We first investigate the interconnection of invariants of certain  group actions and time-reversibility of a class of two-dimensional polynomial systems with 1:-1 resonant singularity at the origin. The time-reversibility is related to the Sibirsky subvariety of the center (integrability) variety and it is known that every time-reversible system has a local analytic first integral at the origin. We propose a new algorithm to obtain a generating set for the Sibirsky ideal of such polynomial systems and investigate some algebraic properties of this ideal. Then, we discuss a generalization of the concept of time-reversibility in the n-dimensional case considering the systems with 1:zeta: ... :zeta^(n-1) resonant singularity at the origin (where zeta is a primitive n-th root of unity) and study a connection of such reversibility with the invariants of some group actions in the space of parameters of the system and Lawrence ideals.

Let G⊆SL2(ℂ) be a finite primitive group. A classical result of Klein states that there exists a hypergeometric equation such that any second order linear ordinary differential equation whose differential Galois group is G is projectively equivalent to the pullback by a rational map of this hypergeometric equation. In this talk I present a generalization of this result. Let G⊆SLn(ℂ) be a finite primitive group. I will show that there is a positive integer d=d(G) and a standard equation such that any linear ordinary differential equation whose differential Galois group is G is gauge equivalent over a field extension of degree d to an equation projectively equivalent to the pullback by a map of this standard equation. For n=3, the standard equations can be chosen so that they are hypergeometric. Implementations of Klein's result exist. If time permits, I will show how the properties of the invariants of the primitive subgroups of SL3(ℂ) can be exploited to aim for an efficient implementation of this generalization for n=3.

This talk is based on a paper written in collaboration with Lewis Marsh, Helen Byrne and Heather Harrington, where we explored the algebra, geometry and topology of ERK kinetics. The MEK/ERK signalling pathway is involved in cell division, cell specialisation, survival and cell death. We studied a polynomial dynamical system describing the dynamics of MEK/ERK proposed by Yeung et al. with their experimental setup, data and known biological information. The experimental dataset is a time-course of ERK measurements in different phosphorylation states following activation of either wild-type MEK or MEK mutations associated with cancer or developmental defects. My focus in this talk will be on identifiability, both structural and practical. Structurally identifiable is concerned with asking whether parameter values can be recovered from perfect data. Practical identifiability addresses the more realistic situation where we assume there is measurement noise. We observe that the original model is structurally but not practically identifiable. We will discuss how algebraic quasi-steady state approximation leads to a smaller simpler model which is both structurally and practically identifiable, while providing a probable explanation for the practical non-identifiability of the original model.

O-minimality is a powerful model-theoretic tool with applications to analysis. For example, Wilkie showed how to construct derivations that respect any o-minimal function, including the complex exponential function. These derivations can then be combined with functional transcendence results like the Ax-Schanuel theorem to prove generic number theoretic transcendence results. In this talk, I will discuss joint work with P. Speissegger in which we prove that the Gamma function, which was known to be o-minimal when restricted to the positive real numbers, is in fact o-minimal on certain unbounded complex domains. I will also discuss one challenge that arises when using o-minimality to build derivations that respect Gamma.

Differentially large fields are a class of differential fields introduced by León Sánchez and Tressl which generalise the notion of largeness to differential fields. In order to study these fields, they construct the Twisted Taylor Morphism, a functor which constructs differential ring homomorphisms into the ring of power series from ring homomorphisms into the base field. In this talk, we will begin by giving a brief introduction to the context of differentially large fields. We will then consider a generalised notion of a Taylor morphism, their relation to differentially large fields, and give a characterisation in terms of a twisted Hurwitz series ring. Time permitting, we will also consider a few categorical properties of this construction. 

The Sasano system of type A^(2)_4 is a time dependent Hamiltonian system which admits affine Weyl group symmetry of type A^(2)_4. In this talk, utilizing the differential Galois approach to the non-integrability of Hamiltonian systems, I will show in a strict way that for all values of the parameters for which the Sasano system of type A^(2)_4 has a particular rational solution, it is not integrable in the Liouville-Arnold sense by meromorphic first integrals which are rational functions in t. 

Deriving insights from Dynamical Systems can be challenging due to the large number of variables involved. To address this, model reduction techniques can be used to project the system onto a lower-dimensional state space. Constrained Lumping can reduce systems of ordinary differential equations of the form x'(t) = f(x(t)) up to linear combination of the variables in x(t). Exact reductions may be too restrictive in practice when the input data of the problem may already include some numerical errors. This may come with the cost of limiting the actual aggregation of exact reductions techniques. In this talk we will present an extension on current lumping algorithms which relaxes the exactness requirement up to a given tolerance parameter epsilon. We will show that this approach allows to recover reductions on systems with errors as well as aggregating systems without incurring in big simulation errors. Furthermore, we will propose a heuristic for finding the biggest possible epsilon for a given accepted simulation error.

This is a joint work with Alexander Leguizamon-Robayo, Mirco Tribastone, Max Tschaikowski and Andrea Vandin. 

Recently Adamczewski, Bell and Delaygue gave an algebraic independence criterion for power series with coefficients in Z that satisfy “p-Lucas congruences” for infinitely many prime numbers p. Most of the powers series satisfying this type of congruences are G-functions. In the first part of the talk, we are going to see how we can obtain this congruences when the power series is a solution of a differential operator. The main tools are, on the one hand, the p-adic study of the differential operator, strong Frobenius structure, and on the other hand, the classical notion of monodromy. In the second part, I introduce a new set of G-functions, denoted MF, and show that the elements of MF also satisfy convenience congruences modulo p. Finally, we will see that in some cases these last congruences allow us to establish the algebraic independence  of G functions that are in MF. 

In this talk, I will give a transverse presentation of Differential Galois Theory over a discrete valuation ring (DVR), call it R. This is based on several works done with P. H. Hai over the last decade.

While in the classical approach we employ linear algebraic groups acting on differential extensions, in the case of a DVR, we are better off with Tannakian categories and affine flat group schemes. Now, the group schemes in question can easily fail to be of finite type over R, therefore making much of the standard theory somewhat unsuitable. I shall then present two structure results and one result allowing calculations.

-- The first structure result centres around the notion of *Néron blowups*: this allows us to modify the special fibre of a group while preserving the generic one. They are analogues of congruence subgroups.

With this tool in hands, we shall see that, in many cases, the Differential Galois groups failing to be of finite type are obtained in an "automatic way" from something of finite type.

-- The second structure result relies on a property called "prudence" of representations; it is to be applied to equations on proper ambient spaces.

-- Finally, I will show how "Schlesinger's Density Theorem" can be recovered, allowing us to compute effectively. In particular, I will show through an example that Galois groups of infinite type are commonplace.

In this talk, we mainly present the work on unirational differential curves and the corresponding differential rational parametrizations. We first investigate the basic properties of proper differential rational parametrizations for unirational differential curves. Then we show the implicitization of proper linear differential rational parametric equations could be solved by means of differential resultants. Furthermore, for linear differential curves, we give a criterion to decide whether an implicitly given linear differential curve is unirational, and  in the affirmative case, to compute a proper differential rational parametrization. This is joint work with Lei Fu.