Linear differential equations with polynomial coefficients in a complex variable appear naturally in fields such as combinatorics, algebraic geometry, and mathematical physics. Their numerical solution, and in particular the numerical solution of connection problems between regular singular points, finds applications that include the high-precision computation of values of Feynmann integrals, volumes of semi-algebraic sets, or more generally periods of algebraic varieties, the asymptotic expansion of linearly recurrent sequences, and the factorization of linear differential operators. In this talk, I will give a demonstration of a numerical ODE solver that focuses on this very special class of equations, supports arbitrary-precision computations, and provides rigorous error bounds along with the numerical results. I will also demonstrate software based on this solver for some of the applications listed above.

This talk presents recent results on the reconstructibility and identifiability of nonlinear dynamical networks. These results rely on the identification of specific variables, called variables of interest, together with local differential relations obtained through elimination procedures applied to the equations governing each node. This framework makes it possible to propagate reconstructibility and identifiability properties across the network structure. As an application, I illustrate the approach on the neuronal network of C. elegans involved in chemotaxis.

We prove an Ax-Lindemann-Weierstrass differential transcendence result for Euler's gamma function, namely that the functions Γ(ν-ζ1(ν)),…, Γ(ν-ζn(ν)) are differentially independent over the field of rational functions in the variable ν, with coefficients in the field k of 1-periodic meromorphic functions over the complex numbers, as soon as ζ1,…, ζn determine a set of algebraic functions over k, stable by conjugation and pairwise distinct modulo the integers. 

To prove this result we use both the Galois theory of difference equations and the theory of a characteristic zero analog of the Carlitz module introduced by the second author in 2013. As an intermediate result we give an explicit description of the Picard-Vessiot rings and of the Galois groups associated to the operators in the image of the Carlitz module, using techniques inspired by the Carlitz-Hayes theory. This is a joint work with Federico Pellarin. 

Consider a subfield of the field of rational functions. It is finitely generated, and a natural question is to find a simple generating set. In this talk, we will present an algorithm to do this. We will also discuss one practical application: the problem of structural parameter identifiability of ODEs. Joint work with Gleb Pogudin.

Difference algebraic groups occur naturally as Galois groups in certain Galois theories, and they can also be used as a tool to study difference Hopf structures. In this talk I will discuss some finiteness properties of partial difference algebraic groups, in particular the existence of a dimension polynomial for any partial difference algebraic group. This allows us to define some invariants of a difference algebraic group such as its difference type and difference dimension.

A rational dynamical system consists of a variety X and a rational (algebraic) map from X to itself. We study a special class of such dynamical systems, the isotrivial ones, showing for example that if the system has no "first integrals", then it has only finitely many maximal proper invariant subvarieties (the Dixmier-Moeglin problem). This result, as well as others, is a corollary of the main theorem, which states that the birational automorphism group of such a system is an algebraic group. This, in turn, is a special case of the "binding group" theorem in model theory. I will explain the definitions and statements involved, as well as the model theoretic point of view on the situation. This is joint work with Rahim Moosa from the University of Waterloo.

Classically, differential Galois theories, including Picard-Vessiot theory for linear differential equations and Kolchin's strongly normal theory, associate an algebraic group to a differential field extension. By using categorical Galois theory and novel techniques of descent, we develop a differential Galois theory that applies to morphisms of differential schemes. We find effective criteria for universality of geometric quotients and thus pave the way to numerous applications of the theory. In particular, we show how it can be used to treat differential equations parametrised over a scheme (including problems of commuting differential operators seen as parametrised over associated spectral curves), and to study Galois groupoids associated to linear differential equations.