Title of project:
Derived Categories and Algebraic Geometry

Code:
TL2122B

Supervisor: 
Dr Timothy Logvinenko

Project description:
Algebraic geometry studies geometrical objects by attaching invariants to them. For example any Riemann surface has its genus — the number of holes in it. More sophisticated examples include Betti and Hodge numbers, cohomology groups and ultimately – the derived category. The latter became over the last two decades the main technical tool of algebraic geometry, a sophisticated, abstract but very rewarding instrument.

This project would first introduce students to the abstract notion of a derived category of an abelian category. The main example we want to study is the bounded derived category of vector bundles on a projective space. Ultimately, the students should learn to compute the derived categories of P^1 and P^2, one- and two-dimensional projective spaces.

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

Students who wish to take this project in Year 4 are strongly recommended to take this supervisor’s Category Theory project in Year 3.

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

Prerequisite 3rd year modules:
MA3013: Algebra II – Rings
MA0322: Algebra III – Fields

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