Speakers

Mini courses

Topic area: Interactive proof verification.

Title: Lean: Filters and limits.

Abstract

In this mini course you'll learn two things. First, I'll teach you about filters, and how they can be used in analysis and geometry to give a better understanding of concepts such as limits and compactness. And secondly, I'll teach you about Lean, an interactive theorem prover. The goal is to get you to be able to use Lean to prove basic results in the theory of filters, and apply them to other areas of mathematics.

Topic area: Numerical Methods.

Title: Low rank matrix approximations in the age of machine learning.

Abstract

Approximation of a given matrix by a low-rank matrix is a famous problem in linear algebra and data compression that can be solved with the singular value decomposition, incomplete LU decomposition (Gaussian elimination) and cross approximation. However, data matrices in practical applications may have missing and noisy elements. This may happen, for example, for user i preference for movie j in a recommendation database, or for absorption of X-ray of energy i at pixel j in spectromicroscopy. We will look at some traditional algorithms for restoring low-rank matrices from data, advance them to new methods of machine learning (such as Adam with automatic differentiation and data splitting for reliable error estimation), and implement them in Python*. Time permitting, we will consider how low-rank matrices can be used for compressed approximation of tensors and multivariate functions.

Topic Area: Partial Differential Equations and Pattern Formation

Title: TBC

Abstract: TBC

Topic area: Number Theory.

Title: Continued Fractions: old and new results.

Abstract

TBC

Topic area: Geometry.

Title: Algebraic Curves.

Abstract

These lectures will be an introduction to algebraic geometry, the study of geometric shapes defined as the zero loci of polynomial equations (these are called algebraic varieties). I will focus on complex algebraic curves (also known as compact Riemann surfaces) which are complex algebraic varieties of dimension one. Using a combination of fairly elementary analytic and geometric techniques, I will present results on the classification of curves, their topology, and their behaviour in families.

Topic area: Knot Theory/Graph Theory.

Title: The graph theory hiding in knot theory.

Abstract

Knot theory is a branch of topology that studies the ways in which circles can sit in 3-dimensional space. Graph theory is an area of combinatorics that studies the properties of networks. Although these topics sound quite distinct from one another, there are strong symbiotic connections between the areas, with knot theory advancing graph theory, and graph theory advancing knot theory. In these lectures I will discuss a selection of the ways in which the two areas interact, demonstrating how the same mathematics arises in knot theory, graph theory and even statistical physics, and how these connections can be used advance each of the areas.

Colloquia

Topic area: Mathematical Education.

Title: Conditional Inference in Undergraduate Mathematics Students.

Abstract

Conditional inference has been studied for decades in cognitive psychology - we know a lot about what inferences people do and do not accept from statements of the form ‘If A then B’. We know much less about conditional inference in mathematics students, despite the obvious importance of this type of reasoning for understanding theorems and proofs. This talk will give a speedy overview of relevant research in cognitive psychology, then present findings from a sequence of studies of conditional inference in mathematics undergraduates. These studies investigate inferences from abstract, everyday and mathematical conditionals, and include evidence on believability effects and some striking individual differences. I will invite discussion of what the findings mean for teaching and learning, particularly at the transition to proof-based undergraduate mathematics.

Topic area: Cryptography.

Title: Public key cryptography and algebra.

Abstract

I will give a brief summary of some of the recent developments in public key cryptography, and why the search for new cryptosystems is so important at the moment. After touching on some of the classical cryptosystems, I will highlight some beautiful newer proposals based on group theory, and say why they are no good as they stand. I will also talk about some of the interesting mathematical problems that the next generation of systems will be based on.

Topic area: Group Theory.

Title: Zeta functions of groups.

Abstract

Group Theory is the area in mathematics that studies symmetries through structures known as groups. Although of very abstract nature, groups are everywhere in science. Ranging from chemistry to physical and dynamical systems, and all the way to cryptography and even music theory. In this talk, we will discuss a tool that is very useful to understand infinite groups; the so-called zeta functions of groups. These are generalisations of the famous Riemann zeta function, that allow us to investigate group properties through analytic properties of the function. The idea is the following: we split the elements of a group into finite pieces according to some interesting property. (For instance, we could split the integer numbers accordingly to being even or odd, getting precisely two pieces). Then, we define a zeta function that encodes arithmetic data of these smaller pieces. One can then study properties of the function, and these will reflect properties of the group.

Topic area: Statistics.

Title: TBC

Abstract

TBC

Topic area: Fluid Dynamics

Title: Life at low Reynolds number

Abstract

Much of life is concerned with fluids -- both within organisms, such as the flow of blood within vessels, passage of air within the lungs, and transport of mucus over membranes; and outwith, for the fish in the ocean, birds and insects in the air, and microbial cells in their environments. While we can generally visualise how a fish swims, and perhaps even how a bird flies, the motion of the microscopic is less intuitive. In this talk, we will briefly introduce the formulation of fluid dynamics, and define the Reynolds number. We will then focus on the world of the very small, where the Reynolds number approaches zero. This is the regime of the Stokes equation, and we shall explore the consequences this has for swimming as a microorganism.

Topic area: Functional Analysis.

Title: Geometric and Functional Inequalities in Sobolev Spaces.

Abstract

TBC

Topic area: Statistical Modelling.

Title: Maths and Stats in Infectious Disease Outbreak Response.

Abstract

My talk will focus on a series of case studies that using statistics and data science to inform real life problems encountered during infectious disease outbreak response. I will begin the talk by introducing different types of infectious disease models. Then I will show how Hawkes Processes can be used to model malaria transmission in countries close to elimination and how similar ideas were built on for COVID-19 modelling. Next I will discuss how it’s not just important to look at primary impacts of disease transmission but how secondary impacts such as orphanhood and caregiver loss are important. Finally, I will draw on experiences from Ebola outbreak response to show how collecting good contact tracing can help inform response.

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