Explorations in Mathematics

Expository notes and essays by W. E. Aitken

Here is an ever expanding collection of essays, reports, reviews, and notes on various mathematical topics. Currently this collection contains about 832 pages of material, with plans to grow in the next few years.

Some of my essays are foundational. These will explore the logical and philosophical foundations of mathematics, starting with various approaches to the paradoxes of set theory. This will be followed by a treatment of the foundations of set theory and a book length treatment on building the number systems from Peano's axioms.

I plan to write reviews of the basics of several foundational areas of mathematics, such as topology, category theory, and geometry. These reviews will outline a rigorous, Bourbaki-like (but contemporary), trail through these foundational subject, but will usually leave routine details to the reader in order to keep these concise. The intention is to cover topics that every graduate student should know, at least in an ideal world.

Much of my writing will be reports of various topics. For example, I have included a report on freely representable groups. Another report examines a more classical topic: the theory of Dedekind domains. So far these have mainly been in algebra, but I hope to write some reports on topics in number theory, and maybe even geometry and mathematical physics (from a mathematical point of view).

Finally, I plan to include translations with commentary of a few classic papers in mathematics. For example, I include here a document called Artin’s First Article on the Artin L-Series (1924): Paraphrasis and Commentary. This is an expansive translation of Emil Artin's Über eine neue Art von L-Reihen (1924) with extensive commentary.

1. Foundations of Mathematics and Number Systems

(2020 - 2022) Commentary on Bourbaki's Chapter I, Description of Formal Mathematics. Describes in detail the formal logical system developed by Bourbaki in the 1950s. (June 11, 2022. 52 pages)

Coming soon (late 2023): Naive class theory. This is designed to be an accessible account of NBG theory of classes and sets.

(2007 - 2019) Number systems: an axiomatic foundation for algebra, number theory, and analysis. A detailed introduction to the number systems from the natural numbers (Peano's axioms) to the complex numbers. (Approximately 300 pages in total)

2. A Concise Review of General Topology

Here are a few documents reviewing the foundations and basics of general topology. The prerequisites are just basic set theory and some material about the real numbers and other number systems (see, for example, my notes on number systems). Later material about other topics such as topological groups or some basic results about topological manifolds will be added as needed.

2.1 Core Topology Series

This seven part series covers the core material including most of the material needed to get started with applications of general topology to other branches of mathematics (analysis, number theory, differential topology, differential geometry, and so on). It is a relatively concise review of the material (around 95 pages.)

(2012 - 2023) General Topology. Part 1: First Concepts. The basic definitions of point-set topology. (31 pages)

(2012 - 2023) General Topology. Part 2: Hausdorff Spaces. A short introduction to Hausdorff spaces (10 pages).

(2017 - 2021) General Topology. Part 3: Sequential Convergence. This document compares continuity with sequential continuity (3 pages)

(2017 - 2020) General Topology. Part 4: Metric Spaces. This document introduces metric spaces (11 pages)

(2012 - 2020) General Topology. Part 5: General Products. This document introduces infinite products. (6 pages)

(2017 - 2021) General Topology. Part 6: Connectedness. This document introduces connectedness and path connectedness. (13 pages)

(2012 - 2021) General Topology. Part 7: Compactness. This document focuses on compactness, but has some material about sequential compactness. (21 pages)

2.2. Additional Topics in General Topology

This section is currently planned to include selected topics building on the seven part series above. Planned topics: topological groups, topological manifolds, paracompactness, nets, filters, uniform structures, local compactness, compactification, and so on.

3. Topics in Algebra

3.1. Foundations of Algebra

(2017-2022) Binary Operations, Monois, and Groups This constitutes the "Ground Floor" of abstract algebra and only relies on set theory and some basics from my number systems notes as a prerequisite. It covers topics such as associativity, general associativity laws, commutativity, general commutative laws, laws of exponents in monoids and groups, identity elements, inverse elements. It covers monoids more than in a typical abstract algebra course, and covers just the very start of group theory. It overlaps common topics that one sees in an introductory abstract algebra class, but perhaps in a more extensive manner. It is the first of a planned series on foundations of algebra. (About 37 pages)

(2010-2021) Notes concerning minimal polynomials and primitive polynomials which serve as a sort of prerequisite to my short Galois theory course below. (It also considers unique factorization in Z[X], the Eisenstein criterion for irreducibility, the irreducibility of cyclotomic polynomials in the prime case. About 19 pages)

(2019) Localization in Integral Domains. This document introduces the concept of localization of rings in the relatively simple situation of integral domains. 11 pages.

3.2. Galois theory

(2003 - 2019) A quick introduction to Galois theory. Consists of an overview essay (about 30 pages) plus 16 one-page worksheets.

3.3. Reports and Essays in Algebra

(2021) Report on Freely Representable Groups. This concerns groups that admit a linear representation acting freely on nonzero vectors. (99 pages)

(2010 - 2019) Noetherian modules and rings. This is a short (5 page) introduction to the topic of Noetherian modules and rings. Both the ascending and descending chain conditions are explored in more detail in my later essays "Chains and Lengths of Modules" and "The Krull-Akizuki Theorem".

(2019) What are discrete valuation rings? What are Dedekind domains? A long 68 page essay on discrete valuations rings, Dedekind domains, and related integral domains. Includes around 60 exercises.

(2019) Chains and Lengths of Modules. This essay explores the descending and ascending chain conditions for modules, and introduces the concept of length for modules. The material here is needed for my later essay "The Krull-Akizuki Theorem." 13 pages.

(2019) The Krull-Akizuki Theorem. This essay gives the background necessarily for the proof of the Krull-Akizuki theorem, including basic results about Noetherian and Artinian rings. It gives a proof of the Krull-Akizuki theorem, and another theorem of Akizuki that asserts that every (commutative) Artinian ring is Noetherian. 16 pages.

(2019) Divisors and Krull Domains. This essay develops the theory of divisors in the context of Krull domains. It was written as a sequel to my earlier essay on Dedekind domains ("What are discrete valuation rings? What are Dedekind domains?"). 26 pages.

4. Translations and Commentary

(2022) Artin’s First Article on the Artin L-Series (1924): Paraphrasis and Commentary (45 pages). This is an expansive translation of Emil Artin's Über eine neue Art von L-Reihen (1924) with extensive commentary.