What is the probability that two random integers are coprime, by Prof Jing Lei
I will first review this classical problem in number theory, with some heuristics and historical accounts. I will also discuss this problem through the lenses of modern probability theory frameworks, reconciling the “probability” claim in the original statement with a countably additive probability space, and some finitely additive probability spaces. Somewhat surprisingly, when finitely additive probabilities are considered, the answer depends on the interpretation of uniformity.
Cops and Robbers, by Prof Imre Leader
Some cops are chasing a robber around a network. The robber runs to an adjacent point, then the cops all run to adjacent points, and so on. For a given network, how many cops are needed to catch the robber?
Expander Graphs and Where To Find Them, by Dr Ana Khukhro
Expander graphs are graphs which enjoy the somewhat contradictory properties of being well-connected and sparse. These properties make them useful objects both in and outside of mathematics, and their various characterisations ensure they appear in interesting ways in many different contexts. We will motivate their definition, look at ways to construct them, and explore how varied the world of expanders really is.
Equidistribution in Number Theory, by Prof Jack Thorne
Equidistribution results aim to bring order to the chaos inherent in number theory. (Example: if q is a prime, then remaining primes are equally distributed in the non-zero residue classes modulo q.) I will introduce some of the basic ideas and discuss some of the outstanding results in this area.
Braid Groups, by Dr Ailsa Keating
‘But I Was Just Trying to Help’, When Irresponsible Mathematics Screws Things Up , by Dr Maurice Chiodo
Mathematics is both the language and the instrument that connects our abstract understanding with the physical world, thus knowledge of mathematics quickly translates to substantial knowledge and influence on the way the world works. But those who have the greatest ability to understand and manipulate the world hold the greatest capacity to do damage and inflict harm, often entirely unintentionally. In this talk I’ll explain that yes, there is ethics in mathematics, and that it is up to us as mathematicians to make good ethical choices in order to prevent our work from becoming harmful.