# 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)

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.
If you need to brush up on these things then the Oxford notes cover everything and more. The first year notes from UEA contain the bare minimum that you need to know before starting. If you have also read the Chinese Remainder Theorem, and seen that (Z/pZ)* is cyclic then so much the better.

Abstract and exercises.

Lecture 1.

Lecture 2.

Lecture 3.

Some solutions to the exercises.

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

## 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.

## Support

The research school is funded by the London Mathematical Society.