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MSc Thesis: Invariant Theory and Graph Theory

Fri, 12 Jan 2024 20:58:44 +0100

On the 11th of January 2024, I defended my MSc thesis in Invariant Theory and Graph Theory. The thesis can be found here or you can read the abstract below.

Abstract

In this thesis, we provide an exposition of the invariant theory of finite groups, with a focus on algorithms and the Hilbert series. We apply the built-up theory to the algebra of invariants of multigraphs, as well as \(s\)-graphs, which are graphs weighted in \(\{0,1,…,s\}\). Utilizing computer exploration on the invariant algebra of \(s\)-graphs, we derive a formula for the Hilbert series of any permutation group acting on a special discrete variety, \(V|_s\). We conjecture that this formula can be generalized to any finite group. Furthermore, we present a version of King’s algorithm for computing a (minimal) generating set for the algebra of invariants on simple graphs. We conjecture the correctness of this algorithm and its potential generalization to any finite group acting on \(V|_s\). Finally, we recreate Thiery’s disproof of Pouzet’s conjecture

On the existance of a closed geodesic on compact Riemannian manifolds

Thu, 29 Jun 2023 10:23:09 +0100

As part of a seminar course on Riemannian geometry, i wrote a small paper on the basics of Lyusternik-Schnirelmann theory, with a goal of proving the Lyusternik-Fet Theorem. The project can be found here.

The theorem concerns closed geodeics, which are defined as follows.

Definition. A closed geodesic in a Riemannian manifold, \(M\), is a non-constant geodesic segment \(c: [0,1] \to M\) such that \(c(0)=c(1)\) and \(c’(0) = c’(1)\).

Intuitively, a closed geodesic is thus a ‘straight’ line, which wraps around the manifold and ends up where it started. I.e. periodic!

A debt game and Riemann-Roch on graphs (draft)

Tue, 17 Jan 2023 14:56:49 +0100

I recently gave a talk at a seminar course (Invitation To Combinatorics), where i was so lucky as to get to talk about the Riemann-Roch theorem for graphs! The theorem and all the underlying theory is marvalous and the analogy to algebraic geometry is really something.

What more the theorem need not much mathematical knowledge to state nor to be interesting. Indeed, it turns out to have a super nice analogy in a simple game anyone can play one graphs!

Cracking a memory

Thu, 29 Dec 2022 01:56:49 +0100

Finding a memory

I was recently back for holidays, vitising at my parents, and found all my old computer games. Among them was ‘Magnus Og Myggen 2: Den Store Skattejagt’ which I loved to play, even though it was among the most difficult games I’ve played to date.

I made an ISO file of it and wanted to boot it up through Wine and play it on my linux machine. However, the game is from 1997, and it ’erroneously’ told me I had no CD.

Nim-Squared

Thu, 17 Feb 2022 14:56:49 +0100

Introduciton & Rules

I recently did a project for the course Experimental Mathematics about an extension the game Nim, called Nim squared. The rules are quite simple; on an n by m chess board there are on each square at most one peg. The two players take turns to remove pegs from the board, following the rules:

You can only remove pegs from one row or column each turn.
You must remove at least one peg.
The player who removes the last peg wins the game.

So for example the following is a game of 3 by 3 Nim squared:

BSc Thesis: Triangulated Categories

Wed, 16 Feb 2022 14:56:49 +0100

On the 21st of June 2021, I defended my BSc thesis in Triangulated Categories. The thesis can be found here or you can read the abstract below.

Abstract

In this paper, we give a short introduction to the theory of triangulated categories. We present the relevant definitions and properties of triangulated categories which we use to investigate the Verdier localisation of triangulated categories. Furthermore, we show that the stable category of a Frobenius category is triangulated. By exhibiting the category of chain complexes as a Frobenius category we may take its stable category, which turns out to coincide with the homotopy category, thus showing that the homotopy category is triangulated. Then, by Verdier localising the homotopy category with respect to the subcategory of acyclic complexes (resp. \(\mathcal{X}\)-acyclic complexes), we obtain the derived category (resp. \(\mathcal{X}\)-relative derived category), proving it to be triangulated. Finally, using the theory of triangulated categories, we prove that the (Gorenstein) derived category of an abelian category A is abelian if and only if A is semisimple.

Contact

Mon, 01 Jan 0001 00:00:00 +0000

You can find and contact me these places:

Email
MagnusRH {at} maraha {dot} dk

GitHub
github.com/ManusRH

LinkedIn
linkedin.com/in/MagnusRHansen

Friends

Mon, 01 Jan 0001 00:00:00 +0000

Check out this little webring I have with all my friends!!!

Christian Bertram
cbertr.am

Søren Lund Skeie
skeie.xyz

Théo Borém Fabris
theobf.github.io

Sophus Valentin Willumsgaard
sophuswald.github.io

Mads Vestergaard Jensen
goodvibescorneroffice.gitlab.io/mads

Amik Raj Behera
sites.google.com/view/amik-raj-behera/home

Good Vibes Corner Office
goodvibescorneroffice.gitlab.io

Research

Mon, 01 Jan 0001 00:00:00 +0000

My papers in reverse chronological order:

Separation Results for Constant-Depth and Multilinear Ideal Proof Systems

Amik Raj Behera, Magnus Rahbek Dalgaard Hansen, Nutan Limaye, Srikanth Srinivasan
Under submission

paper :: ECCC

On Closure Properties of Read-Once Oblivious Algebraic Branching Programs

Robert Andrews, Jules Armand, Prateek Dwivedi, Magnus Hansen, Nutan Limaye, Srikanth Srinivasan, Sébastien Tavenas
17th Innovations in Theoretical Computer Science (ITCS), 2025

paper :: arXiv

Talks

Mon, 01 Jan 0001 00:00:00 +0000

My talks in reverse chronological order:

Workshop on Algebraic Complexity Theory 2026

University of Copenhagen
On Closure Properties of Read-Once Oblivious Algebraic Branching Programs
June 5, 2026

slides :: video