The parameter identifiability problem of a dynamical system is to determine whether the parameters of the system can be found from a given subset of variables of the system. Verifying whether the parameters are indentifiable is a necessary first step before a meaningful parameter estimation can take place. Reparametrization can improve identifiability properties of a dynamical system. We will discuss one of the approaches to do this, which is based on differential algebra and other symbolic computation methods. 

We present a method for computing rigorous enclosures of eigenvalues and eigenfunctions of the Laplacian, both in the plane and on the sphere. We give two different applications of these methods. One giving an explicit counterexample to a conjecture by Payne in 1967 related to the behaviour of the nodal line of the second Dirichlet eigenfunction of the Laplacian. The other about computing approximations of constants appearing in the asymptotic analysis of random walks in 3 dimensions. This is joint work with Bruno Salvy, Javier Gómez-Serrano and Kimberly Hou. 

We study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated using tools from homological algebra and the theory of D-modules. In this talk, I will present an overview of the main tools needed to study these vector spaces, namely twisted de Rham cohomology and Mellin transform. Finally, I will discuss relations between these approaches. This is a joint project with Daniele Agostini, Anna-Laura Sattelberger, and Simon Telen.

In 1873, Georg F. Frobenius, then just 24 years old, published a paper (Crelle 76) in which he proposed a simpler construction of the solutions of an ordinary linear differential equation with regular singularity at 0 - simpler than the one of Fuchs' seminal paper from 1866. Quite miraculously, he starts with a first solution, one without logarithm, then replaces its local exponent by a variable and derives the obtained function with respect to this variable. After this, he replaces the variable again by the local exponent. As a result, he gets a new solution. Iterating the process, he finds and constructs a full basis. Amazing!

In the lecture, we wish to explain Frobenius' method in modern terms, and to show that once the construction of the solutions is put on a more conceptual basis, it pops up almost automatically.

Let R be a commutative (ordinary) differential ring. We consider differential R modules which are projective (and usually finitely generated) as R-modules. An example is R itself, and also direct sums of copies of R. In fact every projective R module is shown to have a differential structure. In differential Galois theory of differential fields differential modules given by matrix differential equations having a full set of solutions are precisely the above direct sums. For rings, these modules have the same interpretation. More generally, a differential module which is a direct summand of these kinds of direct sums can be shown to be induced by tensoring up to R from the ring of constants of R. One way to classify differential modules finitely generated and projective as R modules is to declare two such equivalent if after adding direct sums of copies of R to both they become isomorphic as differential modules. The equivalence classes form a monoid analogous to and linked with the K groups of R and its ring of constants. Examples and consequences of such are examined.

A power series is a k-Mahler function (k>1) if it satisfies a certain difference equation. Such power series were originally studied by Mahler in the context of transcendence theory. Later it was realized that power series whose coefficients are k-automatic, or more generally, k-regular (in the sense of Allouche and Shallit) are always k-Mahler, but it was not clear which k-Mahler series give rise to k-automatic or k-regular sequences. We show that k-Mahler series with algebraic coefficients fall into five distinct classes, based on the asymptotic growth of their coefficients. One of these classes corresponds to k-automatic sequences, and two to k-regular (but not k-automatic) sequences. Moreover, these growth classes are decidable.

Following some earlier results, Schaefke and Singer proved several theorems asserting that a Laurent power series which satisfies two difference (or differential) equations with polynomial coefficients, with respect to independent difference / differential operators, must be a rational function. Recently, Adamczewski, Dreyfus, Hardouin and Wibmer applied parametrized Picard Vessiot theory to obtain a remarkable strengthening of the Schaefke-Singer results, phrased as theorems on algebraic independence over C(x).

We shall report on our work concerning the same questions over fields of elliptic functions. The focus will be on the rather surprising aspects that are new in comparison to the rational case.

In this talk I shall speak about Painleve and quasi-Painleve equations, their differences and similarities. Painleve equations are nonlinear second order differential equations solutions of which have no movable critical points. They appear in many applications. For quasi-Painleve equations movable critical singularities in solutions are allowed. I shall explain some recent results on non-integrability, polynomial Hamiltonian structure and spaces of initial conditions.

In this talk we study an application of differential algebra to machine learning. A machine learning model can be thought of as a function approximator, and we describe a family of models yielding functions satisfying homogeneous linear PDE with constant coefficients. We will be studying Gaussian process models, where the right choice of kernel function yields solutions to PDE. Our kernels are constructed by exploiting the Ehrenpreis-Palamodov fundamental principle to represent solutions as integrals over algebraic varieties. This can be made algorithmic and algebraic using the duality between PDE systems and polynomial modules.

A theorem of Serre says that if a field is bounded, meaning that it has finitely many finite extensions of each degree, then the Galois cohomology of finite algebraic groups over that field can be understood in terms of finite group cohomology. In this talk, an analogous condition for differential fields will yield a theorem relating Kolchin's differential Galois cohomology to the cohomology of algebraic groups. I will describe how this cohomological machinery works and the relation to Galois theory in the differential and algebraic contexts. This is joint work in progress with Anand Pillay.

We will discuss new challenges in the elimination of unknowns to simplify systems of difference-differential equations with parameters. For this task we will analyze how different approaches using differential resultants can help in this process. When considering differential-difference polynomials similar prolongation techniques can be applied, but alternative methods are needed to achieve elimination of the desired variables when applied to models in Biology. This is joint work with A. Jiménez-Pastor and A. Ovchinnikov.