# WMS Talks

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## Past Talks

### Celestial Mechanics: Mathematics of the Solar System

#### Vassili Gelfreich

Celestial mechanics studies the motion of objects in outer space. In this talk we will discuss the equations which governs the motion under the influence of gravitational forces and explore how these equations are used to predict the motion of planets and asteroids in the Solar System.

### Second year essay presentations

Will Cohen: Colouring the Klein Bottle
Jon Cheah: Braid Groups
Sophie Peggs: An Introduction to Modern Forms of Cryptography

### Is mathematics invented or discovered?

#### Minhyong Kim

This is a question that might go back as far as Plato and is regarded as quite difficult by philosophers even now. As a practicing mathematician, I think the answer is quite simple. I will explain my reasons by way of two examples.

### Computers and Mathematics

#### Damiano Testa

Often, I find that I need to perform a calculation. Calculations range from trivial to arbitrarily complex: you barely need to think to know that 3 is prime, you should be able to check in your head that 57 is not prime, Euler found a prime factor of 2^{2^5}+1, but, currently, no one knows if 2^{2^33}+1 is prime (and not for lack of trying!). Computers were invented to perform calculations — and they are very good at it! For instance, they have been used extensively in the search of larger and larger prime numbers. Not entirely clear why, but still interesting! More recently, proofs of abstract results have been encoded into calculations that computers can (and did!) verify. Among them, the Four Colour Theorem, Kepler’s Conjecture, the Continuum Hypothesis,… In my talk, I will discuss formalization and computer verification of mathematics.

### Who Wants to be a Millionaire?

#### Philippe Michaud-Rodgers

Have you ever dreamt of being rich? (really rich?) and famous? Then this talk is for you! Come along to learn how to earn \$1 million by using your maths degree, without having to go into banking! All you need to do is understand congruent numbers: a natural number, n, is said to be a congruent number if it is the area of a right-angled triangle with rational side lengths. Is 3 a congruent number? If not, why not? Despite the seemingly simple nature of this problem, these are deep questions, with a history dating back to Diophantus. In this talk I will explain how congruent numbers are linked with objects known as elliptic curves, and how understanding congruent numbers requires proving a Clay millennium problem: the Birch and Swinnerton-Dyer conjecture for elliptic curves, whose proof would net you the aforementioned cash. This talk should be accessible to all, and I will not assume any background knowledge.

### Mathematical Sciences Research and its Interfaces with Society

#### Colm Connaughton

As a subject of human study, mathematics presents distinct aspects. Firstly, as an academic discipline, we study it for its own sake. Secondly, as the universal language of quantitative science, we develop mathematical tools and methodologies that allow other disciplines to advance. In this talk I will argue that in recent years, mathematics has developed a third aspect: it has become one of the fundamental enabling technologies underpinning the modern world. In universities we think about the first of these aspects a lot, about the second occasionally and about the third hardly at all. I will argue that traditional university mathematics departments should take this third aspect of the subject more seriously in their research and teaching. This is needed both for the benefit of society and for the benefit of the subject. I will finish by discussing a couple of examples from my own experience of mathematical work at the interface between academia and society more broadly: the Data Science for Social Good partnership between Warwick and the Alan Turing Institute and the Ergodicity Economics programme at the London Mathematical Laboratory.

### Continued fractions with bounded digits and dimension

#### Mark Pollicott

Any rational number can be written as a finite continued fraction (with digits that are natural numbers). An unresolved conjecture of Zaremba (1972) describes the rational numbers that arise when the digits are restricted to be from the set {1,2,3,4,5}. Some progress has been made of this problem. Surprisingly, it involves estimating the dimension of Cantor sets which arise for infinite continued fractions. No previous knowledge is required.

### Entropy in the kinetic theory of Gasses

#### Jo Evans

The second law of thermodynamics says entropy always increases. I will explain what we mean by entropy in the context of mathematical analysis and solutions to PDEs. I will explain on of the first places this was introduced which is the Boltzmann equation for the modelling of dilute gasses and Boltzmann’s H-theorem which says for the solution to Boltzmann’s equation entropy always increases. I will also talk about how entropy gives a meaning to the “arrow of time” and how this sense of time moving forward emerges when we move from looking at systems on a very small scale to looking at a larger scale.

### Tropical Geometry

#### Diane Maclagan

Tropical geometry is a combinatorial shadow of algebraic geometry. It is geometry over the tropical semiring, where multiplication is replaced by addition, and addition is replaced by minimum. I will give a gentle introduction to this field, giving some idea of where it can be applied, both inside and outside algebraic geometry. No knowledge of algebraic geometry will be assumed.

### Second year Essay Talks

#### Itamar Aharoni, Thomas Stanford, James Jones

Confused about the second-year essay? Looking for inspiration? In this 1 hour talk our three speakers will introduce their respective essays – hopefully lighting your curiosity.