$$ \usepackage{amssymb} \newcommand{\N}{\mathbb{N}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ZZ}{\ooalign{Z\cr\hidewidth\kern0.1em\raisebox{-0.5ex}{Z}\hidewidth\cr}} \newcommand{\colim}{\text{colim}} \newcommand{\weaktopo}{\tau_\text{weak}} \newcommand{\strongtopo}{\tau_\text{strong}} \newcommand{\normtopo}{\tau_\text{norm}} \newcommand{\green}[1]{\textcolor{ForestGreen}{#1}} \newcommand{\red}[1]{\textcolor{red}{#1}} \newcommand{\blue}[1]{\textcolor{blue}{#1}} \newcommand{\orange}[1]{\textcolor{orange}{#1}} \newcommand{\tr}{\text{tr}} \newcommand{\id}{\text{id}} \newcommand{\im}{\text{im}\>} \newcommand{\res}{\text{res}} \newcommand{\TopTwo}{\underline{\text{Top}^{(2)}}} \newcommand{\CW}[1]{\underline{#1\text{-CW}}} \newcommand{\ZZ}{% \ooalign{Z\cr\hidewidth\raisebox{-0.5ex}{Z}\hidewidth\cr}% } % specific for this document \newcommand{\cellOne}{\textcolor{green}{1}} \newcommand{\cellTwo}{\textcolor{red}{2}} \newcommand{\cellThree}{\textcolor{brown}{3}} \newcommand{\cellFour}{\textcolor{YellowOrange}{4}} $$

Hi, I am Luca Leon Happel. I am a computer scientist/mathematician and B.Sc in mathematics, who works at multiple high class universities and hospitals throughout Germany. I earned multiple awards for my works (e.g. in AI, medical software development,…) and secured funding for my project in cancer treatment. Read more about me…

Publications

Blog Posts

Bitwarden for Git Commit Signing

cyber security
encryption
git
programming

Signing Git commits is a crucial practice for ensuring the authenticity and integrity of your code and the work you have done. It allows you to verify that the commits that were made by you and your team have not been tampered with. In this post, I’ll show you how to use Bitwarden, a popular password manager, to sign your Git commits.

Feb 18, 2026
Luca Leon Happel

Commutator subgroups, derived series and their geometric interpretation

algebra
group theory
Given a group \(G\), the commutator \([G,G]\) is defined as
Feb 8, 2026
Luca Leon Happel

The \(\ell^2\)-Alexander torsion and the Thurston norm

algebraic topology
3-manifolds
Not finished yet…
Feb 7, 2026
Luca Leon Happel

Bachelorarbeit

math
algebra
geometry
I already published my masters thesis here on my website. So why don’t I also publish my bachelors thesis?
Jan 21, 2026
Luca Leon Happel

Masterarbeit

math
algebra
topology
category theory
I am currently writing my masters thesis about the connection between a generalization of the reidemeister-torsion (called \(\ell^2\)-alexander torsion) and the thurston norm…
Jan 21, 2026
Luca Leon Happel

Solving elliptic differential equations

math
algebra
Assume you are given an elliptic differential equaiton possibly of the form:
Oct 31, 2025
Luca Leon Happel

Corne + 2 PSP joysticks/dual trackpoint (Rust/RMK)

computer science
compilers
hardware
esp32
keyboard
rust

I designed, wired, built, and programmed (completely from scratch, because no software previously supported 2 joysticks) an ergonomic keyboard using some trash PSP parts. This led PCBWAY to sponsor me :)

Oct 30, 2025
Luca Leon Happel
 

Types in model theory

math
model theory
I did not finish writing this post. Kinda got busy…
Oct 20, 2025
Luca Leon Happel

Differentials, 0-forms and 1-forms – Placing (real) analysis on a rigorous footing

mathematics
differential geometry
differential forms
I have written about differential form before (What are „differential forms), but there I have never adressed the elephant in the room, of why, we should consider \(df\) as…
Sep 27, 2025
Luca Leon Happel

Affine lineare Transformationen als lineare Transformationen auf dem projektiven Raum

math
algebra

Kurze Notizen, wie man affin-lineare Transformationen auf dem \(\mathbb{R}^n\) als lineare Transformationen auf dem \(\mathbb{R}^{n+1}\) betrachten kann.

Aug 17, 2025
Luca Leon Happel

Onion architecture

Interfaces as List of Propositions

Or: Applying the Howard-Curry correspondence to OOP interfaces using the calculus of constructions
mathematics
category theory
oop

This post came to be due to my close friend Stefan Schulz with whom I studied computer science and mathematics at the University of Düsseldorf, after we had a discussion about the way interfaces are used in object-oriented programming languages like Java. It turns out, that we can regard these as a list of propositions, which can be proven by implementing the interface. This is closely related to the Howard-Curry correspondence and some functional programming concepts.

May 31, 2025
Luca Leon Happel

Automated theorem proving

mathematics
logic
automated theorem proving
lean4
formal verification
\[ \newcommand{\a}{\text{a}} \newcommand{\b}{\text{b}} \newcommand{\f}{\text{f}} \newcommand{\g}{\text{g}} \newcommand{\T}{\mathbb{T}} \]
Feb 21, 2024
Luca Leon Happel

Definition of \TopTwo

\(\ell^2\)-Invariants

mathematics
topology
algebra
group theory
ring theory
measure theory
functional analysis
differential geometry
differential topology
algebraic topology

This article is about the \(\ell^2\)-Invariants of \(G\)-CW-complexes. We will start by constructing the \(\ell^2\)-completion of cellular chain complexes and then take a look at some applications of the \(\ell^2\)-completion of cellular chain complexes. We will calculate the \(\ell^2\)-homology groups and the \(\ell^2\)-betti numbers of \(S^\infty := \colim_n S^n\) with the group \(G:=\prod_{n=0}^\infty \Z/2\Z\) acting on it. We will also take a look at the \(\ell^2\)-betti numbers of the torus and the \(\ell^2\)-betti numbers of the universal covering space of the torus. Finally, we will take a look at Lück’s theorem and its implications for the \(\ell^2\)-betti numbers of the universal covering space of a connected, compact CW-complex whose fundamental group is residually finite.

Feb 4, 2024
Luca Leon Happel

Surreale Zahlen

Seminar
Surreale Zahlen
abstract algebra
ring theory
In diesem Vortrag ging es darum zu zeigen, dass die surrealen Zahlen ein Körper sind. Ich arbeitete dabei unter der Annahme, dass wir bereits gezeigt haben, dass \(\boldsymbo…
Nov 16, 2023
Luca Leon Happel

Knot theory - peripheral system

math
algebra
abstract algebra
knot theory

In knot theory, the peripheral system is an important concept that helps to classify and distinguish different knots. In this video I animated it for an oberseminar at Heinrich Heine University during my masters degree in mathematics.

May 5, 2023
Luca Leon Happel

Go2Jasmin

computer science
compilers

Writing assembly code by hand is tedious and error-prone. Go2Jasmin is a compiler that translates a subset of the Go programming language into Jasmin assembly code for the Java Virtual Machine (JVM). This was a project I did during my bachelor’s degree in computer science for a good grade and for fun. Ultimately I decided to skip my bachelor’s degree in computer science to pursue a PhD in computer science afterwards, so this project was only really for fun (and its good grade never proved useful).

Jan 30, 2023
Luca Leon Happel

Haskell - Angewandter „abstract nonsense“

Seminar
Haskell
programming
Heinrich-Heine Universität
Dieses Semester habe ich einen Kurs in der Heinrich-Heine Universität in Düsseldorf belegt, welcher sich mit verschiedenen Programmiersprachen beschäftigte. Dabei habe ich…
Jan 23, 2023
Luca Leon Happel

What are „differential forms“

mathematics
differential geometry
differential forms
tangent space
cotangent space
manifold
tangent bundle
cotangent bundle
exterior algebra
global section
Currently while visiting the “KoMa” (Konferenz der deutschsprachigen Mathematikfachschaften / conference of German-speaking mathematics students) I got into a lot of…
May 27, 2022
Luca Leon Happel
 

Elliptic curves over the field with four elements (not finished)

mathematics
algebraic geometry
We define the affine space \(\mathbb{A}^n_k\) over a ground field \(k\) to be the set of all tuples \((x_1, \dots, x_n)\) with elements in \(k\). In the case of \(n=2\) we…
Mar 1, 2022
Luca Leon Happel

Sheaves in Minecraft

math
algebraic geometry
sheaves
topology
Minecraft
A few days ago, after my introductory class in topology, and right before my class about algebraic geometry, my friend Anastasia and I were asked by fellow topology students…
Oct 29, 2021
Luca Leon Happel

Prove that \(\mathbb{R}[x]/(x^2+bx+c)\) is isomorphic to \(\mathbb{R}^2\) or \(\mathbb{C}\) - A proof

math
algebra
abstract algebra
ring theory
meme
Around one year ago I tried to solve the following riddle involving abstract algebra, but I failed and did not finish the question at that
Oct 21, 2021
Luca Leon Happel

Prove that \(\mathbb{R}[x]/(x^2+bx+c)\) is isomorphic to \(\mathbb{R}^2\) or \(\mathbb{C}\) - First try

math
algebra
ring theory
abstract algebra
polynomial rings
quotient rings
isomorphisms
A few minutes ago I stumbled upon this meme on some math and science forum:
Dec 3, 2020
Luca Leon Happel

A new method to calculate dimension

math
fractals
dimension
calculus
geometry
A few days ago, I stumbled upon an article titled Finally We May Have a Path to the Fundamental Theory of Physics… and It’s Beautiful by Stephen Wolfram, an American…
May 22, 2020
Luca Leon Happel

Bildbesucher - First time using JSDoc

JSDoc
JavaScript
game
programming
As you can see in the image above, today I’ve added JSDoc support to my game, BildBesucher. This means that by visiting http://quoteme.github.io/bildbesucher/docs/, you can…
Apr 18, 2020
Luca Leon Happel

Bildbesucher - A simple jump and run game

programming
game
Recently, I’ve started to develop a game again, finding a bit of a positive side during the current Corona crisis.
Apr 7, 2020
Luca Leon Happel
 

My first program in the AUR

programming
linux
Recently I discovered some more of the incredibly useful features of my favorite text-editor Neovim:
Mar 28, 2020
Luca Leon Happel

Glider - A simple boids based video game

programming
game
boids
Today, while revisiting my old laptop, I rediscovered a project from just over a year and a month ago:
Feb 27, 2020
Luca Leon Happel
 

intersect.mjs und der Sphären/Boxen-Schnitt

javascript
programming
webgl
math
Momentan arbeite ich an meiner intersect.mjs Bibliothek, welche dafür da ist Kollisionen und Schnitte von verschiedenen geometrischen Strukturen in \(n\) dimensionen zu…
Feb 14, 2020
Luca Leon Happel

WebGLCS

My first online multiplayer 3D game (written from scratch)
computer science
game development

I was kinda obsessed with the game Cave Story for a long time. I wanted to expand on it, make it look better (3D, lighting, …) and also be able to play it with my friends online. Remind you, back then in 2015-2027 the software-development landscape was much harsher than today. There were no game engines that would allow you to all that! (At least I did not know of any?). Also, my budget was 0€, so I needed to get creative. This is the result:

Apr 19, 2017
Luca Leon Happel

The game for which I got a price in Brazil

Relic Hunters Zero on Android and Steam
computer science
game development

This game was a collaborative effort by a lot of people. I contributed by fixing some minor bugs and (this was kinda unthinkable for a LOT of people back then) porthing this game to Android.

Jan 8, 2015
Luca Leon Happel

My second competitive programming challenge

super platform land - Game++ - Community Challange #3
computer science
game development

In this video I showcase my submission for the Game++ Community Challenge #3, my second competitive programming challenge. Back when I was 14, the landscape of game development and competitive programming was quite different from today. Tools and resources were not as readily available, making it a unique experience to participate in such challenges. I was especially proud of integrating the gamepad api into my JavaScript game. This required more “low-level” programming compared to what I was used to previously. In fact, this was (and possibly still is?) kinda unique for web games, as most web games rely on keyboard and mouse inputs.

Jul 15, 2014
Luca Leon Happel

My frist competitive programming challenge

super platform land - Game++ - Community Challange #2
computer science
compilers

From a young age I was fascinated by programming video games. Ultimately this culminated in my participation in various competitive programming challenges (and even winning some high ranking university led challenges many years later!). This video shows my first competitive programming challenge, which I participated in when I was only 14 years old.

May 20, 2014
Luca Leon Happel
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