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”

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”

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”

The Snake Lemma in unflattering detail

“Proving the snake lemma is something that should not be done in public…” ~ Paolo Aluffi. The Snake Lemma is a theorem about exact sequences of modules, and is an important tool in homological algebra. Almost every textbook that includes the Snake Lemma leaves it as an “easy” exercise. Atiyah–MacDonald is kind enough to construct … Continue reading “The Snake Lemma in unflattering detail”