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A dessin denfant, or dessin, is a bicolored graph embedded into a Riemann surface, and the monodromy group is an algebraic invariant of the dessin generated by rotations of edges about black and white vertices. A rational billiards surface is a two dimensional surface that allows one to view the path of a billiards ball as a continuous path. In this paper, we classify the monodromy groups of dessins associated to rational triangular billiards surfaces.
We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic techniques, such as
We study general properties of the dessins denfants associated with the Hecke congruence subgroups $Gamma_0(N)$ of the modular group $mathrm{PSL}_2(mathbb{R})$. The definition of the $Gamma_0(N)$ as the stabilisers of couples of projective lattices i
A short review of the Operator/Feynman diagram/dessin denfants correspondence in the rank 3 tensor model is presented, and the cut & join operation is given in the language of dessin denfants as a straightforward development. We classify operators of
We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manins conjecture for a cubic surface whose singularity type is A_5+A_1.
There have been several constructions of family of varieties with exceptional monodromy group. In most cases, these constructions give Hodge structures with high weight(Hodge numbers spread out). N. Katz was the first to obtain Hodge structures with