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The course will be a collection of lectures on different topics in algebra, geometry and mathematical physics on the style of "What is...?" column of the Bulletin of the AMS. Each topic will be covered in one or two 2-hour sessions.

Each lecture will provide an overview of a concept, including definitions, examples, and examples of applications. A good reference list on the topic will be provided, including at least one reference for general review and a couple of more specialised ones for further reading.

**Lectures [Thursdays, 10:00-12:00]**

- Lectures 1 & 2, 10.01.2019 and 17.01.2019

Symmetry and mechanics: why are the row lengths of the periodic table of the form 2 N^2?

Lecturer: Bernd Schroers (b.j.schroers@hw.ac.uk).

- Lectures 3 & 4, 24.01.2019 and 31.01.2019

The decomposition theorem in complex algebraic geometry.

Lecturer: Ben Davison (ben.davison@ed.ac.uk).

- Lecture 5 & 6, 07.02.2019 and 14.02.2019

What is noncommutative algebraic geometry?

Lecturer: Sue Sierra (s.sierra@ed.ac.uk).

- Lecture 7, 21.02.2019

What are quiver varieties? A differential geometer's viewpoint.

Lecturer: Lorenzo Foscolo (l.foscolo@hw.ac.uk).

- Lecture 8, 28.02.2019

The Hall algebra of an abelian category.

Lecturer: Sjoerd Beentjes (sjoerd.beentjes@ed.ac.uk).

- Lectures 9 & 10, 07.03.2019 and 14.03.2019

Groups, complexes, action!

In this 4h mini-course we will give an introduction to groups as geometric objects, and show how nice actions of groups on spaces such as trees can give lots of information about the group structure. The course will consist of two hours introducing actions on trees and two hours discussing actions on related higher-dimensional complexes.

Lecturers: Laura Ciobanu (l.ciobanu@hw.ac.uk) and Alexandre Martin (alexandre.martin@hw.ac.uk).

**Assessment**

There is a single assessment for this course, in the form of an essay on one of the topics discussed during the semester. The topic of the essay will be suggested and must be agreed with the lecturer who gave the relevant lecture.

The assessment is due on Friday April 19th, 2019.

**Prerequisites**

Topics will vary from week to week; lectures will be aimed at a broad audience of postgraduate students working in algebra, geometry and/or mathematical physics. The final assessment is based on a single topic of your choice amongst the ones introduced during the lectures.

ICMS

Fifth Floor

Bayes Centre

47 Potterrow

Edinburgh

EH8 9BT

Tel : 0131 650 9816

Fax : 0131 651 4381

Mail : johanna.mcbryde@icms.org.uk