The first important theorem a student of algebra learns is the fundamental theorem of homomorphisms (FTH), or more commonly a special case known as the first isomorphism theorem. However, the theorem has more depth than might seem at first look and one cannot avoid seeing it pop up in various places along the journey to mastering algebra. In this post …

Some time ago I finally decided to sit down and work through the details of constructing the Hilbert schemes of projective schemes. I figured it would be nice to try to write down some kind of a "road map" for the construction and that is what I will attempt here. To be clear, the point of this post is not …

I was reading about first order deformations from Hartshorne's Deformation Theory some time ago, and was thrown off by a small mistake in one of the exercises. Exercise 4.1 in section 4 of chapter 1 asks to prove that if a k-algebra endomorphism f: A → A of a local k-algebra with residue field k induces an isomorphism A/m² → A/m², …

Aalto University hosts an annual maths camp for upper secondary school students of ages 16-18, and I was kindly granted two 1 hour sessions in the camp this year. Given this unique and exciting opportunity, I wanted to give my best shot at designing a fun activity for the students. Since accomplishing anything requires a solid plan, I first asked …

One of the hardest modules I took at Imperial was MATH70061 Commutative Algebra taught by Yankı Lekili. What made the module so challenging was the fact that we had to submit solutions to 6-8 tedious homework problems every 2 weeks. In addition, the course covered quite a broad selection of material, which we had to stay on top of. The …

Morphisms between affine varieties are determined simply by specifying a fixed number of polynomials. On the other hand, morphisms between projective varieties are not as easy to deal with in general and may sometimes require construction via gluing. If one tries to understand collections of morphism, it becomes tricky to keep track of all the "gluing data". This problem is …

I decided to start this blog by writing about my favourite proof that can be fit in a blog post and which put me on the track to learn algebraic geometry. I was originally fascinated by the connections that algebraic geometry makes between branches of mathematics and uses these to solve actual problems, which is what I saw nicely demonstrated …