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Project
Azumaya representation varieties and stacks.
A lot has changed in geometry since the study of shapes (like triangles) in ancient Greece. The shapes studied nowadays very often have more than three dimensions and are curved, and can be extremely complicated. This is useful for fields like physics or engineering, which have also evolved drastically since Pythagoras and need geometrical techniques that become more and more involved.In this project we want to investigate a way to stretch the abstraction of geometry still further. To this extent, we will need concepts from algebra, like coordinates and equations, but further elaborated and more abstract. More precisely, we will study 'rings', collections of values that you can multiply or add. One particular kind of rings that will be important are 'Azumaya algebras'. We will need to solve some specific questions about these algebras to get a better understanding of the geometry in question. The study of Azumaya algebras or rings in general is also interesting on its own because they appear everywhere in mathematics, and there are still a lot of unresolved questions about them. Additionally, the proposed research will have implications in physics, more precisely string theory. From a string theorist's perspective, the smallest building blocks of the universe are vibrating strings (like guitar strings). Endpoints of these strings are called 'D-branes', and they are described accurately by the geometry we propose to explore.
Date:1 Oct 2015 → 30 Sep 2017
Keywords:THEORETICAL MATHEMATICS
Disciplines:History and foundations