# Sebastian Ørsted

Position: | PhD in mathematics |

Supervisor: | Jørgen Ellegaard Andersen |

Project supervisor: | Sergey Arkhipov |

Office: | 334, building 1530 |

E-mail: | sorsted@qgm.au.dk |

Field: | Algebra, geometric representation theory |

Research group: | QGM |

Duration: | August 2016–July 2020 |

I am a PhD student of pure mathematics at Aarhus University, specializing in abstract algebra and geometric representation theory. Some vague hint at what I am dealing with may be found in my PhD appilication. My main supervisor is Jørgen Ellegaard Andersen, and I am part of the Centre for Quantum Geometry of Moduli Spaces (QGM). My main project supervisor is Sergey Arkhipov.

This web page will probably remain boring for the time being, but at least I can redirect you to some of the material I have produced so far.

## My bachelor thesis

This paper, available here (in Danish), deals with the classification of the simple, finite-dimensional Lie algebras over arbitrary fields of characteristic zero. My supervisor for this project was Jens Carsten Jantzen.

## Supplementary notes for Calculus

These notes, available here (in Danish), grew out of my job as a teaching assistant of *Calculus* in the Fall of 2014.
Among other things, they cover elementary set theory, a formal definition of the complex numbers, and the theory of real and complex polynomials, including polynomial long division. The chapter on polynomials is also available as a
standalone document.

## Euler’s formula

Euler’s formula $e^{ix} = \cos(x) + i\sin(x)$ is arguably one of the most important, fundamental, and beautiful results in all of mathematics. However, it is not immediately clear what exactly the result contains. First of all, how do we actually define sine and cosine? Appealing to our geometric definition causes problems since this relies on the notion of arc length. But to measure arc lengths along the unit circle, one needs a $C^1$ (and preferably smooth) parametrization of this circle. And it is extremely hard to write down such a smooth parametrization that does not somehow rely on sine and cosine. Therefore, in this paper, I take the stand that Euler’s formula is actually, in a sense, the*definition*of sine and cosine. The “result” contained in Euler’s formula is now more of a justification of this definition, including a verification of the relation to the unit circle.

## Jordan normal form

A note on the Jordan normal form is found here.## Perspectives on Abstract Algebra

These notes were written while being a teaching assistant in introductory abstract agebra. They are available here.## Simplicial methods in homological algebra

This essay, available here, deals with simplicial objects, particularly simplicial sets and Abelian groups, focusing on their applications within homological algebra. It was part of my final exam in Introduction to Homological Algebra.

## L^{a}T_{e}X

I have produced several ressources aimed at helping make L^{a}T_{e}X more accessible. This includes a
general guide for writing hand-ins in L^{a}T_{e}X based on the errors I have encountered so far.
I have also published an example preamble for students to use, available in two versions, a
simple one, covering the most necessary things, and an advanced one, similar to the simple one except for the inclusion of tools for creating theorem environments and bibliographies.

## Lecture slides

The lecture slides for some lectures I have held are available here:

- Zermelo-Fraenkel Axioms
- Non-standard analysis and hyperreal numbers (also has English translation)
- Čech cohomology (contains an error which I have been too lazy to fix)