In the context of (bio)chemical reaction networks, the dynamics of the concentrations of the chemical species over time are often modelled by a system of parameter-dependent ordinary differential equations, which are typically polynomial or described by rational functions. The study of the steady states of the system translates then into the study of the positive solutions to a parametric polynomial system.

In this talk I will start by shortly presenting the formalism of the theory of reaction networks. Afterwards I will focus on the study of the parameter region where the relevant parametric system admits at least two positive solutions (a property termed multistationarity). I will show recent results on how to determine the region and whether it is connected.

The results I present arise from several joint works involving Conradi, Kaihnsa, Mincheva, Telek, Yürük, Wiuf and de Wolff.

We discuss a framework for defining and detecting singularities of arbitrary fully nonlinear systems of ordinary or partial differential equations with polynomial nonlinearities. It combines concepts from differential topology with methods from differential algebra and algebraic geometry. With its help, we provide for the first time a general definition of singularities of partial differential equations and show that it is at least meaningful in the sense that generic points are regular. Our definition is then extended to the notion of a regularity decomposition of a differential equation at a given order and the existence of such decompositions is proven by presenting an algorithm for their effective determination (with an implementation in Maple). Finally, we rigorously define the notion of a regular differential equation (a fundamental concept in the geometric theory of differential equations). We show that our algorithm automatically extracts one provably regular differential equation from each prime component of a given equations and thus provides an effective answer to an old problem in the geometric theory.

The assumption of perfect knowledge of rate parameters in continuous-time Markov chains (CTMCs) is undermined when confronted with reality, where they may be uncertain due to lack of information or because of measurement noise. In this talk we consider uncertain CTMCs, where rates are assumed to vary non-deterministically with time from bounded continuous intervals. This leads to a semantics which associates each state with the reachable set of its probability under all possible choices of the uncertain rates. We develop a notion of lumpability which identifies a partition of states where each block preserves the reachable set of the sum of its probabilities, essentially lifting the well-known CTMC ordinary lumpability to the uncertain setting. We proceed with this analogy with two further contributions: a logical characterization of uncertain CTMC lumping in terms of continuous stochastic logic; and a polynomial time and space algorithm for the minimization of uncertain CTMCs by partition refinement, using the CTMC lumping algorithm as an inner step. As a case study, we show that the minimizations in a substantial number of CTMC models reported in the literature are robust with respect to uncertainties around their original, fixed, rate values.

This is a join work with Luca Cardelli, Radu Grosu, Kim G. Larsen, Mirco Tribastone and Andrea Vandin.

Tropical techniques allow us to know the possible supports of power series solutions to a system of differential equations by putting some bounds on these supports. By tropicalizing a system of differential equations and finding its solution set, we will have information about the set of power series solutions of the original system.

In this talk, I will speak about tropical differential geometry in both ordinary and partial cases and I will mention the fundamental theorem of tropical differential geometry that states equality between the solution set of tropicalizations of the system of differential equations and set of power series solutions of the original system.

The construction of explicit general solutions of (non-linear) algebraic ODEs (AODEs) is a particularly difficult topic. For first-order AODEs defined over a differential field K, adjoining a single arbitrary constant c to K is arguably the simplest extension in which one may hope to find such solutions. The aim of this talk is to discuss a framework for computing general solutions of first-order AODEs contained in K(c). In general, the approach combines an interplay of differential algebra and (birational) algebraic geometry. More precisely, the computation consists of a parametrization step, in which one seeks a rational parametrization of a particular algebraic curve, followed by the deduction of a suitable K-automorphism, a Möbius transformation, of K(c).

For symbolic computations with systems of linear functional equations, like (integro-)differential equations or boundary problems, we need an algebraic framework that enables effective computations in corresponding rings of operators. To represent and compute with concrete linear systems usually matrices of operators with scalar coefficients are used. For statements about whole families of linear functional systems, however, symbolic methods that directly work with operators having undetermined matrix coefficients needs to be used.

In the talk, we discuss a construction of integro-differential operators over an arbitrary integro-differential ring. Allowing noncommutative coefficients, we can treat functional systems of generic size with this approach. We outline how we can find and prove all consequences of the fundamental theorem of calculus in differential rings using computer algebra. Our general approach is based on tensor reduction systems and allows us to find and compute with normal forms for certain rings of linear operators. Normal forms can also be used to prove identities as well as to solve operator equations by ansatz. We illustrate our approach with examples for some classes of linear functional systems. We will also discuss some ongoing and future work on generalized integro-differential rings and symbolic computations for identities of linear operators.

This talk is based on joint work with Jamal Hossein Poor and Clemens G. Raab.

We will give a differential algebraic condition which is sufficient to show that the complex analytic solutions of a nonlinear differential equation are not Pfaffian functions. The class of Pfaffian functions was introduced by Khovanskii in conjunction with various finiteness problems. The class generalizes many well-known classes of functions (e.g. Liouvillian, elementary). In recent years, their connection to diophantine geometry has motivated questions (e.g. by Binyamini and Novikov) about whether certain classical functions are Pfaffian. In this talk, we will resolve these questions and talk about the prospects for a differential algebraic classification of equations whose solutions are Pfaffian.

In this talk, I will introduce some of the verification problems we are interested in, for proving control systems and cyber-physical systems correct. Set-based methods have proven very effective in proving mostly finite time horizon properties (e.g. reachability properties). In this talk I will introduce some other approaches, algebraic (and if time permits, topological), that help understand infinite time properties (e.g. invariant properties) of differential and control systems.

The talk is based on joint work with Sylvie Putot and a number of colleagues over the years.

In this talk I will introduce a problem in arithmetic geometry known as the Existential Closedness Problem, which aims at understanding interesting algebraic properties of transcendental functions. Despite the problem being open in general, I will show how results and techniques from differential algebra allow us to get powerful partial results. During the talk, I will mostly speak about the complex exponential function to help fix ideas, but it will then be easy (hopefully) to see how these ideas can be applied in other contexts.

Many biological systems can be modelled with ordinary differential equations. Sometimes these models admit symmetry transformations, that is, they possess an invariance property with respect to transformations of certain variables. These symmetries can be analysed as Lie groups of transformations. In this talk I will discuss the significance of such invariances and how to detect them. For example, if two parameters can be interchanged without affecting the model output, this means that they are unidentifiable, which is undesirable for the purpose of system identification. On the other hand, this feature may also be interpreted as biological robustness: variations in the value of one parameter, due e.g. to environmental changes, can be compensated by variations in other parameter, leaving the output invariant. If a model has symmetries that make it non-observable (i.e. some state variables cannot be inferred from output measurements), it will yield incorrect simulations of the time course of the non-observable variables. To predict the behaviour of said variables we would need a transformed model that is observable and preserves the meaning of the state variables of interest. I will show how such reparameterization can sometimes be obtained with a recently presented method.

The algebraic identities satisfied by an integral equation can vary greatly depending on the kernels present. In this talk I will discuss the development of an algebraic theory of general integral equations that allows for both arbitrary kernels and limits of integration using bracketed words and decorated rooted trees. As an application, I will show how any separable Volterra integral equation is equivalent to one that is operator linear, that is, contains only iterated integrals. This talk is based on joint work with Li Guo and Yunnan Li.

I will describe recent research aiming at initializing a dynamical aspect of symbolic integration by studying stability problems in differential fields. We present some basic properties of stable elementary functions and D-finite power series that enable us to characterize three special families of stable elementary functions including rational functions, logarithmic functions, and exponential functions. Some problems for future studies are proposed towards deeper dynamical studies in differential and difference algebra.

Signalling pathways in molecular biology can be modelled by polynomial dynamical systems. I will present mathematical models describing two biological systems involved in development and cancer. I will overview approaches to analyse these models with data using computational algebraic geometry, differential algebra and statistics. I will also present how topological data analysis can provide additional information to distinguish wild-type and mutant molecules in one pathway. These case studies showcase how computational algebra, geometry, topology and dynamics can provide new insights to better understand model parameter values as well as biological systems with time course data, specifically how changes at the molecular scale (e.g. molecular mutations) result in kinetic differences that are observed as phenotypic changes (e.g. mutations in fruit fly wings).

To a difference equation we may associate a Galois group that computes that algebraic relations among the solutions. The latter is an algebraic group and a theorem due to van der Put and Singer prove that up to a change of variable, we may assume that the system is a point of the algebraic group. The order two case was quite well understood, and In this talk we will explain how to find this change of variable for order three equations.

This is a joint work with Marina Poulet.

We present a new approach to the problem of quantising integrable systems of differential-difference equations. The main idea is to lift these systems to systems defined on free associative algebras and look for the ideals there that are stabilized by the new dynamics. In a reasonable class of candidate ideals, there are typically very few that are invariant for the first equation in the hierarchy. Once these ideals are picked the challenge is to prove that the whole hierarchy of equations stabilizes them. We will discuss these ideas using as a key example the hierarchy of the Bogoyavlensky equation. This is a joint work with A. Mikhailov (Leeds) and J. P. Wang (U. of Kent) that we put on arXiv last month (2204.03095).