Math 351: Riemann Surfaces and Dessins d'Enfants

Spring 2013


Course Info:



[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.



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
114 Jan(M)IntroductionHW 1 [TeX] [PDF]
216 Jan a.m.(W)1.1.1: Topological spaces and manifoldsHW 2 [TeX] [PDF]
318 Jan(F)1.1.1: Riemann surfacesHW 3 [TeX] [PDF]
21 Jan(M)No class, Martin Luther King Day
423 Jan a.m.(W)1.1.1, 1.1.2: Morphisms of Riemann surfacesHW 4 [TeX] [PDF]
523 Jan p.m.(W)1.1.2HW 5 [TeX] [PDF]
625 Jan(F)1.1.2: Automorphisms of P^1(CC)HW 6 [TeX] [PDF]
728 Jan(M)1.1.2: Automorphisms of HH and DD
830 Jan a.m.(W)1.2: Topological classification by genusHW 8 [TeX] [PDF]
930 Jan p.m.(W)2.1: Uniformization theorem
1 Feb(F)No class
104 Feb(M)Properly discontinuous group actionsHW 10 [TeX] [PDF]
116 Feb a.m.(W)Geometry: Euclidean and sphericalHW 11 [TeX] [PDF]
126 Feb p.m.(W)2.1.1: PSL_2(RR) as isometries of HHHW 12 [TeX] [PDF]
138 Feb(F)2.1.1, 2.4.1: Hyperbolic area and Gauss-BonnetHW 13 [TeX] [PDF]
1411 Feb(M)2.4.2-2.4.3: Triangle groupsHW 14 [TeX] [PDF]
13 Feb(W)No class
1515 Feb(F)2.3: Fuchsian groups
18 Feb(M)No class, Presidents Day
1620 Feb a.m.(W)1.1.2: Degree and ramification of morphismsHW 16 [TeX] [PDF]
1720 Feb p.m.(W)1.1.2HW 17 [TeX] [PDF]
1822 Feb(F)1.2.4: Riemann-Hurwitz formulaHW 18 [TeX] [PDF]
1925 Feb(M)1.2.4
2027 Feb a.m.(W)4.4: Subgroups of triangle groupsHW 20 [TeX] [PDF]
2127 Feb p.m.(W)2.7: Monodromy
221 Mar(F)Projects
4 Mar(M)No class, Spring break
6 Mar(W)No class, Spring break
8 Mar(F)No class, Spring break
11 Mar(M)No class
2313 Mar a.m.(W)1.1.3: Meromorphic differentialsHW 23 [TeX] [PDF]
2413 Mar p.m.(W)1.1.3: Holomorphic differentials
2515 Mar(F)1.1.3: Residue theorem
18 Mar(M)No class
20 Mar(W)No class
22 Mar(F)No class
2625 Mar(M)Degree and the Poincare-Hopf theoremHW 26 [TeX] [PDF]
2727 Mar a.m.(W)Big picture
2827 Mar p.m.(W)Power series expansions of modular forms (Klug)
2929 Mar(F)4.1: DessinsHW 29 [TeX] [PDF]
301 Apr(M)Genus 2 example
313 Apr a.m.(W)4.6: Fermat curves
323 Apr p.m.(W)4.6: Further examplesHW 32 [TeX] [PDF]
335 Apr(F)3.1: Belyi's theorem (a) => (b)
348 Apr(M)3.1
3510 Apr a.m.(W)Escher and the Droste Effect
3610 Apr p.m.(W)3.1: Belyi's theorem (b) => (a)
3712 Apr(F)Descent
15 Apr(M)No class
17 Apr(W)No class
19 Apr(F)No class
3822 Apr(M)"Elliptic" curves
3924 Apr a.m.(W)Elliptic functions
4024 Apr p.m.(W)2.1.1: Meromorphic functions in genus 1
4126 Apr(F)Weierstrass equations
4229 Apr(M)Uniformization in genus 1
431 May a.m.(W)Conclusion