Some applications of parameterized Picard-Vessiot theory (Q2810939)

The author summarizes some of his recent results on the applications of the parameterized Picard-Vessiot theory (PPV-theory), pointing on those obtained in joint work with M. Singer for families of ordinary differential equations having parameterized regular singularities.NEWLINENEWLINEThese results include parameterized analogues of Schlesinger's theorem and of the weak Riemann-Hilbert problem [the author and \textit{M. F. Singer}, Bull. Lond. Math. Soc. 44, No. 5, 913--930 (2012; Zbl 1254.34124)]. The author also exposes an algebraic interpretation in terms of PPV-theory of a special case of the so-called monodromy evolving deformations [the author and \textit{M. F. Singer}, Proc. Am. Math. Soc. 141, No. 2, 605--617 (2013; Zbl 1268.34187)]. The latter is illustrated by the Darboux-Halphen equation, as the author also adds a brief history of the same equation.NEWLINENEWLINEIn addition, the author outlines recent applications of the above results, including the work of \textit{T. Dreyfus} [Pac. J. Math. 271, No. 1, 87--141 (2014; Zbl 1364.12004)] and the joint papers of \textit{A. Minchenko} et al. [J. Inst. Math. Jussieu 13, No. 4, 671--700 (2014; Zbl 1364.12005); Int. Math. Res. Not. 2015, No. 7, 1733--1793 (2015; Zbl 1339.12003)] on the inverse problem of PPV-theory and on the generalization to irregular singularities.