Bers Slices in Families of Univalent Maps

A general coefficient theorem for univalent functions

Using the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces and some of their other complex geometric features, we prove a general theorem on maximization of homogeneous polynomial (in fact, more general holomorphic) coefficient functionals $J(f) = J(a_{m_1}, a_{m_2},\dots, a_{m_n}) $ on some classes of univalent functions in the unit disk naturally connected with the canonical class $S$. The given functional $J$ is lifted to the Teichmuller space $\mathbf T_1$ of the punctured disk $\mathbb{D}_{*} = \{0 < |z| < 1\}$ which is biholomorphically equivalent to the Bers…

Quaternionic analogues of Bers's theorem and Iss'sa's theorem

In their recent work, Gentili and Struppa proposed a different quaternionic analogue of the notion of holomorphic functions in the complex plane, called \textit{slice regular functions}, which has led to several analogues of classical theorems in complex function theory in the quaternionic setting. The quaternionic analogue of meromorphic functions is called \textit{semiregular functions}. Such a function is said to be \textit{slice preserving} if it maps each domain in a complex line to a domain in the same complex line. In this work, we deal with the relation between analytic and algebraic p…

Teichmüller空间、多项式轨迹与代数对应空间中的退化

Teichmüller spaces, polynomial loci, and degeneration in spaces of algebraic correspondences

该研究在代数对应语境下提出特征簇类似物，发现富克斯群与多项式的交配包含其中，通过固定多项式或富克斯群得到类Bers切片，证明其有界性并启动紧化研究，揭示新退化现象且对四次穿孔球面证明紧化自然同胚。

反全纯对应与配对 II：Shabat多项式切片

Antiholomorphic correspondences and mating II: Shabat polynomial slices

研究以有根dessin d'enfants索引的Shabat多项式单叶限制产生的反全纯对应族，证明参数空间为拓扑四边形，设计拼图结构产生组合刚性，发展重整化理论描述Multibrot和Multicorn副本，并构造局部连通的组合模型。