Representation theory via homological algebra (GRANTJU19SCIC)
Representation theory studies algebraic structures by investigating their action on other mathematical objects. We often consider algebras acting on vector spaces, as we understand linear algebra relatively well. A vector space together with such an action is called a module, and one of the goals of representation theory is to understand all the modules of a given algebra. If we understand the smallest building blocks, known as simple modules, then the question becomes: how can we combine them to build more complicated modules? This combination is called an extension, and it describes how the larger module is built from a submodule and a quotient module.
Powerful techniques for studying the extension problem were developed in the 1970s by Auslander and Reiten. Their theory works particularly well when the algebra under investigation is the path algebra of a directed graph known as a quiver. More recently, Iyama and collaborators have developed ways to extend Auslander-Reiten theory to more complicated algebras of higher global dimension. This higher dimensional theory has been widely studied, and is connected to other areas of modern mathematics such as cluster algebras. The tools involved include homological algebra, which is an algebraic theory inspired by algebraic topology and category theory, and the objects of study are closely related to algebraic geometry, braid groups, and Lie algebras. The aim of this PhD project is to apply these techniques to find new results about algebras and their representations.
Project Start Date: Oct 2019
Mode of Study: Full-time
Acceptable First Degree: Normally Mathematics. Other subjects, e.g., Physics, Computer Science, Natural Sciences, may be considered for an exceptional student who has taken various mathematics courses
Minimum Entry Requirements: UK 2:1
Early application is encouraged.
This PhD project is in a Faculty of Science competition for funded studentships. These studentships are funded for 3 years and comprise home/EU fees, an annual stipend of £14,777 and £1,000 per annum to support research training. Overseas applicants may apply but they are required to fund the difference between home/EU and overseas tuition fees (which for 2018-19 are detailed on the University’s fees pages at https://portal.uea.ac.uk/planningoffice/tuition-fees . Please note tuition fees are subject to an annual increase).
i) Ralf Schiffler, "Quiver Representations", Springer, 2014
ii) Maurice Auslander, Idun Reiten, and Sverre Smalø, "Representation theory of Artin algebras", Cambridge University Press, 1997
iii) Osamu Iyama and Steffen Oppermann, "Stable categories of higher preprojective algebras", Adv. Math. 244 (2013), 23-68.
iv) Joseph Grant, "Higher zigzag algebras", arXiv:1711.00794 (2017)