107 Alexander Solynin

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 C with the minimal possible number of