Skip to main content Mathematik Zettelkasten

Welcome to my wonderful Zettelkasten of mathematics definitions! Here you can find a curated list of definitions from various different disciplines in mathematics. The aim is to provide (among many other sites) a centralized repository where I can find definitions with relative ease. I say I and not we because first and foremost, this zettelkasten is for my own personal use for when I need quick access to definitions. You’ll find most of the definitions here line up with my personal research interests and work. However, this work is all in the public domain, and anyone is free to access it; just keep in mind it is not necessarily what you need.

There are some things that need to be moved out the way first:

Definitions only please!
This zettelkasten is exclusively for mathematical definitions, and mathematical definitions only. I will occaisonally leave remarks assuming, or hinting at well known results, but no explicit results or theorems will be mentioned, their proofs, even less. If you want more in-depths results to go along, you have to look else-where. You may look at my notes page, compile my \(\LaTeX\) notes (I DO NOT claim these notes as my own intellectual property), and go on from there.
Zettelkastens and Zibaldones

I do have somewhat of a personal process regarding these. I have also started keeping a zibaldone for which I (at-least I hope I will) jot down some short observations and reflections. No doubt, I will be reflecting on also keeping an online zettelkasten If you want to know more about my personal process, feel free to read the zibaldones I label as zettelkasten here.

I hope to update these zettelkastens on a regular basis, in accordance to my immediate needs given my research work and interests, and given my self study habits. In fact, the zettelkasten has provided me another paradigm of self study in itself.

These zettelkastens lack a structure and are rather free form as of now. This is by design, and I have decided not to group these zettelkastens under different topics. I believe so far that the free form nature of the zettelkasten is a benefit, and there is no need to impose structure where it is not needed. For the most part, the graph imparts and implies most, if not all the structure of the zettelkasten, and if you know any graph theory, you’ll be able to deduce that disciplines of math are represented by cliques in the graph. I encourage you to just pick a definition and get lost in the web of vertices!

On Keeping Up to Date
I hope to update these zettelkastens on a regular basis, in accordance to my immediate needs given my research work and interests, and given my self study habits. In fact, the zettelkasten has provided me another paradigm of self study all in itself.

Acknowledgements

I would like to thank my friend Sona Tau Estrada Rivera for giving me the idea for zettelkastens. You can check out her zettelkasten here, and you can check out some of her other stuff here.

I would also like to thank my friend Sergio Rodriguez, for no particular reason. Go check out his stuff here.

Recent posts

  1. Colinear

    Let \(G\) be an incidence geometry . We call two points \( A,B \in G \) colinear if there exists a line \( l \) containing both \(A\) and \(B\). We call \(A\) and \(B\) non-colinear otherwise. remark (sidenote: By definition of \(G\), any two points are colinear by axiom (1), so that non-colinear points must occur in triples. Indeed axiom (3) for incidence geometry ensures the existence of non-colinear points.

  2. Incidence Geometry

    An incidence geometry is a set \(G\), whose elements are called points, together with a collection of subsets \(\mathcal{L}\), whose elements are called lines, such that the following hold: (1). For any points \( A,B \in G \), there exists a unique line \( l \in \mathcal{L} \) for which \( A,B \in l \). (2). For every line \( l \in \mathcal{L} \), \( |l| \geq 2 \). (3). There exists points \( A,B,C \in G \), and a line \( l \in \mathcal{L} \) such that \( A,B \in l \) but \( C \notin l \).

  3. Matroid (Independence Axioms)

    A matroid is a set called the ground set, together with a collection \(\mathscr{I}\) of subsets called independent sets such that the following hold: (1). \( \emptyset \in \mathscr{I} \) (2). If \( I \in \mathscr{I} \) and \( J \subseteq I \), then \( J \in \mathscr{I} \). (3). If \(I_1, I_2 \in \mathscr{I} \), and \( |I_1|<|I_2| \), then there exists an \( e \in {I_2 \backslash I_1} \) for which \(I_1 \cup \{e\} \in \mathscr{I} \).