Currently I am focusing on the calculus of variations and its interplay with partial differential equations, e.g. topics such as the Iwaniec-Martin conjecture. I am also interested in various topics in harmonic analysis, and I try to sneak them into my research as often as I can (see for example how UMD spaces make an appearance here). If there's any maths you'd like to discuss with me, send me an email!
Recently, I have gotten some results related to Morrey's problem (see here). The technique used in the paper is, I believe, quite interesting: in essence, the rank-one convexity side of the problem gets translated (via the laminate-martingale bridge that I have been developing) into one of martingale transforms in Banach spaces, and hence perfectly suitable to the UMD machinery; this also allows one to construct a lot of new counterexamples to Morrey's problem, even very simple ones (the paper contains two homogeneous fourth degree polynomial counterexamples). I'm still trying to understand more deeply the structure behind this, so if you are interested, want to have a chat on the topic, etc, by all means feel free to contact me. Similarly, almost all the dimensions in the paper are "large enough": they haven't been optimised in the slightest, partly due to me being too busy/lazy/not good at optimisation (take your pick or mix and match), and partly because the most straightforward approach, i.e. a computational one, seems to require more resources than I have at my disposal. If you are interested in this, and/or have some interesting observations on the topic, feel free (and encouraged!) to contact me.