In this paper we study the singularities of holomorphic functions of bicomplex variables introduced by G. B. Price ($An Introduction to Multicomplex Spaces and Functions$, Dekker, New York,1991). In particular, we use computational algebra techniques to show that even in the case of one bicomplex variable, there cannot be compact singularities. The same techniques allow us to prove a duality theorem for such functions.

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We show that Thurston’s skinning maps of Teichmüller space have finite fibers. The proof centers around a study of two subvarieties of the $${{\rm SL}_2(\mathbb{C})}$$ character variety of a surface—one associated with complex projective structures, and the other associated with a 3-manifold. Using the Morgan–Shalen compactification of the character variety and author’s results on holonomy limits of complex projective structures, we show that these subvarieties have only a discrete set of intersections.

This note considers the hyper-contractivity and the uniqueness of the weak solutions to the two dimensional Keller–Segel equations. We prove a refined hyper-contractive property and consequently obtain the uniqueness of global weak solutions provided that the initial data satisfy , and . We also extend the result to higher dimensions.

Due to the length this work is published in two parts. The second part will appear in Vol 23: 1 of this journal.