Quadratic differentials in complex analysis and beyond

Abstract: I will discuss the role of quadratic differentials in the extremal problems in Complex Analysis and beyond. We start with main definitions, then discuss Jenkins' theory of extremal partitioning, and then I will mention main results of the differentiation theory for the Jenkins' weighted sum of moduli suggested by this speaker in 1985-2000.

Turning to applications, I show first how quadratic differentials can be used to study fingerprints of (complex) polynomial lemniscates. The main result here includes, as special cases, Ebenfelt-Khavinson-Shapiro characterization of fingerprints of polynomial lemniscates as well as Younsi characterization of rational lemniscates. Then I will show that every real algebraic curve can be treated as a trajectory of a quadratic differential defined on a certain Riemann surface.

After that, we will discuss how quadratic differentials on $\overline{\mathbf{C}}$ with the minimal possible number of poles (that is $4$) can be used to solve the problem on the canonical embeddings of pairs of arcs, studied recently by M. Bonk and A. Eremenko, and in several other extremal problems on ring domains.

complex variablesdynamical systems

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

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