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.
Instructor Contact Information
- Edray Herber Goins Ph.D.
- Office: MATH 612
- Extension: 4-1936
- E-Mail: egoins@math.purdue.edu
- Web Site: http://www.math.purdue.edu/~egoins/site
- Office Hours on Mondays, Wednesdays, and Fridays from 11:00 AM — 12:00 PM
Prerequisites
- MA 52500: Introduction to Complex Analysis
- MA 55300: Introduction to Abstract Algebra
Optional Texts
- The Grothendieck Theory of Dessins d’Enfants (London Mathematical Society Lecture Note Series, No. 200) by Leila Schneps (editor) as published by the Cambridge University Press (1994)
- Graphs on Surfaces and Their Applications (Encyclopaedia of Mathematical Sciences, Vol. 141) by Sergei K. Lando and Alexander K. Zvonkin as published by Springer-Verlag (2004)
- Introduction to Compact Riemann Surfaces and Dessins d’Enfants (London Mathematical Society Student Texts, No. 79) by Ernesto Girondo and Gabino González-Diez as published by the Cambridge University Press (2012)
- Introduction to Dessins d’Enfants by Edray Herber Goins as published online at http://www.math.purdue.edu/~egoins/notes/reuf4.pdf
Links to all other required reading will be available via the web page http://www.math.purdue.edu/~egoins/site//Dessins%20d’Enfants.html.
Grading Scheme
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 homogeneous polynomials , the set is called an algebraic variety because it is defined by algebraic equations . 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 is actually an algebraic variety. To be more precise, is a smooth, irreducible, projective variety of dimension 1 corresponding to a single equation . The French mathematician André Weil proved in 1956 that if there exists rational function which has at most three critical values, then can be defined by a a polynomial equation where the coefficients are not transcendental. Conversely — and surprisingly — the Russian mathematician Gennadiĭ Vladimirovich Belyĭ showed in 1979 that if can be defined by a polynomial equation where the coefficients are not transcendental, then there exists a rational function 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 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.
More information about the course can be found online: http://www.math.purdue.edu/academic/grad/courses/fall13mathad.pdf.
Lecture Calendar
Lecture | Date | Topics Covered |
---|---|---|
Lecture 1 | Monday, August 19, 2013 | Beal’s Conjecture |
Lecture 2 | Wednesday, August 21, 2013 | Affine and Projective -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 |
Lecture 9 | Monday, September 9, 2013 | Rigid Rotations of |
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 |
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 | — |