I've been studying properties of the Leray transform (a fairly natural analogue of the Cauchy transform in one variable) in the setting of convex Reinhardt domain in C^2, which have rotational symmetry with respect to each variable. I'm mostly interested in the singular spectrum of the Leray transform and how it classifies the underlying convex Reinhardt domain. Can one hear the shape of such a domain? The answer probably depends on the exact meaning of "hearing". "Shape" is only defined up to dilations, variable swaps and taking the projective dual, all of which preserve the spectrum. One of my current results is that in two dimensions, you can distinguish (up to dilations and duality) two "almost" C^2 smooth, convex Reinhardt domains (satisfying one more condition) by looking at each spectrum as a labeled set. The unlabeled case seems more complicated, and might not have a straightforward statement. I was also able to compute the Leray norm for l_p balls and their dilations (given by a|z_1|^2+b|z_2|^p<1 for p>1 and a,b>0) for a special measure, and more generally for domains in this setting, the essential norm for a family of measures.
I've also been studying the setting of rigid Hartogs domains in C^2, which have one rotational symmetry in addition to translation invariance (in the real part of one of the variables), so the spectrum generally lacks eigenvalues (it is generally continuous due to the translation invariance). Still, many similarities between the two settings arise. In both settings, asymptotic analysis plays a special role in establishing sufficient conditions (some are necessary) on the domain for the Leray operator to be L^2 bounded with respect to various measures. Moreover, the inverse spectral problem can be solved in essentially the same way, with the same kind of restriction as before.
My main future direction is to understand these settings in higher dimensions. In fact, the higher the dimension, the more such settings there are. At the same time, there are still things to do in C^2 such as norm computations.
Going back to an earlier period, I have a Bachelor's and Master's in Mathematics from Tel Aviv University. My advisor was Daniel Alpay and the title of my thesis is Quaternionic Wiener Algebras, Factorization and the Corona Theorem. The thesis extends theorems from complex analysis to the quaternionic setting, with the Corona theorem in question being an analogue of the matrix-valued one. I published one related paper, regarding Wiener algebras and factorization.