Mathematics

Here, you will find work I have done concerning mathematics, as well as practical advice on studying it. My biggest project has been the mathematics listing which I invite you to visit if you’re interested in mathematics.

It is a listing of textbooks and lecture notes, roughly from high school level mathematics all the way up to gradute-level research topics such as algebraic geometry, symplectic topology and harmonic analysis. I have also written a variety of lecture notes myself, although they are not yet available for a variety of reasons.

What is mathematics about? Is it about numbers?

Mathematics in its purest form is the study of patterns. In my opinion, the most fundamental work of mathematics is equivalence, by which I mean, to prove that two things that at first seem widely dissimilar actually turn out to be related in particular ways.

In set theory, this is called a bijection. In algebra, this is called an isomorphism. In topology, this is called a homeomorphism. In differential geometry, this is called a diffeomorphism. In all of these branches of mathematics, we look for patterns among seemingly different things.

To me, this is the essence of mathematics: Finding and classifying structures. Some other mathematicians might have other opinions about this. It turns out that numbers are useful structures as well, and the branch of mathematics that studies the properties of numbers (especially the integers and their generalizations) is aptly called number theory, but it is less about specific numbers and computations as it is patterns in numbers at large.

Can I study mathematics even if I sucked at it in school?

Well, there is nothing preventing you from trying. Mathematics in its purest form is not very different from artistic endeavors. It requires creativity, patience, ingenuity and a certain sense of curiosity. If you really like puzzles, you will love modern mathematics.

On top of this, previous performance in mathematics, especially in a school setting, does not really mean much. I know plenty of people who were average at best in school and ended up becoming outstanding Ph.D. students in mathematics. You do not need to compute numbers fast to be good at mathematics.

At some point, I intend to write a textbook on elementary mathematics (primary and middle school) for adults at some point, but as you might have noticed, I tend to work on a lot of things at once.

What kind of mathematics do you like?

I thoroughly enjoy a large portion of modern mathematics, but the fields I have studied and appreciated the most tend to start with the adjective ‘algebraic’: algebraic geometry, algebraic number theory, algebraic topology, algebraic coding theory, etc.

At the same time, my early love for mathematics emerged through mathematical physics and differential geometry, in particular after I watched Frederic Schuller’s Lectures on the Geometrical Anatomy of Theoretical Physics.

I am also interested in applications of mathematics to other sciences, most notably finance, cryptography and ecology (in the biological sense).

How can I study mathematics?

Mathematics is not a spectator’s sport. You do not learn mathematics; you do mathematics, and through this, you learn. Practically speaking, this means your focus should generally be on solving problems in order to learn mathematics, not the other way around.

If you’re a total beginner, start by going through Khan Academy. Once your bases are covered (at the very least, up to precalculus), you can start going through the mathematics listing.

Most everyone interested in a basic education in undergraduate mathematics should cover the entirety of Part I: Core Subjects. After this, you can pick and choose. If you’re only interested in a specific portion of mathematics and do not care about general mathematics, then feel free to disregard this and go straight to what you want, keeping in mind you might struggle more than expected.