Dynamical Systems and Analysis Seminar - Nikolai Prokhorov
| Dates: | 23 March 2026 |
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| Times: | 14:00 - 14:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Current University students |
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Speaker: Nikolai Prokhorov (University of Manchester)
Title: Graphs and meromorphic functions
Abstract: Let f be a polynomial (in one variable) with complex coefficients. It can be viewed as a self-map of the complex plane C. Suppose that f has exactly two critical values, namely -1 and 1. Consider the segment -1,1 and its preimage under f, i.e., f^{-1}(-1,1). This set forms a planar tree in C. Moreover, the isomorphism class of this tree as a planar graph determines the polynomial f almost uniquely, up to pre-composition with an affine map. Such trees are known as dessins d'enfants and serve as a starting point for a rich theory developed by Grothendieck, Belyi, and others. This combinatorial construction extends naturally to polynomials with more than two critical values. Analogous constructions also exist for large classes of entire or meromorphic functions on C. For instance, it applies to maps of Speiser class - that is maps with finitely many singular values, i.e., values where not all inverse branches can be defined; examples include the functions cos, exp, and tan, and many others. In this broader setting, analogous combinatorial objects are often referred to as line complexes. In this talk, I will discuss how this combinatorial approach can be used to analyze function-theoretic and dynamical properties of broad classes of entire and meromorphic functions.
Room: Frank Adams 1
Further information: https://personalpages.manchester.ac.uk/staff/yotam.smilansky/dynamics_analysis
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Frank Adams 1
Alan Turing Building
Manchester
