# PRiME 2012: Purdue Research in Mathematics Experience

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The Department of Mathematics at Purdue University will 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).

## Background and Project Description

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.

## 2013 Research Questions

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:

• Graph Problem: Given a bipartite graph $$\Gamma$$ on $$X$$, exhibit a Belyĭ map $$\beta: X \to \mathbb P^1(\mathbb C)$$ such that $$\Gamma \simeq \Delta_\beta$$.
• Group Problem: Given a finite group $$G$$, exhibit a Belyĭ map $$\beta: X \to \mathbb P^1(\mathbb C)$$ such that $$G \simeq \text{Aut}(\beta)$$.
In order to solve these two problems, students will be divided into two research groups.

## Group #1: Drawing Planar Graphs via Dessins d'Enfants

Abstract. There are many examples of finite, connected planar graphs $$\Gamma$$: for instance, there are paths, trees, cycles, webs, and prisms to name a few. In 1984, Alexander Grothendieck, inspired by a result of Gennadiĭ Belyĭ from 1979, constructed a finite, connected planar graph $$\Delta_\beta$$ via certain rational functions $$\beta(z) = p(z) / q(z)$$ by looking at the inverse image of the interval from 0 to 1. In this project, we investigate how restrictive Grothendieck's concept of a Dessin d'Enfant is in generating all planar graphs. We show that certain trees (such as stars, paths, and caterpillars), certain webs (such as cycles, dipoles, wheels, and prisms), and each of the Platonic solids (the tetrahedron, cube, octahedron, icosahedron, and dodecahedron) can all be generated as Dessins d'Enfants by exhibiting explicit Belyĭ maps $$\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$.

## Group #2: Associating Finite Groups with Dessins d'Enfants

Abstract. Each finite, connected planar graph has an automorphism group $$G$$; such permutations can be extended to automorphisms of the Riemann sphere $$S^2(\mathbb R) \simeq \mathbb P^1(\mathbb C)$$. In 1984, Alexander Grothendieck, inspired by a result of Gennadiĭ Belyĭ from 1979, constructed a finite, connected planar graph $$\Delta_\beta$$ via certain rational functions $$\beta(z) = p(z) / q(z)$$ by looking at the inverse image of the interval from 0 to 1. The automorphisms of such a graph can be identified with the Galois group $$\text{Aut}(\beta)$$ of the associated rational function $$\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$. In this project, we investigate how restrictive Grothendieck's concept of a Dessin d'Enfant is in generating all automorphisms of planar graphs. We discuss the rigid rotations of the Platonic solids (the tetrahedron, cube, octahedron, icosahedron, and dodecahedron), the Archimedean solids, and the Catalan solids via explicit Belyĭ maps. Conversely, we enumerate groups of small order and discuss which groups can -- and cannot -- be realized as Galois groups of Belyĭ maps.

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