Title of project:
Category Theory and Homological Algebra

Code:
TL2122A

Supervisor: 
Dr. T. Logvinenko

Project description:
Category theory enjoys an unfortunate reputation of a somewhat dry and abstract subject. Yet it is firmly established as the main language of modern pure mathematics, just as homological algebra is established as its main toolset.

Most new developments in algebraic and differential geometry, number theory and theoretical physics are stated in the language of category theory, and it is essential for anyone interested in these subjects to learn this language.

The project would consist of a general introduction to category theory and homological algebra, focusing on the elements which enjoy broad applications mentioned above. The students would learn the following notions: categories, functors and natural transformations, examples of commonly occuring categories, adjunctions, representable functors and the Yoneda Lemma, additive and abelian categories. The students would study in detail the category of modules over a ring and prove it to be abelian. They would then study complexes of objects in an abelian category, their cohomology groups, short and long exact sequences, snake lemma and five lemma, long exact sequence of cohomology.

The project will involve guided reading, review of literature and writing up a report. There will be also be a minor computational element to the project.

Project offered as double module, single module, or both:
Double

Prerequisite 2nd year modules:
MA2008: Linear Algebra II
MA0213 Algebra I – Groups

Recommended 3rd year modules for concurrent study:
MA3013: Algebra II – Rings
MA0322: Algebra III – Fields

Number of students who could be supervised for this project:
1