When you ask someone what maths is, their answer will massively depend on how they’ve been exposed to maths up until that point. From a 10-year-old who will tell you it’s adding up numbers, to a Fields medalist who may say to you about the idea of abstraction of logical ideas, there is no clear consensus as to the “right” answer to this question. Our individual journeys as mathematicians give us a clear idea about what it means to us, and this affects how we then communicate our ideas to an audience of other mathematicians and the general public. However, a pitfall that we easily fall into as a result is forgetting that others can understand maths in a different way to ourselves, and by only offering our preferred perspective, we are missing out on the chance to effectively communicate our ideas.

In this talk, I will explore how our individual understanding of what mathematics is can shape our methods of communication. I will review which methods of communication mathematicians utilise, and show examples where each method does well, and not so well.  Examples of communication methods include writing equations, plotting graphs, creating diagrams and storytelling. Given this, I will cover how by using a collection of these different methods, you can increase the impact of your research by engaging with the various different mindsets your audience may have on what mathematics is.

In this talk I will review some of the key ideas behind
the study of entanglement measures in 1+1D quantum field theories employing
the so-called branch point twist field approach. This method is based on the
existence of a one-to-one correspondence between different entanglement
measures and different multi-point functions of a particular type of
symmetry field. It is then possible to employ standard methods for the
evaluation of correlation functions to understand properties of entanglement
in bipartite systems. Time permitting, I will then present a recent
application of this approach to the study of a new entanglement measure: the
symmetry resolved entanglement entropy.

Modular forms of slow growth admit a decomposition in terms of the eigenfunctions of the Laplacian operator in the Upper Half Plane. Whilst this technology has been used for many years in the context of Number Theory, it has only recently been used to further understand the partition function and the spectrum of Conformal Field Theories in 2d. In this talk, we’ll review the technology and how it has been applied to CFTs by several authors, as well as present a few new results.