First week in PhD

I just officially started my PhD last week, but I already started 3 weeks ago. Here are some of the topics and cool facts I have been reading:

Frobenius Algebra (strongly separable condition, knowledgeable Frob. algebra)

Sheaf: not much apart from allowing me to make some jokes.

Geometric group theory

• Fix k. Let G be a finitely-generated group. There are finitely many subgroups index k.
• Every subgroup of finite index of a finitely-generated group is finitely generated.
• Let G be a free group of finite rank. Then a normal subgroup of G is of finite index if and only if it is finitely generated.
• Table-Tennis lemma: more jokes for me.
• Kurosh's theorem

Reidemeister-Schreier method:  I use this to prove the first point

1. Fix k. Let G be a finitely-generated group. Let H be a subgroup of index k. Then there exists a system of representatives such that the length of each representative is not exceeding k.
2. Let G be a finitely-presented group. Then every finite index subgroup is finitely presented. 

Transportation Network: Coordinated vs Anarchy

Recently I wrote an article for Paradox (Melbourne University Maths and Stats  society's magazine). It can be found on page 21-24 on this issue. Below are the first few paragraphs.

In the "real" world, many systems can be characterised by a network with nodes and paths joining them. Here, we will consider traffic flows of a decentralised transport system for personalised vehicles. It is natural to ask ourselves whether or not this network system is the most efficient one, alternatively, on average, does this road network allows commuters to get from A to B most efficiently?

On the surface, this question seems easy to answer; one can

• set up a model of the system;
• find the global minimum using the convex minimum cost flow algorithm.  

In reality, individual commuters do not collectively opt for the most optimal strategy, but their own optimal strategy. Hence, the actual performance of the network is often far from its best, even if all individuals choose the quickest route and all information is available to them. The key question here is to understand how far the actual performance is from the most optimal one.