Getting into my project with a little representation theory

As the new semester dawns, so begins my final/masters year. This includes a year-long project, for me on Grassmannians, though I’m still not sure on the details. I’ll be sharing some of the things I learn on this site, beginning with some elementary representation theory of finite groups. I didn’t study the representation theory module … Continue reading “Getting into my project with a little representation theory”

Pop maths: A sliver of algebraic geometry – Part 3

This is the final post in this series, where I give a layperson’s explanation of a result from algebraic geometry, using only high school–level algebra as a starting point. If you’ve not read the first two in the series, you might want to start here. In this post, we’re going to need a couple of … Continue reading “Pop maths: A sliver of algebraic geometry – Part 3”

Pop maths: A sliver of algebraic geometry – Part 2

You might want to check out part 1 before reading this. A little motivation Last week, we took the Euclidean plane and forgot what angles and distances were, while retaining the concept of parallel lines. This gave us a new space called the affine plane, . Our mission today is to describe an extension of … Continue reading “Pop maths: A sliver of algebraic geometry – Part 2”

Pop maths: A sliver of algebraic geometry – Part 1

This 3-part series attempts to explain a classical result in algebraic geometry to a lay-audience, using only high-school level mathematics. From ancient Greece to the modern era: Cones, curves, and coordinates In the 2nd century BC, Apollonius of Perga studied and classified the so-called conic sections: geometric curves obtained by intersecting a flat 2-dimensional plane … Continue reading “Pop maths: A sliver of algebraic geometry – Part 1”

Brief update

This week I’ve been super busy revising since all of my exams are in the upcoming week. I’ve not really had a chance to think much about writing anything here. However, I will say that I have a sort of popular mathematics post in the pipeline, where I’ll try to explain the classification of projective … Continue reading “Brief update”

Some LaTeX beginners’ mistakes and tips

I didn’t post last week and can’t really guarantee consistent posts for the rest of the month; it’s exam period now. Normal service will resume in June. For today, I’m just going to go over a few things that can make early attempts at LaTeX typesetting look much more professional. At my university, there’s virtually … Continue reading “Some LaTeX beginners’ mistakes and tips”

Elementary proof of Krull’s intersection theorem

When doing my 3rd-year project, I found I needed a classical theorem in commutative algebra called Krull’s Intersection Theorem. It turned out to be very useful; I used it in three proofs involving formal power series, and it also gave me some intuition for DVRs. However, despite being an intuitive result, the proofs I found … Continue reading “Elementary proof of Krull’s intersection theorem”

Make time to be bored

The last couple of year, I’ve increasingly scrutinized my relationship wtih technology, and the relationship we seem to cultivate in our culture. We instinctively react to its beeps and buzzes, we whip it out whenever there’s a quite moment, and we sometimes read texts out the corner of our eye during real-life conversations. It can … Continue reading “Make time to be bored”

Algebraic derivatives

In commutative algebra and algebraic geometry, a common operation is to take derivatives of polynomials. This would seem a fairly straightforward thing to do, but in commutative algebra/geometry, we study polynomials over arbitrary rings and fields, and the good old derivative from standard calculus is quite dependent on the metric structure of or , since … Continue reading “Algebraic derivatives”