## Math 351: Riemann Surfaces and Dessins d'Enfants

### Spring 2013

**Course Info:**

**Course:**Riemann Surfaces and Dessins d'Enfants**Lectures:**Monday, Wednesday, Friday, 10:40 - 11:30 a.m.; Wednesdays, 3:00 - 3:50 p.m.**Dates:**14 January 2013 - 1 May 2013**Room:**Votey 223**Instructor:**John Voight**Office:**16 Colchester Ave, Room 207C**Phone:**(802) 656-2271**E-mail:**jvoight@gmail.com**Office hours:**By appointment**Course Web Page:**[http://www.cems.uvm.edu/~jvoight/351/**Prerequisites:**Math (241, 242, and) 333, Math (251 and) 252, Math 331 (corequisite is OK), or permission. This may sound like a long list, but to get permission, all you need is some advanced coursework and a healthy dose of curiosity!**Required Texts:**Ernesto Girondo and Gabino Gonzalez-Diez, Introduction to Compact Riemann Surfaces and Dessins d'Enfants, 2012.**Grading:**Homework will count for 35% of the grade. A final project will count for 65% of the grade.

**Syllabus:**

[**PDF**] **Syllabus**

We will present the theory of three-point branched covers of the complex projective line and its connection with the geometry and arithmetic of algebraic curves defined over number fields.

**Homework:**

The homework assignments will be assigned on a varying basis and is posted below. It will be due in one week, but late homework will be accepted.

Cooperation on homework is permitted (and encouraged), but if you work together, do not take any paper away with you--in other words, you can share your thoughts (say on a blackboard), but you have to walk away with only your understanding. In particular, you must write the solution up on your own.

Plagiarism, collusion, or other violations of the Code of Academic Integrity will be referred to the The Center for Student Ethics and Standards.

[**PDF**] **Homework Submission Guidlines**

Chapter 1: Compact Riemann Surfaces and Algebraic Curves |
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1 | 14 Jan | (M) | Introduction | HW 1 [TeX] [PDF] |

2 | 16 Jan a.m. | (W) | 1.1.1: Topological spaces and manifolds | HW 2 [TeX] [PDF] |

3 | 18 Jan | (F) | 1.1.1: Riemann surfaces | HW 3 [TeX] [PDF] |

21 Jan | (M) | No class, Martin Luther King Day
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4 | 23 Jan a.m. | (W) | 1.1.1, 1.1.2: Morphisms of Riemann surfaces | HW 4 [TeX] [PDF] |

5 | 23 Jan p.m. | (W) | 1.1.2 | HW 5 [TeX] [PDF] |

6 | 25 Jan | (F) | 1.1.2: Automorphisms of P^1(CC) | HW 6 [TeX] [PDF] |

7 | 28 Jan | (M) | 1.1.2: Automorphisms of HH and DD | |

8 | 30 Jan a.m. | (W) | 1.2: Topological classification by genus | HW 8 [TeX] [PDF] |

9 | 30 Jan p.m. | (W) | 2.1: Uniformization theorem | |

1 Feb | (F) | No class |
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10 | 4 Feb | (M) | Properly discontinuous group actions | HW 10 [TeX] [PDF] |

11 | 6 Feb a.m. | (W) | Geometry: Euclidean and spherical | HW 11 [TeX] [PDF] |

12 | 6 Feb p.m. | (W) | 2.1.1: PSL_2(RR) as isometries of HH | HW 12 [TeX] [PDF] |

13 | 8 Feb | (F) | 2.1.1, 2.4.1: Hyperbolic area and Gauss-Bonnet | HW 13 [TeX] [PDF] |

14 | 11 Feb | (M) | 2.4.2-2.4.3: Triangle groups | HW 14 [TeX] [PDF] |

13 Feb | (W) | No class |
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15 | 15 Feb | (F) | 2.3: Fuchsian groups | |

18 Feb | (M) | No class, Presidents Day |
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16 | 20 Feb a.m. | (W) | 1.1.2: Degree and ramification of morphisms | HW 16 [TeX] [PDF] |

17 | 20 Feb p.m. | (W) | 1.1.2 | HW 17 [TeX] [PDF] |

18 | 22 Feb | (F) | 1.2.4: Riemann-Hurwitz formula | HW 18 [TeX] [PDF] |

19 | 25 Feb | (M) | 1.2.4 | |

20 | 27 Feb a.m. | (W) | 4.4: Subgroups of triangle groups | HW 20 [TeX] [PDF] |

21 | 27 Feb p.m. | (W) | 2.7: Monodromy | |

22 | 1 Mar | (F) | Projects | |

4 Mar | (M) | No class, Spring break
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6 Mar | (W) | No class, Spring break
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8 Mar | (F) | No class, Spring break
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11 Mar | (M) | No class
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23 | 13 Mar a.m. | (W) | 1.1.3: Meromorphic differentials | HW 23 [TeX] [PDF] |

24 | 13 Mar p.m. | (W) | 1.1.3: Holomorphic differentials | |

25 | 15 Mar | (F) | 1.1.3: Residue theorem | |

18 Mar | (M) | No class
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20 Mar | (W) | No class
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22 Mar | (F) | No class
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26 | 25 Mar | (M) | Degree and the Poincare-Hopf theorem | HW 26 [TeX] [PDF] |

27 | 27 Mar a.m. | (W) | Big picture | |

28 | 27 Mar p.m. | (W) | Power series expansions of modular forms (Klug) | |

29 | 29 Mar | (F) | 4.1: Dessins | HW 29 [TeX] [PDF] |

30 | 1 Apr | (M) | Genus 2 example | |

31 | 3 Apr a.m. | (W) | 4.6: Fermat curves | |

32 | 3 Apr p.m. | (W) | 4.6: Further examples | HW 32 [TeX] [PDF] |

33 | 5 Apr | (F) | 3.1: Belyi's theorem (a) => (b) | |

34 | 8 Apr | (M) | 3.1 | |

35 | 10 Apr a.m. | (W) | Escher and the Droste Effect | |

36 | 10 Apr p.m. | (W) | 3.1: Belyi's theorem (b) => (a) | |

37 | 12 Apr | (F) | Descent | |

15 Apr | (M) | No class
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17 Apr | (W) | No class
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19 Apr | (F) | No class
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38 | 22 Apr | (M) | "Elliptic" curves | |

39 | 24 Apr a.m. | (W) | Elliptic functions | |

40 | 24 Apr p.m. | (W) | 2.1.1: Meromorphic functions in genus 1 | |

41 | 26 Apr | (F) | Weierstrass equations | |

42 | 29 Apr | (M) | Uniformization in genus 1 | |

43 | 1 May a.m. | (W) | Conclusion |