Research Projects

Our plan is to concentrate on two types of problems-(1) obtaining information about the zeroes and the critical points of complex polynomials in one variable, (2) the Toeplitz operators-or, more accurately, the Toeplitzness of composition operators. Speaking of our goal (1), many beautiful and important results regarding the zeroes of complex polynomials and their derivatives are known nowadays. We emphasize that one major source of motivation for us is a famous conjecture of B. Sendov from 1958, which states that: " If all zeros of a complex polynomial F with degree at least 2 lie in the closed unit disk, then there exists a critical point of F in every disk of radius one centered at a zero of F. " It is worth mentioning that significant progress on this problem have been made during the last sixty-seven years. Yet, this conjecture continues to be a topic of interest due to the unavailability of a complete solution. Our target (2) is concerned with the theory of Toeplitz operators, and the theory of composition operators, both defined on the Hardy space. At the center of our focus lies the result of F. Nazarov and J. H. Shapiro from 2007, stating that a non-identity, non-compact composition operator can never be a compact perturbation of a Toeplitz operator. Let us also mention that Toeplitz operators keep occurring frequently in the theory of composition operators. While there are volumes of literature on both of these century-old objects, not much has been done after the Nazarov-Shapiro paper to understand how they actually interact with each other. In terms of significance, these questions are known to be difficult and are open for quite some time. Thus, any progress will turn out to be important and will possibly bring new insights. Indeed, the study of zeroes and critical points of a polynomial is a very classical problem, and over the years Sendov’s conjecture has gained a lot of popularity among the mathematicians. It is also worth noting that results obtained from this line of research might have potential real-life applications. Namely, the problem of counting the zeroes of harmonic polynomials and rational functions has applications to the “gravitational lensing problem” in astrophysics. Furthermore, the substantiality of the " Toeplitz product problem " in the framework of composition operators is manifold. Even a partial solution of this question could pave the way for progress on bigger problems-such as the question posed by D. Sarason, which asks for a classification of composition operators belonging to the Toeplitz algebra. We also believe that such results will have a sizable amount of practical impact in future, like the Toeplitz operators do (e.g. applications in the Lenz-Ising Model or in the realm of singular integral operators). Finally, let us conclude this discussion by pointing out that a link between these two above-mentioned, seemingly unrelated areas do exist, which will be discussed in the uploaded .pdf.