# PRiME 2013: Purdue Research in Mathematics Experience

[Flyer] | [Program Overview] | [Planned Activities] | [Project Description] | [Application] | [Summary of Results]

# Program Overview

## General Information

The Department of Mathematics at Purdue University will offer an 8-week residential program to conduct research in pure mathematics. The program, entitled PRiME (Purdue Research in Mathematics Experience). will run from June 10, 2013 through August 2, 2013. The program is being 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).

Between 6 and 8 undergraduate students will be selected to conduct research with Dr. Edray Goins, Associate Professor of Mathematics. Students will choose from two different projects in Number Theory. These projects will focus on the theory of Dessins d'Enfants, an emerging field which combines ideas of Abstract Algebra, Complex Analysis, Differential Geometry, Graph Theory, and Number Theory.

## Goals and Expectations

During the summer, each of the undergraduate participants will:

• Complete a research project done in collaboration with other PRiME students.
• Give a presentation and write a technical report.
• Attend a series of colloquium talks given by leading researches in their fields.
• Attend workshops aimed at developing skills and techniques needed for research careers in the mathematical sciences.

In order to successfully complete this project, participants will:

• Meet at least 10 hours every week for a minimum of 8 weeks.
• Be introduced to Abstract Algebra, Complex Analysis, Differential Geometry, Graph Theory, and Number Theory.
• Learn how to use an advanced symbolic computational package, such as Sage or Mathematica.
• Learn how to use $$\LaTeX{}$$, a mathematical typesetting language.
• Write a technical paper explaining the details of the project.
• Design a poster giving an overview of the project.

## Stipend and Travel

Upon the successful completion of the 8-week program, participants will receive a $4,000 USD stipend. They will also have room covered for the eight weeks from Sunday, June 9, 2013 and through Saturday, August 3. (Meals will not be covered for the participants.) Travel expenses up to$500 USD to and from the Purdue West Lafayette campus will also be covered.

## Prerequisites

Students must be undergraduates in good standing, although preference will be given to applicants who will begin either their Junior or Senior year in the Fall of 2013. Applicants must have taken a proof-based course in Abstract Algebra, Discrete Mathematics, and/or Number Theory. It is not required that participants be US Citizens.

## Application Instructions

You can fill out the application online at the bottom of this page by clicking here. Applicants should have the following items sent via e-mail to egoins@math.purdue.edu:

• Current resume or curriculum vitae
• Unofficial transcript
• Three letters of recommendation

Complete applications should be received no later than Friday, April 26, 2013. Decisions will be announced by Monday, April 29, 2013.

[Flyer] | [Program Overview] | [Planned Activities] | [Project Description] | [Application] | [Summary of Results]

# Planned Activities

## Field Trips

Participants will join in weekly activities to local outings in Indiana. Some potential locations include:

In cooperation with the Center for Faculty Success, PRiME will host a speaker series entitled "Negotiating a Career in the Mathematical Sciences". Dr. Goins and Dr. Alejandra Alvarado, working in conjunction with the Purdue Student Chapter of the Association of Women in Mathematics (AWM), will bring in five outside speakers to discuss their professional journey from being an undergraduate student to being a member of the professoriate. There will be a dinner following each speaker on Friday evenings. These meals will be free for PRiME participants.

Here are the confirmed speakers:

Invited speakers will be underrepresented minorities who are young faculty at institutions of higher learning having significant underrepresented student populations. This will be a continuation of the series from "2011 Midwest Crossroads AGEP PRiME: The Journey from Undergraduate to The Professoriate" and 2012 ADVANCE PRiME Speaker Series: Women of Color in the Mathematical Sciences".

## Conferences

From August 9 - 11, 2013, students will have the opportunity to take a field trip to Columbus, OH to attend the Young Mathematicians Conference (YMC). This undergraduate-friendly conference provides an opportunity for undergraduate students around the country to present their independent research in mathematics, and find new research opportunities through interaction with their peers as well as a graduate school orientation. Further it serves as a forum for faculty who are involved in mentoring undergraduate research. PRiME will cover all local expenses associated with travel to this conference.

The program will provide funding to attend the National Conference for the Society for the Advancement of Chicanos and Native Americans in the Sciences (SACNAS) to be held from from October 3 - 6, 2013 in San Antonio, TX. We plan to join over 3,700 attendees for four days of scientific research presentations, professional development, networking, exhibits, culture, and community. One of the largest annual gatherings of minority scientists in the country, the interdisciplinary, and interactive SACNAS National Conference supports its diverse membership showcasing cutting-edge science and features mentoring and training sessions. PRiME participants will apply for a travel scholarship; the applications are due by June 14. This will cover travel and housing costs for the conference. PRiME will cover up to \$400 USD for the registration fee. Summer abstract submissions will be due by July 3, 3013.

[Flyer] | [Program Overview] | [Planned Activities] | [Project Description] | [Application] | [Summary of Results]

# Project Description

## Background

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.

## Topics to be Discussed

Here is a list of the topics we will cover. It will not be assumed that participants know this material, although it will be useful to be somewhat familiar with some of the topics before the program begins:
• Projective Geometry
• Affine and Projective $$n$$-Space
• Projective Line $$\mathbb P^1$$
• Projective Varieties
• Galois Theory of Transcendental Extensions
• Groups
• Addition and Multiplication in $$\mathbb C$$
• Cyclic Groups
• Dihedral Groups
• Subgroups
• Morphisms
• Group Actions
• Symmetric and Alternating Groups
• Matrix Groups
• Fields and Extensions
• Complex Analysis
• Möbius Transformations
• Riemann Surfaces
• Riemann Sphere
• Elliptic Modular Functions
• Dessins d'Enfant
• Belyĭ Maps
• Mason-Stothers Theorem
• Bipartite Graphs
• Trees
• Platonic Solids
• Truncated Platonic Solids

## Sample Problems

Given any loopless, connected, planar, bipartite graph $$\Gamma$$, we wish to use properties of the symmetry group $$G$$ to construct a Belyĭ map $$\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$ such that $$\Gamma$$ arises as its Dessin d'Enfant.
1. Consider four distinct complex numbers $$z^{(-1)}$$, $$z^{(0)}$$, $$z^{(+1)}$$, and $$z^{(\infty)}$$ with cross-ratio $$\bigl( z^{(-1)}, \, z^{(0)}; \, z^{(+1)}, \, z^{(\infty)} \bigr) = -1$$ and define the rational function $$\beta(z) = \left[ \dfrac {2 \, \bigl( z^{(0)} - z^{(1)} \bigr) \, \bigl( z^{(\infty)} - z^{(1)} \bigr) \, \bigl( z - z^{(0)} \bigr) \, \bigl( z - z^{(\infty)} \bigr)}{ \bigl( z^{(0)} - z^{(1)} \bigr)^2 \, \bigl(z - z^{(\infty)} \bigr)^2 + \bigl( z^{(\infty)} - z^{(1)} \bigr)^2 \, \bigl(z - z^{(0)} \bigr)^2} \right]^2.$$ Then $$\beta(z)$$ is a Belyĭ map whose associated Dessin d'Enfant $$K_{2,2}$$ has vertices $$B = \bigl \{ z^{(0)}, \, z^{(\infty)} \bigr \}$$ and $$W = \bigl \{ z^{(-1)}, \, z^{(+1)} \bigr \}$$. Given any four distinct complex numbers $$z^{(-1)}$$, $$z^{(0)}$$, $$z^{(+1)}$$, and $$z^{(\infty)}$$, prove that we cannot always find a Belyĭ map $$\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$ such that the associated Dessin d'Enfant $$\Gamma = K_{2,2}$$ is the complete bipartite graph having $$B = f^{-1}(0) = \bigl \{ z^{(0)}, \, z^{(\infty)} \bigr \}$$ and $$W = f^{-1}(1) = \bigl \{ z^{(-1)}, \, z^{(+1)} \bigr \}$$.
2. There exist Belyĭ maps $$\phi: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$ which are invariant under the group $$S_3 = \left \langle r, \, s \, \bigl | \, r^3 = s^2 = (s \, r)^2 = 1 \right \rangle$$ as generated by the Möbius Transformations $$r(z) = \dfrac {z-1}{z} \qquad \text{and} \qquad s(z) = \dfrac {z}{z-1}.$$ How does the Dessin d'Enfant of a given Belyĭ map $$\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$ compare to the Dessin d'Enfant of the composition $$\phi \circ \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$?
3. For each subgroup $$H \subseteq S_3$$, find all Belyĭ maps $$\phi_H$$ which are invariant under $$H$$, that is, $$\phi_H \bigl( \gamma(z) \bigr) = \phi_H(z)$$ for all $$\gamma \in H$$.
4. The families of graphs $$K_{1,n}$$, $$K_{2,n}$$, $$P_{n+1}$$, and $$C_{2n}$$ may each be realized as the Dessin d'Enfant of some Belyĭ map. That is, every planar complete bipartite graph $$K_{m,n}$$ be realized as the Dessin d'Enfant of some Belyĭ map, namely either $$\beta(z) = z^n$$ or $$\beta(z) = 4 \, z^n/(z^n + 1)^2$$. Similarly, every path graph be realized as the Dessin d'Enfant of some Belyĭ map, namely $$\beta(z) = \bigl( 1 + \cos \, ( n \, \arccos z) \bigr)/2$$. Moreover, every bipartite cycle graph be realized as the Dessin d'Enfant of some Belyĭ map, namely $$\beta(z) = (z^n + 1)^2/ (4 \, z^n)$$. Are there other families of graphs that may be realized as Dessins d'Enfants?
5. Is the Dessin d'Enfant of $$\beta(z) = \bigl( z^{2n} + 6 \, z^n - 3 \bigr)^3 / \bigl( 36 \, z^n \, ( z^{2n} + 3 )^2 \bigr)$$ the prism $$P_2 \times C_{2n}$$? Conversely, can every bipartite prism graph be realized as the dessin of some Belyĭ map? What about the Cartesian product $$P_m \times C_{2n}$$?
6. Each of the Platonic Solids may be realized as a Dessin d'Enfant. Which of the truncated Platonic Solids may be realized as a Dessin d'Enfant?
7. Let $$\Gamma = \bigl( B \cup W, \, E \bigr)$$ be the Dessin d'Enfant associated to a Belyĭ map $$\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$. For each $$\gamma(z) \in S_3$$, let $$\Gamma_\gamma = \bigl( B_\gamma \cup W_\gamma, \, E_\gamma \bigr)$$ be the Dessin d'Enfant associated to the composition $$\gamma^{-1} \circ \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$. That is, $$B_\gamma = \beta^{-1}\bigl( \gamma(0) \bigr), \qquad W_\gamma = \beta^{-1} \bigl( \gamma(1) \bigr), \qquad \text{and} \qquad E_\gamma = \beta^{-1} \bigl( \gamma([0,1]) \bigr).$$
• If $$\gamma = 1$$, then $$\Gamma_1 = \Gamma$$ is the original Dessin d'Enfant.
• If $$\gamma = s$$, then $$\Gamma_s$$ can be obtained from $$\Gamma$$ by interchanging the white vertices $$W$$ with the midpoints of the faces $$F$$.
• If $$\gamma = s \, r$$, then $$\Gamma_s$$ can be obtained from $$\Gamma$$ by interchanging the black vertices $$B$$ with the white vertices $$W$$. In other words, $$\Gamma_s$$ is the dual graph to $$\Gamma$$.
• If $$\gamma = r \, s$$, then $$\Gamma_s$$ can be obtained from $$\Gamma$$ by interchanging the black vertices $$B$$ with the midpoints of the faces $$F$$.
• If $$\gamma = r$$, then $$\Gamma_r$$ can be obtained from $$\Gamma$$ by cyclically rotating the black vertices $$B$$ to the midpoints of the faces $$F$$ to the white vertices $$W$$.
• If $$\gamma = r^2$$, then $$\Gamma_{r^2}$$ can be obtained from $$\Gamma$$ by cyclically rotating the black vertices $$B$$ to the white vertices $$W$$ to the midpoints of the faces $$F$$.
Given any Dessin d'Enfant $$\Gamma = \bigl( B \cup W, \, E \bigr)$$ associated to a Belyĭ map $$\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$, there are several operations one can perform:
• delete edges from $$E$$
• change colors of the vertices $$V = B \cup W$$
• etc.
For each such operation, find a corresponding Belyĭ map $$\phi: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$ such that the Dessin d'Enfant $$\Gamma'$$ associated to the composition $$\phi \circ \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$$ is the operation on $$\Gamma$$.

[Flyer] | [Program Overview] | [Planned Activities] | [Project Description] | [Application] | [Summary of Results]

# Application

The application process is closed for 2013.

[Flyer] | [Program Overview] | [Planned Activities] | [Project Description] | [Application] | [Summary of Results]

# Summary of Results

## Main 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

This group will meet from 10:00 AM -- 12:00 PM in REC 302. It will consist of the following students and graduate assistant.

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

This group will meet from 3:00 PM -- 5:00 PM in REC 302. It will consist of the following students and graduate assistant.

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.

[Flyer] | [Program Overview] | [Planned Activities] | [Project Description] | [Application] | [Summary of Results]