27-29 mai 2020 Strasbourg (France)

Titres et résumés

Annette Bachmayr, Free differential Galois groups

   I will talk about recent progress towards a conjecture of Matzat that states that the absolute differential Galois group of C(x) is free as a proalgebraic group, if C is an algebraically closed field of characteristic zero. This is joint work with David Harbater, Julia Hartmann and Michael Wibmer.

Thomas Blossier,  Autour du groupe d'automorphismes du corps des nombres complexes

  Au début des années 90, Daniel Lascar a montré que le groupe des automorphismes du corps des nombres complexes laissant les nombres algébriques fixes est simple. Je présenterai dans cet exposé des éléments de sa preuve et j'évoquerai un travail commun avec Zoé Chatzidakis, Charlotte Hardouin et Amador Martin-Pizarro, qui consiste à généraliser ce résultat à des corps différentiels et à des corps aux différences.

Thomas Cluzeau, A constructive approach to the presentation of isomorphic finitely presented modules by equivalent matrices

   We use the algebraic analysis approach to linear systems theory to handle problems related to the so-called equivalence problem in systems/modules theory. In the first part of the talk, we develop a constructive version of a theorem due to Fitting which asserts that the presentation matrices of isomorphic finitely presented left D-modules can be inflated by blocks of zeros and identity so that the resulting matrices are presentation matrices of the same D-modules and are equivalent. In practice, Fitting’s theorem can provide large equivalent matrices. In the second part of the talk, we propose a constructive version of Warfield’s results which imply that under some condition on the stable rank of the base ring D, we can reduce the size of the blocks of zeros and identity obtained via Fitting’s theorem in order to get smaller presentation matrices of the same D-modules that are still equivalent. The different results will be illustrated by examples computed using our Maple implementation of the distinct algorithms described in this talk. This is a joint work with Cyrille Chenavier and Alban Quadrat.

Gaël Cousin, Towards effective Liouvillian integration

Viktoria Heu, Confluence from discrete to differential Okamoto spaces for Painlevé 6

Martin Klimes, Singularities of meromorphic sl_2(C)-connections over Riemann surfaces and their deformations

   I will revisit some aspect of the local theory of singularities of meromorphic linear systems in a traceless situation in dimension 2, and of meromorphic quadratic differentials, and talk about how things work in certain parametric settings, namely in problems of confluence and of isomonodromic deformations.

Gabriel Lepetit, Structure des G-opérateurs au sens large

  Les G-fonctions de Siegel peuvent être définis de deux manières - au sens large ou au sens strict - qui sont conjecturalement équivalentes. Dans cet exposé, je présenterai les principaux résultats connus sur la structure de l'espace des solutions d'une équation différentielle linéaire satisfaite par une G-fonction au sens strict. Je montrerai comment ils peuvent être partiellement généralisés aux G-fonctions au sens large. Ceci nous permet d'obtenir des résultats diophantiens sur les valeurs des E-fonctions.

Andrew Elvey Price, The six vertex model on random lattices using Jacobi theta functions

  I will describe our solution to the six vertex model on random lattices. Our solution to this problem is a purely combinatorial reformulation of the non-rigorous solution of Kostov in mathematical physics literature, along with a proof that this it is the solution. The method involves exactly solving a system of functional equations for the associated generating function in terms of Jacobi theta functions. We observe that the solution satisfies certain modular properties implying that it simplifies when the ratio of the two weights is 2\cos(\alpha) with \alpha/\pi rational, thereby explaining the simple expressions found by E.P. and Bousquet-Mélou two such cases. This is joint work with Paul Zinn-Justin.

Maja Resman, Complex dimensions and lengths of epsilon-neighborhoods of orbits

  We consider orbits of model tangent to the identity germs on the real line and compute: (1) length of the epsilon-neighborhood of their orbits, (2) meromorphic extension of the zeta function of the fractal string generated by its orbits (see [2]). We compute complex dimensions of the orbit (among which box dimension is the first one), given as poles and residues of the meromorphic extension. It is known [1] that the length of the epsilon-neighborhood does not admit asymptotic expansion in power-logarithmic scale after first finitely many terms, due to the integer critical time in the definition. We have shown in [1] one way to circumvent this, by introducing the continuous length of epsilon-neighborhoods of orbits. It relies on embedding of a germ in a flow and using continuous critical time instead of discrete. Here we state the hypothesis that, at least in model cases, the contiuous length is related to the expansion of the standard length of the epsilon neighborhood in the sense of Schwarz distributions, which is obtained by inverse Mellin transform of the zeta function of the orbit. The coefficients and exponents of the expansion are determined by the complex dimensions of the orbit [3]. This is a joint work with Goran Radunović, University of Zagreb, and Pavao Mardešić, Université de Bourgogne.

[1] P. Mardešić, M. Resman, J.-P. Rolin, V. Županović, Tubular neighborhoods of orbits of power- logarithmic germs, Journal of dynamics and differential equations,1 (2019), 1; 1-49

[2] Lapidus, Michel L.; van Frankenhuijsen, MachielFractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings. Second edition. Springer Monographs in Mathematics.Springer, New York, 2013. xxvi+567 pp.

[3] Lapidus, Michel L.; Radunović, Goran; Žubrinić, DarkoFractal zeta functions and fractal drums. Higher-dimensional theory of complex dimensions. Springer Monographs in Mathematics.Springer, Cham, 2017. xl+655 pp.

Amélie Trotignon, Discrete harmonic functions in the three-quarter plane

  Harmonic functions play an important role in probability theory and are strongly related to the enumeration of walks. Doob h-transform is a way to build conditioned random processes in cones from a random process and a positive harmonic function vanishing on the boundary of the cone. Finding positive harmonic functions for random processes is therefore a natural objective in the study of confined random walks. There are very few ways to compute discrete harmonic functions. In this talk we are interested in positive discrete harmonic functions with Dirichlet conditions in three quadrants. Whereas planar lattice (random) walks in the quadrant have been well studied, the case of walks avoiding a quadrant has been developed lately. We extend the method in the quarter plane – resolution of a functional equation via boundary value problem using a conformal mapping – to the three-quarter plane applying the strategy of splitting the domain into two symmetric convex cones. We obtain a simple explicit expression for the algebraic generating function of harmonic functions associated to random walks avoiding a quadrant.

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