The Department of Mathematics at Purdue University offered an 8-week residential program to conduct research in pure mathematics from June 10, 2013 through August 2, 2013. The program was sponsored by the National Science Foundation (NSF) as well as generous gifts from Ruth and Joel Spira (BS '48, Physics) and Andris "Andy" Zoltners (MS '69, Mathematics).
Let \(\beta(z) = p(z)/q(z)\) be the ratio of two polynomials. We can view this as a map which takes complex numbers \(z\) to complex numbers \(w = \beta(z)\). We say that \(z\) is a critical point if the derivative \(\beta'(z) = 0\). If \(z\) is a critical point, we say that \(w = \beta(z)\) is a critical value. Such a rational map \(\beta(z)\) is said to be a Belyĭ map if it has at most three critical values \(w_0\), \(w_1\), and \(w_\infty\). We can always choose these critical values to be \(w_0 = 0\), \(w_1 = 1\), and \(w_\infty = \infty\). Are there lots of examples of such maps? How can we construct them efficiently?
There are groups associated with such maps. Let \(G\) be the collection of maps in the form \(\sigma(z) = (a \, z + b) / (c \, z + d)\) such that \(\beta \bigl( \sigma(z) \bigr) = \beta(z)\). Then \(G\) is a finite group under composition. Moreover, if \(z\) is a critical point, then \(\sigma(z)\) is also a critical point for any \(\sigma \in G\). But what kinds of groups \(G\) occur? Must \(G\) be abelian? Or solvable? Given any group \(G\), how do we construct a Belyĭ map associated with it?
We can associate graphs with such maps as well. Let \(B\) denote the collection of all complex numbers \(z\) such that \(\beta(z) = 0\), and let \(W\) denote the collection of all complex numbers such that \(\beta(z) = 1\). This gives a collection of vertices; we color those in \(B\) to be "black" and those in \(W\) to be "white." As we draw a path in the complex plane for the line segment \(0 \leq w \leq 1\), we find a collection of paths from the points in \(B\) to the points in \(W\). We denote these paths by \(E\); they form edges between the "black" vertices and the "white" vertices. Such a graphs \(\Delta_\beta\) is called a Dessin d'Enfant. What do such graphs look like? Can we draw examples? Given any graph \(\Gamma\), can we find a Belyĭ map \(\beta\) such that \(\Gamma = \Delta_\beta\)? And how do the groups \(G\) act on these graphs? Is there some special symmetry we can see?
Much more detailed information and discussion can be found here.
Let \(X = \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R)\) denote the Riemann sphere. A Belyĭ Map \(\beta: X \to \mathbb P^1(\mathbb C)\) is a rational function \(\beta(z) = p(z) / q(z)\) with critical values $$ \left \{ (\omega_1 : \omega_0) \in \mathbb P^1(\mathbb C) \, \biggl| \, \text{disc}\bigl( \omega_1 \, q(z) - \omega_0 \, p(z) \bigr) = 0 \right \} \subseteq \bigl \{ (0:1), \, (1:1), \, (1:0) \bigr \}.$$
A dessin d'enfant \(\Delta_\beta = \bigl( B \cup W, \, E \bigr)\) is a bipartite graph with ``black'' vertices \(B = \beta^{-1}(0)\), ``white'' vertices \(W = \beta^{-1}(1)\), and edges \(E = \beta^{-1} \bigl( [0,1] \bigr)\). The M\"{o}bius Transformations \(\gamma: X \to X\) act on such graphs, so we consider the Galois group $$ \text{Aut}(\beta) = \left \{ \gamma(z) = \dfrac {a \, z + b}{c \, z + d} \in \text{Aut}(X) \ \biggl| \ \beta \left( \dfrac {a z +b}{c z + d} \right) = \beta(z) \right \}.$$
We are motivated by two research problems: