### Week 1 – Problems, Pizza and the Electric Guitar

Dr. David Acheson, Jesus College What is it, exactly, that makes for a good maths problem? Is there a place for pizza in serious mathematics? And what has all this got to do with playing the guitar? These are some of the questions we will consider in an off-beat look at some of the highlights of mathematics.

### Week 2 – Great Mathematicians

Dr Raymond Flood (formerly of Kellogg College) and Prof Robin Wilson (Pembroke College and the Open University) (abstract to follow)

### Week 3 Monday 4.00 – Dirichlet’s Theorem on Arithmetic Progressions

Phil Tootill – (Undergraduate Maths Seminar) One of the best known proofs in mathematics is Euclid’s proof that there are infinitely many prime numbers. Arguments similar to Euclid’s can be used to show that infinitely many primes lie in certain progressions, but fail to show the result in full generality. Using the methods of analysis, Dirichlet’s theorem shows the result for any progression (a+cn), with a and c coprime.

### Week 3 – Can computers deal with infinite objects in finite time?

Martin Escardo, University of Birmingham For example, can they calculate with real numbers represented as infinite sequences of digits, so that all digits printed at any given time will be correct, and so that they print digit after digit in a never ending fashion? Can they check an infinite number of possibilities and answer yes or no after calculating for a finite amount of time? Sometimes they can, and sometimes they cannot. And sometimes we are able to tell whether they can or cannot. I will explore the mathematical tools and foundations for computing with infinite objects, including general topology and constructive logic (and excluding many useful branches of mathematics that time will not permit to cover). There will be theory and examples.

### Week 3 Thursday – Careers Talk

IMA Someone from the IMA will talk to us about careers at 4.00.

### Week 4 Monday 4.00 – Quantum Cryptography

Matt Williams – (Undergraduate Maths Seminar) Throughout history the desire to securely share secret messages has driven research into cryptography; from the simple Caesar cipher to the Public Key Cryptography systems that protect our credit cards every time we buy textbooks on Amazon. The discovery of quantum mechanics at the start of the last century fundamentally changed the way we think about the world around us. More recently it also changed the way we think about computing and information theory. Quantum computers will render useless many of the codes upon which we currently depend. Fortunately some of the weird aspects of quantum theory can be put to use to do cryptographic tasks with security guaranteed not by computational complexity but by the laws of physics. Assuming only a reasonable knowledge of linear algebra I hope to introduce some of these phenomena then look at how QM can be used to securely distribute a key and help us to be truly random.

### Week 4 – Maths In and Out of the Zoo

Prof. Chris Budd Is maths a tame animal, confined to the classroom or university, or is it a wild animal, free to roam in the jungles of the worlds problems. My talk will try to answer this question by taking maths on a tour around the zoo. We will discover the maths of fish, penguins and many other exotic creatures.Maths will be used to peer inside the brain of a bee and also to soar with the starlings. I will even show you how maths can help at the gift shop and at the turnstiles to get in. There will be a mix of maths on show. Bring your imagination!

### Week 5 Monday 4.00 – Contact Line Dynamics of an Evaporating Droplet

Matt Saxton – UG Maths Seminar The dynamic contact angle is the angle between a moving liquid/vapour interface and a solid surface, measured within the liquid at the contact line where the three phases (solid, liquid, gas) meet. There is empirical evidence that the contact angle is related to the velocity of the contact line by a so-called contact line law. In this talk we will first explain how to derive a very simple model governing the height of an evaporating liquid drop. We will then demonstrate how perturbation theory can be used to derive from this an ODE problem for the contact line law and discuss how to solve this problem numerically. We will also mention some of the applications of this work. The talk should be accessible to everyone and will be of particular interest to anyone who likes fluid dynamics. This material is also relevant to anyone who wears contact lenses!

### Week 5 – The Mathematics of Ageing

Dr. David Steinsaltz Why do humans and many (but not all!) other organisms build up elaborate bodies over the course of months or years or even decades, but are then incapable of maintaining what they have built up, so that they gradually (or in some organisms very rapidly) become weaker and more susceptible to disease and death as they age?In the half century since evolutionary biologist George Williams formulated this question, there has been quite a lot of progress toward answering this question. Biologists and medical scientists have played an important role, of course, but so have mathematicians and statisticians.The talk will present a few examples of how this biological question gets transformed into mathematical and statistical questions, and how very concrete questions about the natural world force us to address questions of very abstract mathematics, in areas such as: Dynamical systems on general metric spaces Dynamic programming Asymptotics of products of random matrices Eigenvalues of self-adjoint differential operators (It is not assumed that any of these topics are already familiar to the audience.)

### Week 6 Monday 4.00 – D-modules and Automated Proof

Chris Campbell -(Undergraduate Maths Seminar) The theory of D-modules is an rich and deep algebraic theory that describes differential equations that, since it’s inception in the 60’s, has left few areas of mathematics untouched. The algebraic structure allows for the use of algorithms designed for attacking systems of polynomial equations to be used for differential equations. I will talk about applications in the field of automated proof, specifically proving combinatorial identities and differential equation identities. There will be little previous knowledge required beyond 1st year algebra and basic differential equations.

### Week 6 – The Friendship Theorem

Martin Liebeck This theorem that tells you that if you go to a party where any two people have exactly one mutual acquaintance, then there is someone who knows everybody. Apart from its obvious fundamental applications to the theory of going to parties, it also has a wonderful proof which uses a completely different and unexpected part of maths. The talk should be comprehensible to all undergraduates, including those who have just started.

### Week 7 Monday 4.00 – A Logic with Computable Infinite Expressions, for Foundational Study

Catrin Campbell-Moore – (UGMS) It would be good to find a logic which can characterise the natural numbers. In order to give a logic that can both express the natural numbers and have certain other desirable properties we need to allow infinite conjunctions. In this talk I will explain why and will describe the logic which only allows sentences that have computable syntax.

### Week 7 – Hypercomputation: exploring the extreme theoretical limits of computing

Dr. Toby Ord In computer science, there are many theoretical machines in a hierarchy of increasing power. The most powerful is the Turing machine, which is roughly like a desktop PC with unlimited time and memory. No matter how you try to extend the power of a Turing machine, you never get anything more powerful – well, at least that is what people are usually taught. In fact, there are definitely theoretical machines more powerful than Turing machines, it is possible that some of these may be physically realisable, they have serious implications for the philosophy of mathematics, and they are really interesting. It doesn’t matter if you haven’t studied computer science before – I’ll get you quickly up to speed, then push on into the relatively unknown.

### Week 8 Monday 4.00 – Knot theory and the Jones polynomial

Jakob Blaavand – (UGMS) In this talk we will introduce the basic notions in knot theory. We will start by defining what we mean by a knot and a link. A mathematical knot is almost the same as knots in the real world. If you tie a knot on a rope, you have to glue the ends of the rope back together again, so that your knot is on a circle. A link is just several knots, which might be linked together. We will also define, what it means that two knots are isomorphic. Whenever you have defined some objects and isomorphisms between them, you want to classify them. That is, split the objects into disjoint classes, such that all objects of a class are isomorphic. In our case, given two knots can we specify whether they are isomorphic or not? We can define functions on the quotient space of isomorphic objects, and if the function gives different values on the two knots, they are definitely not isomorphic. A function defined on this moduli space, is called an invariant. We will discuss one of the most famous invariants, the Jones polynomial. We will look at some basic properties of the Jones polynomial, and If time permits, we will discuss how to make the Jones polynomial into an invariant, which can separate the unknot from all other knots. This talk will be understandable for everyone.

### Week 8 – Christmas Party!

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