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The Hurwitz problem asks which ramification data are realizable, that is appear as the ramification type of a covering. We use dessins denfant to show that families of genus 1 regular ramification data with small changes are realizable with the exception of four families which were recently shown to be nonrealizable. A similar description holds in the case of genus 0 ramification data.
We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by Arnold. We si
We show that any embedding $mathbb{R}^d to mathbb{R}^{2d+rho(d)-1}$ inscribes a trapezoid or maps three points to a line, where $rho(d)$ denotes the Radon-Hurwitz function. This strengthens earlier results on the nonexistence of affinely $3$-regular
We reinterpret ideas in Kleins paper on transformations of degree~$11$ from the modern point of view of dessins denfants, and extend his results by considering dessins of type $(3,2,p)$ and degree $p$ or $p+1$, where $p$ is prime. In many cases we de
We prove on the 2D sphere and on the 2D torus the Lieb-Thirring inequalities with improved constants for orthonormal families of scalar and vector functions.
A vertex-transitive map $X$ is a map on a closed surface on which the automorphism group ${rm Aut}(X)$ acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equive