Málstofa í stærðfræði Í DAG 3 Desember, George Shabat

[aronlagerberg]

Fyrirlesari: George Shabat, Moscow State University, Institute of Theoretical and Experimental Physics and Independent University of Moscow

Dagsetning: Mánudagur,  3 Desember, 2012, 14.00 & 15.00

Staður: VRII herbergi 157 

Titill: Dessins d'enfants and algebraic curves

Ágrip:

Colloquium. A dessin d'enfant is such a graph on a compact oriented surface that its complement is homeomorphic to a disjoint union of 2-cells. It turns out (understood by Grothendieck in 1970's) that the category of dessins d'enfants, being appropriately defined, is equivalent to the category of BELYI PAIRS, i.e. the category of smooth complete curves over the field of algebraic numbers together with a covering of the projective line ramified only over a three points. Therefore the combinatorial topology of graphs on surfaces is somewhat equivalent to a part of arithmetic geometry; the precise statements will be given and some examples presented. The action of the absolute Galois group   on the set of isomorphism classes of dessins will be defined and illustrated.

Seminar. The deeper relations of the dessins d'enfants theory with complex algebraic curves will be considered. We’ll start with the equilateral triangulations of Riemann surfaces. Then the classical result of Strebel-Jenkins concerning quadratic differentials on Riemann surfaces, whose horisontal trajectories are generically closed, will be reminded. It will be applied to the Penner-Kontzevich dessins-labelled stratification of moduli spaces of curves with marked points. The implications of this construction will be discussed and open problems mentioned. 

R e f e r e n c e s

Voevodsky V., Shabat G., Drawing curves over number fields. The Grothendieck Festschrift, vol. III, pp. 199–227. Birkhauser, 1990. 

Shabat G., Zvonkin A. Plane trees and algebraic numbers. In: “Jerusalem combinatorics ‘93”. AMS, 1994, Contemporary mathematics, vol. 176, p. 233–275.

Shabat, G.B.  Visualizing algebraic Curves: from Riemann to Grothendieck. Journal of Siberian Federal University, Mathematics & Physics. vol. 1, 2008, pp. 42- 51.