# The lecture courses.

The following is a preliminary outline of each course. Each course will include exercise sessions supervised by the lecturers, together with PhD students## The Compactness Theorem

Mike Prest (University of Manchester)

Slides,Lecture notes.

## The word problem in combinatorial group and semigroup theory

Robert Gray (University of East Anglia)

Abstract and reading list.Exercises.

Lecture notes.

## Local-Global Principles in Number Theory

Shaun Stevens (University of East Anglia)

Background reading - if you are unfamiliar (or a bit rusty) with congruences, rings or groups, then please read up on them before the summer school. In partciular, make sure you are familiar with:- The Euclidean algorithm, modular arithmetic using numbers, including that a is invertible mod n iff gcd(a,n)=1.

- The basic language of rings and fields, including units in a ring.

- Modular arithmetic with the viewpoint of rings & fields: that Z/nZ is a ring, that its group of units is the classes of m coprime to n... so that Z/nZ is a field iff n is prime.

Abstract and exercises.

Lecture 1.

Lecture 2.

Lecture 3.

Some solutions to the exercises.

Here are some links to more information about quadratic residues and Legendre symbols:

http://math453spring2009.wikidot.com/chapter-4

https://www.maths.tcd.ie/pub/Maths/Courseware/NumberTheory/QuadraticResidues.pdf

https://www.maths.tcd.ie/pub/Maths/Courseware/NumberTheory/QuadraticReciprocity.pdf

## Introduction to Schubert varieties

Martina Lanini (University of Università di Roma Tor Vergata)

Lecture notes.

Exercise 1.

Exercise 2.

## Fun with solitons

Derek Harland (University of Leeds)

Exercise sheet 1.

Exercise sheet 2.

## Optimal Transport Theory

David Bourne (Durham University)

Background reading - none, but if you're feeling particularly enthusiastic, then have a look at Chapters 1 and 2 of this book https://optimaltransport.github.io/book/, which is free to download from the arXiv.Lecture notes.

Exercise sheet.

Exercise sheet - solutions.