I'm an adjunct lecturer at Tel Aviv University and Ben Gurion University.
I received my Ph.D disseration in Mathematics from the University of Michigan, where my advisor was David Barrett. My specialization is complex analysis in several variables, with focus on special operators and spectral data.
E-mail: yonshel at umich.edu
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.
The Leray transform: Distinguished measures, symmetries and polygamma inequalities Joint with Luke Edholm
Journal of Functional Analysis, February 2025, Volume 288, Issue 3, Article 110746
Advances in Applied Clifford Algebras Advances in Applied Clifford Algebras, September 2017, Volume 27, Issue 3, pp 2805–2840,
(available online via Springer).
Algebraic geometry of Abel differential equation Joint with Shira Giat, Clara Shikhelman and Yosef Yomdin
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, March 2014, Volume 108, Issue 1, pp 193-210
More papers are currently in preparation in connection to my dissertation, titled A Spectral Exploration of the Leray Transform in Two Different Settings in C^2.
The inverse spectral problem for the Leray transform in two settings in C^2 Virtual East-West Several Complex Variables Seminar, September 2021.
An inverse spectral problem for the Leray transform in two settings AMS Special Session on Several Complex Variables, October 2020.
Exploring the Leray spectrum of convex Reinhardt domains Operators in Complex Analysis Seminar (University of Michigan), December 2019.
I've been an adjunct lecturer at Tel Aviv University and Ben Gurion University since 2024, following my postdoc at the University of Ljubljana (2022-2024). As of July 2025, I will also be writing a Linear Algebra coursebook (for Enigneering students) and proposing AI-related tools for Ben Gurion University.