today in research on the 'hot topic' of elliptic curves. In this seminar we will introduce some concepts from algebraic geometry (divisors, meromorphic functions on curves, descent), and see how closely related these are to Fermat's ideas.

### Tuesday Week 8, 6th Mar - AGM

#### None

As well as eating Jaffa cakes and discussing the past and future activities of the Society, we will be electing the new committee.

### Wednesday Week 8, 7th Mar - Graph Theory: an Introduction to Cycle Spectra

#### Lauren Kutler

One attribute of graphs that can be investigated is cycle length. Roughly, a graph has a cycle of length n if you can pick some vertex, then travel along n edges back to that initial vertex without going through the same vertex twice (excepting, of course, the initial vertex). One 'interesting' kind of graph has all possible different cycle lengths. We call such a graph pancyclic. Graph Theorists have been studying these graphs for the past 40 years or so, asking questions such as what conditions guarantee pancyclicity in an arbitrary graph. Recently, there has been some interest in a modification of this kind of question, asking instead what conditions guarantee that a graph has at least p different cycles lengths, where |p| has been dubbed the 'cycle spectrum' of the graph. I had the privilege of investigating this question during a summer research project. I will discuss our findings along with some of the background preceding our research, as well as other current work. The talk should be very accessible to mathematicians in any year (and hopefully interesting too).