When I was accepted into AMSI’s summer research scholarship program, I confess I didn’t really know what I had signed up for. When my supervisors told me that I would be studying the properties of symmetric random walks, I felt none the wiser. My supervisor Andrea then described to me an ingenious proof contained in the book I would be studying, and I immediately got excited about the what the next six weeks had in store for me.
It was a proof of the following fact: If you place a particle on the number line, and have it move one unit to the left or right with equal probability, then the particle is guaranteed to eventually visit every other point. The proof works on some very simple reasoning about the shape of the graph of the probability of reaching one point from another, and through this reasoning eliminating all possibilities other than a uniform probability one of reaching any other state.
For me, this proof exemplifies a lot of what I love about mathematics. I thought that it was elegant, beautiful, and shockingly simple. The content of my research turned out to be reading an old Russian book about random walks. It’s an awesome book, full of problems and proofs with the same qualities as that first proof that I was introduced to. It was also full of weird wording, ambiguous notation, and massive leaps in logic with no explanation for the reader – or at least for a reader of my mathematical ability!
After 6 weeks, I have rendered the first chapter of the book into what I hope is a crisp, clear, and understandable form. It was a lot of work, over the course of which I learned an enormous amount. If you want to learn some really interesting mathematics, I hope you’ll go find my paper and dive in!
By Liam Wood-Baker,