In this post, I will (try to) explain my research to non-mathematicians. I apologise in advance to the experts that at many points, I will be a bit imprecise to simplify matters.
I’m interested in a field that is called ‘Algebraic Topology’ or more precisely, ‘Betti number gradients’. I’ll explain these terms below.
What is Topology?
Topology is a branch of mathematics where we study geometric objects, which we call ‘(topological) spaces’. We’re only interested in the ‘rough’ shape of these objects, meaning that we allow certain deformations like e.g. stretching, twisting, bending, but not opening or closing holes. We will regard two objects as ’equivalent’ if we can deform one to the other using these rules. For example, a tea mug can be deformed into a doughnut-shaped object (mathematicians call this object a ‘(full) torus’). In a similar way, the animation below shows that a cow is equivalent to a ball.
(Lucas Vieira, Mug and Torus morph, marked as public domain, see Wikimedia Commons)
What is Algebraic Topology?
Now, one can ask ‘Is the doughnut equivalent to the cow?’. One might suspect this not to be the case, but how can we prove it? So far we only have methods to prove that two objects are equivalent. In order to show that two objects are not equivalent, it is not feasible to try out all possibilities for deformations because there are infinitely many such possibilities. So at no point in time, we could have checked all of them. Instead, we need to find some property that is invariant under deformations. In this instance, a suitable invariant is the ‘(first) Betti number’. It measures the number of holes in an object (of a certain type). So, the doughnut has one hole in the centre. We would say that its first Betti number is equal to one. On the other hand, the cow has no holes in it, i.e. its first Betti number is equal to zero. Now, if the doughnut and cow were equivalent, they would have to have the same first Betti number. Since they do not, they cannot be equivalent.
What we’ve just seen is an instance of algebraic topology: we solved a problem in topology by transforming it into some object in algebra (here: the first Betti number, which is a natural number), where we can solve the problem of equivalence/equality.
What are Betti number gradients?
In the above paragraph, we got to know Betti numbers. These are well-studied and have nice properties. However, as often in mathematics, it is not only interesting to understand single objects but also their asymptotic behaviour. Concretely, call the topological space in question \(X\). If we are given a natural number \(d\) (i.e. 1,2,3,…), we often can build a ’larger’ space that winds around \(X\) exactly \(d\) times. We call this new space \(X_d\) and think of \(X_d\) as a space ‘above’ \(X\). We call \(X_d\) a covering of \(X\) and say that \(d\) is its degree. If there are multiple ways to pick such a space, let’s pick one of them (for the sake of this argument). Here are two examples of such spaces: If we take \(X\) to be a circle, and \(d=3\), a covering looks like this.
We denote this ‘winding around’ by an arrow from the covering to the original space. By untangling, we see that \(X_d\) is again (equivalent to) a circle.
Let’s take \(Y\) to be a figure 8 shape and again \(d=3\). A covering may look like this.
We notice that \(Y_d\) looks like two figure 8 shapes glued together or, alternatively, a figure 8 shape with four holes instead of two.
Recall that we mentioned above that we’re not interested in the particular case \(d=3\), rather we care about the behaviour in the limit, i.e. what happens if we make \(d\) larger and larger. In our first example, every covering will be equivalent to a circle, no matter how large we make the degree. As a circle, its first Betti number is equal to one. We now say that \(X\) has vanishing first Betti number gradient because the first Betti number of coverings (i.e. \(1\)) becomes arbitrarily small compared to the degree \(d\) when \(d\) grows larger. In other words, for large \(d\), the first Betti number of a corresponding covering will be close (in comparison to \(d\)) to \(0\cdot d\). We denote this as \[ \widehat b_1(X) = 0. \] On the other hand, in the second example (the figure 8 shape), the general formula for the first Betti number of a covering is as follows: \[ b_1(Y_d) = 1\cdot d+1. \] For large \(d\), the first Betti number of a corresponding covering will thus be close (again - in comparison to \(d\)) to the degree \(1\cdot d\) . We denote this fact as \[ \widehat b_1(Y) = 1. \] In a similar fashion, we can define the first Betti number gradient for any topological space.
What I do in my PhD
In my PhD, I’m mainly interested in the case of vanishing Betti number gradients. For instance, in the article ‘Torsion homology growth and cheap rebuilding of inner-amenable groups’, I was able to show vanishing of the first Betti number gradient for certain geometric objects that come from algebraic structures of a ‘small’ type. In another article on the ‘algebraic cheap rebuilding property’, together with my coauthors, we establish some criteria that are easier to verify in order to show vanishing of Betti number gradients.
What is all of this good for in my everyday life?
I’m not (yet) aware of a real-world applications of Betti number gradients and my research. However, this is a common phenomenon in pure maths, which is the study of mathematical objects independent of potential applications, just out of curiosity.
Undoubtedly though, pure maths has had a large influcence on all STEM subjects, which have produced lots of useful applications. Often enough, applications came at unexpected moments and surprising fields: e.g. since the ancient greeks, mathematicians in the field of number theory have studied prime numbers and their behaviour. Only this research has enabled the creation of asymmetric cryptography with methods like RSA. This has greatly changed the internet: without it, there would not be online banking, online shopping or end-to-end encrypted private messaging.
Interested in more?
If this post sparked your interest in Algebraic Topology (or mathematics in general), and you’d like to know more about my research, please get in touch. I’m happy about any question or comment.