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Mathematics Colloquium

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Date:  Tuesday, January 10, 2006

Title:  Chebysheff and Belyi Polynomials, Dessins d enfants, Beauville Surfaces and Group Theory

Abstract:  This talk describes a generalization of the Grothendieck program of Dessins d'enfants from curves to surfaces. Beauville surfaces are surfaces that are isogenous to a product of two curves and are strongly rigid. Beauville in 1978 found the first example, as a quotient of the product of two Fermat curves. The moduli space of such surfaces carries an action of the absolute Galois group of the rationals Q, which changes the topology of the corresponding surfaces. The talk will first describe Grothendieck's program via elementary examples. I first consider polynomials with two critical values, under automorphisms of R and C, which lead to the polynomials of Chebysheff and Belyi, for which there are explicit formulae. Then I describe Grothendieck's program, "Dessin's d'enfants", and the relation between triangular curves and Beauville surfaces. Finally I give results and conjectures which relate the classification of Beauville surfaces with questions in finite groups. The new results are obtained in cooperation with Ingrid Bauer and Fritz Gruenwald.


Speaker:  Lehrstuhl Mathematik VII
Institution:  Universität Bayreuth

 

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