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Nous montrons que les equations du rep`ere mobile des surfaces de Bonnet conduisent `a une paire de Lax matricielle isomonodromique dordre deux pour la sixi`eme equation de Painleve. We show that the moving frame equations of Bonnet surfaces can be extrapolated to a second order, isomonodromic matrix Lax pair of the sixth Painleve equation.
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We propose a formula expressing Perron - Frobenius eigenvectors of Cartan matrices in terms of products of values of the Gamma function.
The goal of this Habilitation `a diriger des recherches is to present two different applications, namely computations of certain partition functions in probability and applications to integrable systems, of the topological recursion developed by B. Eynard and N. Orantin in 2007. Since its creation, the range of applications of the topological recursion has been growing and many results in different fields have been obtained. The first aspect that I will develop deals with the historical domain of the topological recursion: random matrix integrals. I will review the formalism of the topological recursion as well as how it can be used to obtain asymptotic $frac{1}{N}$ series expansion of various matrix integrals. In particular, a key feature of the topological recursion is that it can recover from the leading order of the asymptotic all sub-leading orders with elementary computations. This method is particularly well known and fruitful in the case of hermitian matrix integrals, but I will also show that the general method can be used to cover integrals with hard edges, integrals over unitary matrices and much more. In the end, I will also briefly mention the generalization to $beta$-ensembles. In a second chapter, I will review the connection between the topological recursion and the study of integrable systems having a Lax pair representation. Most of the results presented there will be illustrated by the case of the famous six Painleve equations. Though the formalism used in this chapter may look completely disconnected from the previous one, it is well known that the local statistics of eigenvalues in random matrix theory exhibit a universality phenomenon and that the encountered universal systems are precisely driven by some solutions of the Painlev{e} equations. As I will show, the connection can be made very explicit with the topological recursion formalism.
Let $X$ be a real-analytic manifold and $gcolon Xto{mathbf R}^n$ a proper triangulable subanalytic map. Given a subanalytic $r$-form $omega$ on $X$ whose pull-back to every non singular fiber of $g$ is exact, we show tha $omega$ has a relative primitive: there is a subanalytic $(r-1)$-form $Omega$ such that $dgLambda (omega-dOmega)=0$. The proof uses a subanalytic triangulation to translate the problem in terms of relative Whitney forms associated to prisms. Using the combinatorics of Whitney forms, we show that the result ultimately follows from the subanaliticity of solutions of a special linear partial differential equation. The work was inspired by a question of Franc{c}ois Treves.
We express discrete Painleve equations as discrete Hamiltonian systems. The discrete Hamiltonian systems here mean the canonical transformations defined by generating functions. Our construction relies on the classification of the discrete Painleve equations based on the surface-type. The discrete Hamiltonians we obtain are written in the logarithm and dilogarithm functions.
We describe the behaviour of the rank of the Mordell-Weil group of the Picard variety of the generic fibre of a fibration in terms of local contributions given by averaging traces of Frobenius acting on the fibres. The results give a reinterpretation of Tates conjecture (for divisors) and generalises previous results of Nagao, Rosen-Silverman and the authors.
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