Quelques contributions au carrefour de la géométrie, de la combinatoire et des probabilités.

Abstract : The present text is a synthesis of research papers in mathematics, dealing with algebraic geometry, analytic combinatorics and probabilities. The first part is about three-dimensional complex algebraic varieties. It begins with the computation of the singular cohomology of non complete smooth toric varieties under some topological assumption on their fans. Afterwards, we construct a toroidal model for any quotient-singularity, whose computation requires a precise combinatorial study of the action of all finite unitary groups on the projectif plane. The second part develops a "hybrid" adaptation of Darboux's method and of singularity analysis for the coefficients' asymptotic expansion of power series that admit a natural boundary. Numerous applications in analytic combinatorics are given, including the analysis of factorization algorithms for polynomials on finite fields that are used in symbolic computation and for error-correcting codes. The third part gives an answer to a conjecture on $m$-ary search trees that are fundamental data structures in computer science used in searching and sorting. To this end, we consider them as urn processes that can be generalized to so called Pòlya processes, whose general asymptotics is studied. In the last part, we give the construction of a random tree associated with the \emph{Chaos Game Representation} of DNA sequences used in bioinformatics and biomathematics. Results on the height's and insertion depth's asymptotics are established.
Document type :
Habilitation à diriger des recherches
Mathématiques [math]. Université de Versailles-Saint Quentin en Yvelines, 2006
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  • HAL Id : tel-00403659, version 2



Nicolas Pouyanne. Quelques contributions au carrefour de la géométrie, de la combinatoire et des probabilités.. Mathématiques [math]. Université de Versailles-Saint Quentin en Yvelines, 2006. 〈tel-00403659v2〉



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