Informal writings
Current work
I'm currently working on results related to an apparent generalisation of the cyclic insertion conjecture, as mentioned on my research page. The result concerns various families of identities that can be proven using Brown's MZV coproduct; the identities give `more symmetric' versions of the general cyclic insertion identities, like the `symmetric insertion' result I found earlier.
I also have results about symmetries and relations between weight 5 (or higher) iterated integrals (versions of multiple polylogarithms) which can be found using the symbol map, and coproduct. These include candidates for numerically testable functional equations for genuine multiple polylogarithms, and identities reducing a general weight 5 multiple polylogarithm to an explicit combination of indices 5, 32, and 311, using Dan's reduction method.
PhD Progression reports
Towards the end of my first PhD year, I had to write a (lengthy) progression report in order to continue. The department also asks for (shorter) reports mid-April each year.
More primes of the form
I've recently got interested in extending my fourth year project work a little further. Mainly going back to do things I didn't have time to write up at the time.
- Homogeneous splitting of primes in \( \mathbb{Q}(\sqrt[3]{7}) \), and other cubic fields with \( h_K = 3 \).
- Representation of primes by other cubic forms from \( \mathbb{Q}(\sqrt[3]{11}) \).
- Binary quadratic forms of discriminant \( D = -47 \) and Ramanujan \(\tau \)-convolutions.
Self-similar sums of squares (originally square sum concatenation) - Number Theory 3/4 Challenge
Dr Gangl was very fond of setting challenges during his lectures in Number Theory 3/4 (and Algebra & Number Theory 2). One such (very open ended) challenge was as follows.
Observe that \( 12^2 + 33^2 = 1233 \), find as many such pairs \( (x,y) \), with \( x^2 + y^2 = x \text{ concat } y \), as possible.
- You can read more about this challenge, and view the solutions and infinite families I found.