Schedule for: 20w5204 - Model Theory of Differential Equations, Algebraic Geometry, and their Applications to Modeling (Online)

A brief introduction and welcome by the BIRS Programme Coordinator.

I will introduce the basic notions on model theory focusing on effective methods such as quantifier elimination and discuss applications to algebraic theory of differential equations.
Pre-talk survey

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 polynomial structure of the system allows the use of techniques from algebra (e.g., real algebraic geometry) to study properties of the system around steady states, for all parameter values. In this talk I will start by presenting the formalism of the theory of reaction networks. Afterwards I will outline the qualitative questions one would like to address, which include deciding upon the existence of multiple equilibrium points or periodic orbits, and their stability. If time permits, I will discuss selected methods with emphasis on their limitations.
Pre-talk survey

Recent work links certain aspects of the second part of Hilbert’s 16th problem (H16) to the theory of o-minimality. One of these aspects is the generation and destruction of limit cycles in families of planar vector fields, commonly referred to as ”bifurcations”. I will outline the significance of bifurcations for H16 and explain how logic–in particular, o-minimality–can be used to understand them well enough to be able to count limit cycles.
Pre-talk survey

For mass-action kinetics, a common model for biochemistry, much work has gone into relating network structure to the possible dynamics of the resulting systems of polynomial ODEs. A family of mass-action systems, complex-balancing, is defined by having a positive equilibrium that balances monomials across vertices. Surprisingly, every positive equilibrium of such a system similarly balance monomials across vertices. These systems enjoy a variety of algebraic and stability properties: toricity in the steady state variety and in parameter space; Lyapunov and conjectured global stability. Unfortunately, most systems are vertex-balanced if and only if the parameters come from a toric ideal. By searching for different graphs representing the same ODEs, we can expand the parameter region for which the system is dynamically equivalent to a complex-balanced system. The expanded region is defined in the space of states and parameters, and the challenge is to eliminate the state variables to obtain explicit conditions on parameters (that is, to perform quantifier elimination over the reals). In this talk, I will introduce and set up the problem via examples.
Pre-talk survey

I will talk about the convex hulls of trajectories of polynomial dynamical systems. Such trajectories also include real algebraic curves. The main problem is to describe the boundary of the resulting convex hulls. The motivation to describe these convex hulls comes from attainable region theory in chemistry, where taking convex combinations of points corresponds to mixing results of reactions. We stratify the boundary into families of faces comprised of patches. We define patches using the notion of normal cycles from integral geometry. I will discuss the numerical algorithms we developed for identifying these patches. This is a joint work with Daniel Ciripoi, Andreas Loehne, and Bernd Sturmfels.
Pre-talk survey

Get ready to turn your cameras on to be included in the official "group photo" -- a screenshot of the Zoom participants in Gallery view!

I will introduce the basic notions on model theory focusing on effective methods such as quantifier elimination and discuss applications to algebraic theory of differential equations.

A dynamic model is structurally identifiable (respectively, observable) if it is theoretically possible to infer its unknown parameters (respectively, states) by observing its output over time. The two properties, structural identifiability and observability, are completely determined by the model equations. Their analysis is of interest for modellers because it informs about the possibility of gaining insight about the unmeasured variables of a model. Here we cast the problem of analysing structural identifiability and observability as that of finding Lie symmetries. We build on previous results that showed that structural unidentifiability amounts to the existence of Lie symmetries. We consider nonlinear models described by ordinary differential equations and restrict ourselves to rational functions. We revisit a method for finding symmetries by transforming rational expressions into linear systems, and extend it by enabling it to provide symmetry-breaking transformations. This extension allows for a semi-automatic model reformulation that renders a non-observable model observable. We have implemented the methodology in MATLAB, as part of the STRIKE-GOLDD toolbox for observability and identifiability analysis. We illustrate its use in the context of biological modelling by applying it to a set of problems taken from the literature, which also allow us to discuss the implications of (non)observability.
Pre-talk survey

To any algebraic differential equation, one can associate a first-order structure which encodes some of the properties of algebraic integrability and of algebraic independence of its solutions. To describe the structure associated to a given algebraic (non linear) differential equation (E), typical questions are:

• Is it possible to express the general solutions of (E) from successive resolutions of linear differential equations?
• Is it possible to express the general solutions of (E) from successive resolutions of algebraic differential equations of lower order than (E)?
• Given distinct initial conditions for (E), under which conditions are the solutions associated to these initial conditions algebraically independent?

In my talk, I will discuss in this setting one of the simplest examples of non completely integrable Hamiltonian systems: the geodesic motion on an algebraically presented compact Riemannian surface with negative curvature. I will explain a qualitative model-theoretic description of the associated structure (and its content in the differential algebraic language used above) based on the global hyperbolic dynamical properties identified by Anosov in the 70’s (today called Anosov flows) for the geodesic motion in negative curvature.
Pre-talk survey

This talk offers a brief and introductory overview of tropical algebraic geometry with a heavy emphasis on computations. We introduce the notions of tropical semirings and tropical varieties, and discuss some of the algorithms surrounding them. Finally, we will highlight recent and ongoing works on the frontiers of tropical differential algebra.
Pre-talk survey

The irreducibility of an ODE is a notion that was introduce by P. Painlevé at the turn of the 20th century and later refined by H. Umemura. Roughly, an ODE is irreducible if all of its solutions are ‘new’ functions. This notion is also almost equivalent to strong minimality, a central notion in model theory. In this talk we will go over the definitions of these concepts and discuss new methods to prove that ODEs with generic constant parameters are irreducible. We use the Painlevé equations as examples.
Pre-talk survey

Dynamical systems arising from chemical reactions can be generated by finite directed graphs embedded in the Euclidean space, called Euclidean embedded graphs (E-graphs). These dynamical systems have polynomial right-hand-side, which creates a strong connection between real algebraic geometry and reaction network theory. In this talk, we will focus on complex-balanced systems, which have been also called “toric dynamical systems" by Craciun, Dickenstein, Shiu and Sturmfels. Toric dynamical systems are known or conjectured to enjoy exceptionally strong dynamical properties, such as existence and uniqueness of positive equilibria, as well as local and global stability. We will discuss the use of E-graphs and algebraic geometry in understanding how the same is true for a larger class of systems. Inspired by work done in [Craciun, Jin, Yu, "An efficient characterization of complex-balanced, detailed-balanced, and weakly reversible systems”], we further analyse from an algebraic perspective the property of being dynamically equivalent to a complex balanced system, which we call "disguised toric dynamical systems". This is based on joint work with Gheorghe Craciun and Laura Brustenga i Moncusí.
Pre-talk survey
The paper with the result

Discrete models, such as Boolean networks, are an increasingly popular modeling framework in systems biology, with many hundreds of published models. The advantages are, among others, that they are intuitive and don't require detailed quantitative knowledge such as kinetic parameters. One disadvantage is that there are relatively few mathematical and computational tools available for this model type. As a basic example, given a model, how can we compute all its steady states? The basic mathematical framework they can be cast in is polynomial dynamical systems over finite fields. There is a rich convergence of dynamic, algebraic, combinatorial, and graph-theoretic features that come together within this type of mathematical object. Yet very little of this convergence has been used to study a mathematically rich class of objects, with important applications to problems in the life sciences and elsewhere. This talk will discuss several mathematical and computational problems, inspired but not directly connected to applications in biology, that can stimulate interesting research in algebra, broadly defined.