## MA 59800 Course Syllabus

### Course Title

MA 59800-509-63571: Introduction to Dessins d’Enfants

### Meeting Times

Class will meet on MWF from 3:30 — 4:20 PM in REC 302.

### Optional Texts

Links to all other required reading will be available via the web page http://www.math.purdue.edu/~egoins/site//Dessins%20d’Enfants.html.

There will be weekly homework assignments due every other Friday at the start of class. There will be final presentations at the end of the course; each will be graded on the combination of an oral presentation and a written paper. Grades will be distributed as follows:

 Homework 70% 7 Assignments Final Presentation 15% 1 Presentation Final Paper 15% 1 Paper

### Due Dates

Assignment Due Date
Homework Set 1 Friday, September 6, 2013
Homework Set 2 Friday, September 20, 2013
Homework Set 3 Friday, October 4, 2013 (*)
Homework Set 4 Friday, October 18, 2013
Homework Set 5 Friday, November 1, 2013 (*)
Homework Set 6 Friday, November 15, 2013
Homework Set 7 Friday, December 6, 2013
Final Presentations TBA

(*) = There will be no class on these days as the Instructor will be out of town, so place homework in the Instructor’s mailbox in MATH 630.

### Course Overview

Given a collection of $m$ homogeneous polynomials $F_k$, the set $V(\mathbb C) = \bigl \{ P \in \mathbb P^n(\mathbb C) \ \bigl| \ F_k(P) = 0 \ \text{for} \ k = 1, \, 2, \, \dots, \, m \bigr \}$ is called an algebraic variety because it is defined by algebraic equations $V: F_1 = F_2 = \cdots = F_m = 0$. If instead of using polynomials, we use analytic functions, we have an analytic variety. Indeed, Riemann surfaces are examples of analytic varieties. There are cases there these two concepts coincide: a compact, connected Riemann surface $X$ is actually an algebraic variety. To be more precise, $X = V(\mathbb C)$ is a smooth, irreducible, projective variety of dimension 1 corresponding to a single equation $V: \sum_{i,j} a_{ij} \, z^i \, w^j = 0$. The French mathematician André Weil proved in 1956 that if there exists rational function $\beta: X \to \mathbb P^1(\mathbb C)$ which has at most three critical values, then $X$ can be defined by a a polynomial equation where the coefficients $a_{ij}$ are not transcendental. Conversely — and surprisingly — the Russian mathematician Gennadiĭ Vladimirovich Belyĭ showed in 1979 that if $X$ can be defined by a polynomial equation $\sum_{i,j} a_{ij} \, z^i \, w^j = 0$ where the coefficients $a_{ij}$ are not transcendental, then there exists a rational function $\beta: X \to \mathbb P^1(\mathbb C)$ which has at most three critical values.

Motived by Belyĭ’s discovery, the French mathematician Alexander Grothendieck wrote a letter in 1984 outlining several new directions for his research.

This discovery, which is technically so simple, made a very strong impression on me, and it represents a decisive turning point in the course of my reflections, a shift in particular of my centre of interest in mathematics, which suddenly found itself strongly focused. […] This is surely because of the very familiar, non-technical nature of the objects considered, of which any child’s drawing scrawled on a bit of paper (at least if the drawing is made without lifting the pencil) gives a perfectly explicit example. To such a `dessin’ we find associated subtle arithmetic invariants, which are completely turned topsy-turvy as soon as we add one more stroke.

He realized that maps $\beta: X \to \mathbb P^1(\mathbb C)$ which have at most three critical values give graphs — called “Dessins d’Enfants” — which contain useful arithmetic information.

In this course, we discuss the budding theory behind Dessins d’Enfants. We will cover the text “Introduction to Compact Riemann Surfaces and Dessins d’Enfants” (London Mathematical Society Student Texts) by Ernesto Girondo and Gabino González-Diez. We will discuss the Riemann surfaces, algebraic curves, the Riemann-Roch Theorem, Fuchsian groups, monodromy, Galois groups, algebraic varieties, elliptic curves, and modular functions.

### Lecture Calendar

Lecture Date Topics Covered
Lecture 1 Monday, August 19, 2013 Beal’s Conjecture
Lecture 2 Wednesday, August 21, 2013 Affine and Projective $n$-Space
Lecture 3 Friday, August 23, 2013 Projective Line and Projective Plane
Lecture 4 Monday, August 26, 2013 Smooth, Projective Varieties
Lecture 5 Wednesday, August 28, 2013 Cubic and Quartic Curves as Smooth Varieties
Lecture 6 Friday, August 30, 2013 Transcendental Galois Theory
Labor Day Monday, September 2, 2013
Lecture 7 Wednesday, September 4, 2013 Inverse Galois Problem
Lecture 8 Friday, September 6, 2013 Rigid Rotations of $A^2(\mathbb R)$
Lecture 9 Monday, September 9, 2013 Rigid Rotations of $S^2(\mathbb R)$
Lecture 10 Wednesday, September 11, 2013 Riemann Surfaces
Lecture 11 Friday, September 13, 2013 Some Elliptic Modular Functions
Lecture 12 Monday, September 16, 2013 Some Modular Curves
Lecture 13 Wednesday, September 18, 2013 Covering Spaces and Deck Transformations
Lecture 14 Friday, September 20, 2013 Non-Singular Algebraic Curves and Dedekind Domains
Lecture 15 Monday, September 23, 2013 Cubic and Quartic Curves as Riemann Surfaces
Lecture 16 Wednesday, September 25, 2013 Meromorphic Functions and Differentials
Lecture 17 Friday, September 27, 2013 Riemann-Roch Theorem
Lecture 18 Monday, September 30, 2013 Curves of Genus 0 and Genus 1
No Class Wednesday, October 2, 2013
No Class Friday, October 4, 2013
October Break Monday, October 7, 2013
Lecture 19 Wednesday, October 9, 2013 Belyĭ’s Theorem: Proof
Lecture 20 Friday, October 11, 2013 Belyĭ’s Theorem: Applications
Lecture 21 Monday, October 14, 2013 Belyĭ Maps
Lecture 22 Wednesday, October 16, 2013 Mason-Stothers Theorem: Proof
Lecture 23 Friday, October 18, 2013 Mason-Stothers Theorem: Applications
Lecture 24 Monday, October 21, 2013 Bipartite Graphs
Lecture 25 Wednesday, October 23, 2013 Riemann’s Existence Theorem
Lecture 26 Friday, October 25, 2013 Ramified Coverings of $S^2(\mathbb R)$
Lecture 27 Monday, October 28, 2013 Bipartite Graphs
Lecture 28 Wednesday, October 30, 2013 Trees and Shabat Polynomials
No Class Friday, November 1, 2013
Lecture 29 Monday, November 4, 2013 Platonic Solids
Lecture 30 Wednesday, November 6, 2013 Archimedean Solids, Part I
No Class Friday, November 8, 2013
Lecture 31 Monday, November 11, 2013 Archimedean Solids, Part II
Lecture 32 Wednesday, November 13, 2013 Johnson Solids, Part I
Lecture 33 Friday, November 15, 2013 Johnson Solids, Part II
Lecture 34 Monday, November 18, 2013 Octahedral Belyĭ Maps, Part I
Lecture 35 Wednesday, November 20, 2013 Octahedral Belyĭ Maps, Part II
Lecture 36 Friday, November 22, 2013 Icosahedral Belyĭ Maps
Thanksgiving Vacation Monday, November 25, 2013
Thanksgiving Vacation Wednesday, November 27, 2013
Thanksgiving Vacation Friday, November 29, 2013
Final Presentations Monday, December 2, 2013
Final Presentations Wednesday, December 4, 2013
Final Presentations Friday, December 6, 2013