I’m a mathematician that dabbles. I wrote my honours thesis in graph theory, my PhD thesis in analytic number theory (gotta love those prime numbers) before working as an options trader at Optiver for five years.

These days I live in Queensland with my wife and our at-most-three children. I have two roles: I work in the Education team at Optiver Sydney and I am an academic at UQ.

I spend my spare time playing soccer and chess and chasing my three tiny children around the backyard.

Supervision

I am always keen to speak to potential honours and PhD students. If you would like to do a project in number theory or graph theory, then feel free to send me an email so that we can have a chat.

Writing to Me

You can write to me at awdudek[AT]gmail.com or a.dudek[AT]uq.edu.au.

My Research Rant

I enjoy working on problems in number theory, particularly anything to do with the distribution of prime numbers, and graph theory, particularly anything to do with the spectrum of a graph.

Number theory is a beautiful area. The primes are a puzzling sequence and the endless array of mathematical machinery we throw at them yields limited information. I am an analytic number theorist, which means I mostly throw tools from analysis (typically real and complex calculus) at the primes. It still floors me that we can use such tools to study discrete sets and indeed the actual mathematics is quite striking.

Some of my number theory research focusses on furnishing explicit estimates in the theory of prime numbers.

For example, it has been known since 1937 that there is a prime between $n^3$ and $(n+1)^3$ for all sufficiently large values of $n$ (this is a step towards Legendre’s Conjecture). One result from my PhD thesis was to make this explicit by showing that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.3))$.

Another interesting area of research to me surrounds the well-known Riemann Hypothesis and its consequences to the distribution of prime numbers. Some of my work revolves around assuming that the Riemann Hypothesis is true and then seeing what we can prove about the primes from this assumption. See here and here.

I also like additive problems such as weak versions of Goldbach’s Conjecture that every even number greater than two can be written as the sum of two prime numbers. In my research, I have shown that every integer greater than two can be written as the sum of a prime and a square-free number (see here).

I also proved a more interesting result with Dave Platt that all integers greater than or equal to 10 (and provided they are not congruent to 1 mod 4) can be written as the sum of the square of a prime and a square-free number.

As mentioned, I am also interested in graph theory, specifically the theory of expander graphs, and I think it’s neat to use prime numbers and number theory to prove results in this area. In one of my papers, I used a simple result from number theory to show that a class of graphs were poor expander graphs.

You can see a complete list of my papers on my Publications page.