## Contact
Dr. Matthias Seiß
Room: 3319 Office hours by appointment. |

- Galois Theories of Differential Equations and Differential Algebra.
- Analysis of Singularities of General Systems of Differential Equations Using Techniques from Differential Algebra, Algebraic Geometry and Differential Topology.

- Reduction of Chemical Reaction Networks with Approximate Conservation Laws with Aurélien Desoeuvres, Alexandru Iosif, Christoph Lüders, Ovidiu Radulescu, Hamid Rahkooy and Thomas Sturm.

Model reduction of fast-slow chemical reaction networks based on the quasi-steady state approximation fails when the fast subsystem has first integrals. We call these first integrals approximate conservation laws. In order to define fast subsystems and identify approximate conservation laws, we use ideas from tropical geometry. We prove that any approximate conservation law evolves slower than all the species involved in it and therefore represents a supplementary slow variable in an extended system. By elimination of some variables of the extended system, we obtain networks without approximate conservation laws, which can be reduced by standard singular perturbation methods. The field of applications of approximate conservation laws covers the quasi-equilibrium approximation, well known in biochemistry. We discuss both two timescale reductions of fast-slow systems and multiple timescale reductions of multiscale networks. Networks with multiple timescales have hierarchical relaxation. At a given timescale, our multiple timescale reduction method defines three subsystems composed of (i) slaved fast variables satisfying algebraic equations, (ii) slow driving variables satisfying reduced ordinary differential equations, and (iii) quenched much slower variables that are constant. The algebraic equations satisfied by fast variables define chains of nested normally hyberbolic invariant manifolds. In such chains, faster manifolds are of higher dimension and contain the slower manifolds. Our reduction methods are introduced algorithmically for networks with linear, monomial or polynomial approximate conservation laws.

- A Computational Approach to Complete Exact and Approximate Conservation Laws of Chemical Reaction Networks with Aurélien Desoeuvres, Alexandru Iosif, Christoph Lüders, Ovidiu Radulescu, Hamid Rahkooy and Thomas Sturm.

We consider chemical reaction networks (CRN) with monomial reaction rates. For these models, we introduce and discuss the concepts of exact and approximate conservation laws, that are first integrals of the full and truncated sets of ODEs. For fast-slow CRNs, truncated ODEs describe the fast dynamics. We define compatibility classes as subsets of the species concentration space, obtained by equating the conservation laws to constants. A set of conservation laws is complete when the corresponding compatibility classes contain a finite number of steady states (quasi-steady states for approximate conservation laws of fast-slow systems). Complete sets of conservation laws can be used for reducing CRNs by variable pooling. We provide algorithmic methods for computing linear, monomial and polynomial conservation laws of CRNs and for testing their completeness. The resulting conservation laws and their completeness are either independent or dependent on the parameters. In the latter case, we provide parametric case distinctions. In particular, we propose a new method to compute polynomial conservation laws by comprehensive Gröbner systems and syzygies.

- On the Numerical Integration of Singular Initial and Boundary Value Problems for Generalised Lane-Emden and Thomas-Fermi Equations with Werner M. Seiler.

We propose a geometric approach for the numerical integration of singular initial value problems for (systems of) quasi-linear differential equations. It transforms the original problem into the problem of computing the unstable manifold at a stationary point of an associated vector field and thus into one which can be solved in an efficient and robust manner. Using the shooting method, our approach also works well for boundary value problems. As examples, we treat some (generalised) Lane-Emden equations and the Thomas-Fermi equation.

- Normal Forms in Differential Galois Theory for the Classical Groups with Daniel Robertz.

Communications in Algebra, published online: 31 Oct 2022, https://doi.org/10.1080/00927872.2022.2137520.

Let G be a classical group of dimension d and let a=(a_{1},…,a_{d}) be differential indeterminates over a differential field F of characteristic zero with algebraically closed field of constants C. Further let A(a) be a generic element in the Lie algebra 𝔤(F⟨a⟩) of G obtained from parametrizing a basis of 𝔤 with the indeterminates a. It is known (cf. work of Juan) that the differential Galois group of ∂(y)=A(a)y over F⟨a⟩ is G(C). In this paper we construct a differential field extension L of F⟨a⟩ such that the field of constants of L is C, the differential Galois group of ∂(y)=A(a)y over L is still the full group G(C) and A(a) is gauge equivalent over L to a matrix in normal form which we introduced in work of Seiss. We also consider specializations of the coefficients of A(a).

- On General Extension Fields for the Classical Groups in Differential Galois Theory.

Let G be one of the classical groups of Lie rank l. We make a similar construction of a general extension field in differential Galois theory for G as E. Noether did in classical Galois theory for finite groups. More precisely, we build a differential field E of differential transcendence degree l over the constants on which the group G acts and show that it is a Picard-Vessiot extension of the field of invariants E^{G}. The field E^{G}is differentially generated by l differential polynomials which are differentially algebraically independent over the constants. They are the coefficients of the defining equation of the extension. Finally we show that our construction satisfies generic properties for a specific kind of G-primitive Picard-Vessiot extensions.

- Algebraic and Geometric Analysis of Singularities of Implicit Differential Equations with Werner M. Seiler.

Computer Algebra in Scientific Computing - CASC 2020, F. Boulier, M. England, T. Sadykov, E.V. Vorozhtsov (eds.), LNCS 12291, Springer-Verlag, Cham 2020, pp. 14-41.

We review our recent works on singularities of implicit ordinary or partial differential equations. This includes firstly the development of a general framework combining algebraic and geometric methods for dealing with general systems of ordinary or partial differential equations and for defining the type of singularities considered here. We also present an algorithm for detecting all singularities of an algebraic differential equation over the complex numbers. We then discuss the adaptions required for the analysis over the real numbers. We further outline for a class of singular initial value problems for a second-order ordinary differential equation how geometric methods allow us to determine the local solution behaviour in the neighbourhood of a singularity including the regularity of the solution. Finally, we show for some simple cases of algebraic singularities how there such an analysis can be performed.

- Singularities of Algebraic Differential Equations with Markus Lange-Hegermann, Daniel Robertz and Werner M. Seiler.

Advances in Applied Mathematics, Volume 131, October 2021, 102266, https://doi.org/10.1016/j.aam.2021.102266.

We combine algebraic and geometric approaches to general systems of algebraic ordinary or partial differential equations to provide a unified framework for the definition and detection of singularities of a given system at a fixed order. Our three main results are firstly a proof that even in the case of partial differential equations regular points are generic. Secondly, we present an algorithm for the effective detection of all singularities at a given order or, more precisely, for the determination of a regularity decomposition. Finally, we give a rigorous definition of a regular differential equation, a notion that is ubiquitous in the geometric theory of differential equations, and show that our algorithm extracts from each prime component a regular differential equation. Our main algorithmic tools are on the one hand the algebraic resp. differential Thomas decomposition and on the other hand the Vessiot theory of differential equations.

- A Logic Based Approach to Finding Real Singularities of Implicit Ordinary Differential Equations with Thomas Sturm and Werner M. Seiler.

Mathematics in Computer Science 15, 333-352 (2021), https://doi.org/10.1007/s11786-020-00485-x .

We discuss the effective computation of geometric singularities of implicit ordinary differential equations over the real numbers using methods from logic. Via the Vessiot theory of differential equations, geometric singularities can be characterised as points where the behaviour of a certain linear system of equations changes. These points can be discovered using a specifically adapted parametric generalisation of Gaussian elimination combined with heuristic simplification techniques and real quantifier elimination methods. We demonstrate the relevance and applicability of our approach with computational experiments using a prototypical implementation in Reduce.

- No Chaos in Dixon's System with Werner M. Seiler.

International Journal of Bifurcation and Chaos, Vol. 31, No. 03, 2150044 (2021), https://doi.org/10.1142/S0218127421500449.

The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behaviour, if its two parameters take their value in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon's system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into sixteen different regions in each of which the system exhibits qualitatively the same behaviour. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist which can easily create in numerical computations the impression of chaotic behaviour.

- Singular Initial Value Problems for Scalar Quasi-Linear Ordinary Differential Equations with Werner M. Seiler.

Journal of Differential Equations, Volume 281, 25 April 2021, Pages 258-288, https://doi.org/10.1016/j.jde.2021.02.010.

We discuss existence, non-uniqueness and regularity of one- and two-sided solutions of initial value problems for scalar quasi-linear ordinary differential equations where the initial condition corresponds to an impasse point of the equation. With a differential geometric approach, we reduce the problem to questions in dynamical systems theory. As an application, we discuss in detail second-order equations of the form g(x)u''=f(x,u,u') with an initial condition imposed at a simple zero of g. This generalises results by Liang and also makes them more transparent via our geometric approach.

- On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations with Elishan Braun and Werner M. Seiler.

Mathematics in Computer Science 14, 281-293 (2020), https://doi.org/10.1007/s11786-019-00423-6.

We discuss how the geometric theory of differential equations can be used for the numerical integration and visualisation of implicit ordinary differential equations, in particular around singularities of the equation. The Vessiot theory automatically transforms an implicit differential equation into a vector field distribution on a manifold and thus reduces its analysis to standard problems in dynamical systems theory like the integration of a vector field and the determination of invariant manifolds. For the visualisation of low-dimensional situations we adapt the streamlines algorithm of Jobard and Lefer to 2.5 and 3 dimensions. A concrete implementation in Matlab is discussed and some concrete examples are presented.

- Root Parametrized Differential Equations for the Classical Groups.

Let C⟨t⟩ be the differential field generated by l differential indeterminates t=(t_{1},…,t_{l}) over an algebraically closed field C of characteristic zero. We develop a lower bound criterion for the differential Galois group G(C) of a matrix parameter differential equation ∂(y)=A(t)y over C⟨t⟩ and we prove that every connected linear algebraic group is the Galois group of a linear parameter differential equation over C⟨t_{1}⟩. As a second application we compute explicit and nice linear parameter differential equations over C⟨t⟩ for the groups SL_{l+1}(C), SP_{2l}(C), SO_{2l+1}(C), SO_{2l}(C), i.e. for the classical groups of type A_{l}, B_{l}, C_{l}, D_{l}, and for G_{2}(here l=2).

- A Root Parametrized Differential Equation for the Special Linear Group.

Let C⟨t⟩ be the differential field generated by l differential indeterminates t=(t_{1},…,t_{l}) over an algebraically closed field C of characteristic zero. In this article we present an explicit linear parameter differential equation over C⟨t⟩ with differential Galois group SL_{l+1}(C) and show that it is a generic equation in the following sense: If F is an algebraically closed differential field with constants C and E/F is a Picard-Vessiot extension with differential Galois group H(C) ⊆ SL_{l+1}(C), then a specialization of our equation defines a Picard-Vessiot extension differentially isomorphic to E/F.

- Root Parametrized Differential Equations, Dissertation, Universität Kassel (better see https://arxiv.org/abs/1609.05535).