Dessins d'Enfants

Undergraduate Seminars, UN 3952, Spring 2022

Welcome to the undergraduate seminar!

In this seminar, we will study dessins d'enfants (children's drawings). These are topological graphs that provide combinatorial information about Riemann surfaces, algebraic curves, and certain Galois actions. Students will encounter concepts from a breadth of mathematical disciplines, including:

Algebraic topology
Algebraic geometry
Complex analysis
Graph theory
Number theory
Riemannian geometry

Throughout the semester, students will study and present talks on the theory of compact Riemann surfaces and dessins d'enfants. References for the talks include the books Introduction to Compact Riemann Surfaces and Dessins D'Enfants by Girondo—González-Diez and Dessins d'Enfants on Riemann Surfaces by Jones—Wolfart, as well as other books and articles.

Students should be familiar with abstract algebra (MATH 4041 and 4042) and complex analysis (MATH 3007 or 4065). Additional background, such as in the aforementioned branches of mathematics, will be helpful in exploring the discipline.

Grading is based on presentation quality and active participation during other students' presentations during the semester. This semester, each student presents four 45-minute talks or two 90-minute talks, or some combination thereof.

The seminar meets in Math 622 from 4:00PM—6:00PM (ET) on Mondays.

A general overview of this semester's undergraduate seminars is available here.
To contact me, please email me directly using your Columbia UNI email.

Seminar schedule

Date	Speaker	Title	References
September 19 (7:30pm, Zoom)	Dave Bayer Raymond Cheng Hindy Drillick Emily Saunders Robin Zhang	Organizational meeting (all sections)	—
January 31	Robin Zhang	Organizational meeting and overview of dessins d'enfants	[GG]§Preface [Zap]
February 7	Sage (Shanzheng) Ba Nicholas Lillis	Connectedness and compactness in point-set topology Holomorphic and meromorphic functions in complex analysis	[Cza]§Topology [Hat2]§1-3 [May]§1-5 [Mun]§23-29 [Cza]§Complex analysis [Dat] [SS]§1.1-1.3, 3.1-3.3
February 14	Elena Gribelyuk Sid Mane	Affine curves Basic definitions & examples of Riemann surfaces	[Ful]§1.1-1.7, 2.1-2.4, 3.1 [Gat]§1 [Cza]§3-4 [GG]§1.1.1-1.1.2
February 21	Sophia Fanelle David Fang	Projective space and projective curves Morphisms between Riemann surfaces and differentials	[Ful]§4.1-4.3, 5.1, 5.3 [GG]§1.3 [Sch] [Ber] [Cza]§4 [GG]§1.1.2-1.1.3
February 28	Sage (Shanzheng) Ba Nicholas Lillis	The classical Euler characteristic and fundamental group The Euler characteristic and fundamental group of a Riemann surface	[Ear]§1-2 [Hat1]§1.1 [Mun]§9.1-9.2 [Ric]§7 [GG]§1.2.1-1.2.3
March 7	David Fang Elena Gribelyuk	Ramified coverings of the sphere and the Riemann–Hurwitz formula Definition of dessins d'enfants and permutation pairs	[GG]§1.2.4-1.2.6, 3.1 [JW]§1.2.1 [LZ]§1.2 [Oor]1-4 [GG]§4.1 [JW]§2.1
March 14	Spring Break (no meeting)	Spring Break (no meeting)	—
March 21	Elena Gribelyuk Sophia Fanelle	Belyi's theorem and triangle decompositions of dessins d'enfants Absolute Galois groups and their actions	[GG]§3.1, 4.2 [JW]§1.4.1-1.4.3 [GG]§3.3 [JW]§4.1
March 28	Sid Mane Sophia Fanelle	Properties and applications of bipartite graphs Galois covers of Riemann surfaces	[ADH]§2.2-4, 11 [JW]§3.3 [Dem]§4 [GG]§2.8 [Sza]§3.3
April 4	Sid Mane Nicholas Lillis	Chebyshev polynomials Hypermaps	[JW]§1.4.2 [Mas] [Moh] [OMS]§22 [Jon]§1-2 [JW]§2.1.3-2.1.6
April 11	David Fang	The abc conjecture and its analogues for polynomials & Riemann surfaces	[Gol1] [Gol2] [GT] [JW]§10.1 [Wal]§9 [Wol]
April 18	Sage (Shanzheng) Ba	The absolute Galois group of the rationals in number theory	[Mac] [Wei]§1-2
April 25	Elena Gribelyuk Sid Mane	Trees and Shabat polynomials Galois actions on dessins d'enfants	[LZ]§2.2 [GG]§4.5 [JW]§4.2
May 2	Nicholas Lillis Sophia Fanelle	Zeta functions and relations to Riemann surfaces Dessins d'enfants from a categorical perspective	[Apo] [Edw] [SS] [Gui]

Books

[Apo] Tom M. Apostol. Zeta and Related Functions. In Digital Library of Mathematical Functions, NIST, 2022.
[ADH] Armen S. Asratian, Tristan M. J. Denley, and Roland Häggkvist. Bipartite Graphs and their Applications. Cambridge Tracts in Mathematics, 131. Cambridge University Press, 1998.
[Edw] Harold M. Edwards. Riemann's Zeta Function. Dover Publications, 2001.
[Ful] William E. Fulton. Algebraic Curves. 3rd Edition, 2008.
[GG] Ernesto Girondo and Gabino González-Diez. Introduction to Compact Riemann Surfaces and Dessins d'Enfants. London Mathematical Society Student Texts, 79. Cambridge University Press, 2012.
[Gol2] Dorian Goldfeld. Modular forms, elliptic curves and the ABC-conjecture. In Gisbert Wüstholz (ed.), A panorama of number theory or the view from Baker's garden, pages 128–147. Cambridge University Press, 2002.
[Hat1] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002.
[JW] Gareth A. Jones and Jürgen Wolfart. Dessins d'Enfants on Riemann Surfaces. Springer Monographs in Mathematics. Springer, 2016.
[LZ] Sergei K. Lando and Alexander K. Zvonkin. Graphs on Surfaces and Their Applications. Encyclopaedia of Mathematical Sciences, 141. Springer, 2004.
[Mas] J. C. Mason. Some properties and applications of Chebyshev polynomial and rational approximation. In Peter Russell Graves-Morris, Edward B. Saff, and Richard S. Varga (eds.), Rational Approximation and Interpolation, pages 27–48. Lecture Notes in Mathematics, 1105. Springer, 1984.
[Mun] James R. Munkres. Topology. 2nd Edition. Pearson, 2000.
[OMS] Keit Oldham, Jan Myland, and Jerome Spanier. An Atlas of Functions. Springer, 2009.
[Oor] Frans Oort. The Riemann–Hurwitz formula. Lizhen Ji, Frans Oort, and Shing-Tung Yau (eds.). The Legacy of Bernhard Riemann After One Hundred and Fifty Years, Vol. II. Advanced Lectures in Mathematics, 35.2. International Press of Boston, 2016.
[Ric] David S. Richeson. Euler's Gem. Princeton University Press, 2008.
[SS] Elias M. Stein and Rami Shakarchi. Complex Analysis. Princeton Lectures in Analysis, II. Princeton University Press, 2003.
[Sza] Tamás Szamuely. Galois Groups and Fundamental Groups. Cambridge Studies in Advanced Mathematics, 117. Cambridge University Press, 2009.
[Wal] Michel Waldschmidt. Lecture on the abc conjecture and some of its consequences. In Pierre Cartier, A.D.R. Choudary, and Michel Waldschmidt (eds.), Mathematics in the 21st Century: 6th World Conference, pages 211–230. Springer Proc. Math. Stat., 98. Springer, 2015.
[Wol] Jürgen Wolfart. ABC for polynomials, dessins d'enfants, and uniformization — a survey. In Wolfgang Schwarz and Jörn Steuding (eds.), Elementare und Analytische Zahlentheorie, pages 313–345. Steiner Verlag, 2006.

Articles

[Ear] Edward Early. On the Euler characteristic. MIT Undergr. J. Math, 1:37–48, 1999.
[Gol1] Dorian Goldfeld. Beyond the Last Theorem. Math Horizons, 4(1):26–31, 1996.
[GT] Andrew Granvile and Thomas J. Tucker. It's as easy as abc. Not. Am. Math. Soc., 49(10):1224–1231, 2002.
[Gui] Pierre Guillot. An elementary approach to dessins d'enfants and the Grothendieck-Teichmüller group. Enseign. Math., 60(3/4):293–375, 2014.
[Jon] Gareth A. Jones. Hypermaps and multiply quasiplatonic Riemann surfaces. European J. Combin., 33(7):1588–1605, 2012.
[Zap] Leonardo Zapponi. What is a dessin d'enfant?. Not. Am. Math. Soc., 50(7):788–789, 2003.

Notes

[Ber] Richard Evan Schwartz. Meromorphic functions and differentials on Riemann surfaces. 2020.
[Cza] Maciej Czarnecki. Introduction to Riemann surfaces. 2015.
[Dat] Ved V. Datar. Meromorphic functions. 2016.
[Dem] Spencer Dembner. Galois groups and branched covers of Riemann surfaces. 2019.
[Gat] Andreas Gathmann. Plane algebraic curves. 2018.
[Hat2] Allen Hatcher. Notes on introductory point-set topology. 2005.
[Mac] Sander Mack-Crane. Where to find Galois representations, and why. 2016.
[May] J. Peter May. An outline summary of basic point set topology. 2000.
[Moh] Pranab Mohapatra. Minimax approximation, optimal interpolation, Chebyshev polynomials. 2013.
[Sch] Richard Evan Schwartz. Some projective geometry. 2019.
[Wei] Jared Weinstein. Exploring the Galois group of the rational numbers: recent breakthroughs. 2015.

YouTube videos

Dessins d'Enfants — Gareth Jones (13 minutes). Serious Science, 2020.
Extra History: The History of Non-Euclidean Geometry (5 videos, 42 minutes). Extra Credits, 2018.