This is the webpage for the LMS lecture series on solitons. This is a five part lecture series designed to be an introduction to topological solitons for new PhD students. After the first lecture, aimed broadly at all students, the remaining lectures each focus on a specific topic.

The lectures are supported by the London Mathematical Society and the Yorkshire Durham Geometry Day, and are organised by Chris Wood.

The lectures are pre-recorded but will be broadcast weekly. During the broadcast session, the lecturer will be present and participants can ask questions. You can join the session by going to https://tinyurl.com/lms-soliton. The meeting password is lms. Recordings of the lectures will be available afterwards.

We also encourage interested readers to watch the lectures by Calum Ross, also organised by the LMS, which can be found here. Calum's lectures cover an introduction to solitons and their application to magnetic Skyrmions.

Our lecture calendar can be found below.

### 12th November 2020, 3pm GMT

#### Chris Halcrow - Introduction to Solitons

Watch on YouTube.

A broad introduction to solitons aimed at students from a mathematical or physical background. We'll start with the topology of belt and use this to introduce sine-Gordon theory. Using this simple theory as a basis we will discuss some basic concepts from solitons which will appear in future lectures: topological degree, the Bogomolny bound, the moduli space, low energy dynamics and transformations.

### 19th November 2020, 3pm GMT

#### Thomas Winyard - Vortices in superconductors

Watch on YouTube.

This lecture will focus on vortices (or vortex strings) in Ginzburg-Landau theory, a 2D effective model of superconductors. We will extend the concepts of the first lecture to this 2 dimensional U(1) gauged theory and in applying the Bogomolny bound, will classify the solutions into three "types". By linearising the model, we will discover that the "type" determines the vortex interactions and hence the possible higher degree solutions. Finally we will apply what we have learned to actual physical systems, explaining some famous results from physics.

### 26th November 2020, 3pm GMT

#### Josh Cork - Yang-Mills theory and instantons

Watch on YouTube.

You can find accompanying notes and exercises here.

This lecture is an introduction to Yang-Mills gauge theory, and the self-dual Yang-Mills equations. Solutions to these equations which have "finite energy" are called instantons. As we shall see, the condition of finite energy determines a topological classification of instantons, and we shall explore this in some detail via the example of instantons on R^4.

### 3rd December 2020, 3pm GMT

#### Josh Cork - The Nahm transform

Watch on YouTube.

You can find accompanying notes and exercises here, starting on page 20.

In the previous lecture we met instantons, and studied qualitative features about their topology. In this lecture we shall talk about a general process used to construct instantons called the Nahm transform. Heuristically, a Nahm transform is a correspondence between solutions to the self-duality equations in one space with solutions to the self-duality equations in a certain "reciprocal space". We shall look at this for the case of instantons on R^4 (where the Nahm transform is called the ADHM construction), and discuss the more general cases of periodic instantons, and generalisations to instantons on non-flat manifolds.

### 10th December 2020, 3pm GMT

#### Chris Halcrow - Skyrmions and nuclei

Watch on YouTube.

Here is a list of references which accompany the talk.

In this lecture we meet Skyrmions. These are solitons in 3D which model nuclei, such as protons and netrons. We'll study the topology and clasical solutions of the theory. To make a comparison to nuclei data we must quantise the solitons. This means we must study quantum mechanics on the moduli space of solitons. We'll do this in some simple cases and make comparisons to nuclear data.