Classifying regular systems of first order linear ordinary equations is a classical subject going back to Poincare and Dulac. There is a gauge group whose action can be described and an integrable normal form produced. A similar problem for higher order differential equations was never addressed, perhaps because the corresponding equivalence relationship is not induced by any group action. Still one can develop a reasonable classification theory, largely parallel to the classical theory. This is a joint work with Shira Tanny from the Weizmann Institiute, see http://arxiv.org/abs/1412.7830.[-]

We identify the persistence probability for the zero-temperature non-equilibrium Glauber dynamics of the half-space Ising chain as a particular Painlevé VI transcendent, with monodromy exponents (1/2,1/2,0,0). Among other things, this characterization a la Tracy-Widom permits to relate our specific Bonnet-Painlevé VI to the one found by Jimbo & Miwa and characterizing the diagonal correlation functions for the planar static Ising model. In particular, in terms of the standard critical exponents eta=1/4 and beta=1/8 for the latter, this implies that the probability that the limiting Gaussian real Kac's polynomial has no real root decays with an exponent 4(eta+beta)=3/4.[-]

In 1978 Roger Apéry proved irrationality of zeta(3) approximating it by ratios of terms of two sequences of rational numbers both satisfying the same recurrence relation. His study of the growth of denominators in these sequences involved complicated explicit formulas for both via sums of binomial coefficients. Subsequently, Frits Beukers gave a more enlightening proof of their properties, in which zeta(3) can be seen as an entry in a monodromy matrix for a differential equation arising from a one-parametric family of K3 surfaces. In the talk I will define Apéry constants for Fuchsian differential operators and explain the generalized Frobenius method due to Golyshev and Zagier which produces an infinite sequence of Apéry constants starting from a single differential equation. I will then show a surprising property of their generating function and conclude that the Apéry constants for a geometric differential operator are periods.
This is work in progress with Spencer Bloch and Francis Brown.[-]