Hi all! It’s been a while since I’ve written. Truthfully, I’ve been wanting to introduce some “advanced” concepts while keeping the material readable. Today we’re going to talk about homology! Homology is a tool we use to count holes in a space. It’s a very smart way to count holes, because as we’ll see, defining a hole is kind of hard. Unfortunately, the “output” of homology are groups or vector spaces. But we’ll deal with this by only looking at the “rank” of the group or vector space, which is a number. …

I’ve recently gotten my drawing tablet and it’s made me way more motivated to work on this blog, so I guess we’re cranking out a Part 3 at 8:21 am!

Today we’re going to talk about knots. These are a fun topic for many reasons. They’re low-dimensional, so we can visualize them easily; they’re very intuitive, since most of us have played with strings at some point; and they have a surprisingly rich theory that starts at a fairly basic level!

But we had an issue last time. **All knots are homeomorphic**. Which is no good. Homeomorphism is what mathematicians…

Last time, I talked about how to think of two shapes being the same. If we can deform one into another, they’re homeomorphic! Seems easy enough? Let’s go through some examples of shapes that may not be obviously homeomorphic, or maybe shapes that you thought were the same but are actually different (at least for homeomorphisms)

The verdict is in and the answer is nope, they’re not the same shape! “But Osama”, you ask, “can’t you just smush the legs of the A in? You could if it was clay”. This is one of those instances where it’s best to…

Hey everyone! I’m trying to pen a series of posts of things I love in Math. My self-professed area is Algebraic Topology, but I’ll be talking about ideas from all sorts of areas. In this post, I’m planning to give a brief introduction to the subject, and meet our first invariant (who we’ll see in various guises), the Euler Characteristic.

Algebraic Topology aims to create a tunnel between shapes and algebraic objects (numbers, polynomials, groups, rings, vector spaces, modules and a lot more). These are called **invariants**¹.

One of the main things we want out of this tunnel is that…