The main topic of my research is the unified and constructive treatment
of systems of algebraic and differential equations (in particular under
and overdetermined systems) based on the central notion
of involution (see my monograph with the same title)
combining algebraic, combinatorial, homological and geometric ideas. I
am interested in both obtaining a deeper theoretical understanding of
this concept and applying it to various fields. On the theoretical side,
I am studying questions in the formal theory of differential equations
like existence and uniqueness of solutions, completion to involution or
geometric singularities and constructive problems in commutative algebra
(in particular, the use of Pommaret bases for the study of polynomial
modules). The applications concern mainly mathematical physics
(geometric mechanics, systems with constraints, gauge theories) and the
analysis of differential algebraic equations with a current emphasis on
systems arising in biochemical reaction networks (in particular,
stability, oscillation and bifurcation questions). More details can be
found in my publications.
The following short survey over some recent works may give an
impression of the main emphasis of our current research:
 We study singularities of general systems of differential
equations combining techniques from differential algebra, algebraic
geometry and differential topology. In the recent preprint
[94]
a
general framework for the algorithmic detection of all singularities of a
given equation over the complex numbers (a real version can be found in
[45]
) has been developed. Before, we analysed in detail the
existence, (non)uniqueness and regularity of solutions of a
class of quasilinear secondorder ordinary differential equations for
initial data prescribed at a singularity in
[93]
. As support for this research, we developed a Matlab package for
the numerical visualisation of some lowdimensional situations
[42]
.
 A key topic in our biological applications is the detection of
oscillations. In an earlier project, we developed algorithms
for effectively proving the appearance of Hopf bifurcations
[33]
. Within
the Symbiont project, we
are generally concerned with the development of symbolic methods for
biological networks. Our focus is here mainly on methods based on
symmetries. A general description of the project was published in
[38]
.
 We are much interested in using Pommaret bases for the
determination and analysis of free resolutions of polynomial
modules. In
[32]
we combined them with discrete
Morse theory to obtain efficient algorithms capable of determining
individual Betti numbers. Some further results in this direction can be
found in
[36,
43,
79]
.
 A general problem in the use of Pommaret bases is the fact
that they only exist in generic coordinates. An indepth study of this
problem can be found in
[35]
, but it was also important
in
[39,
78]
. A recent highlight
[41]
was the use of Pommaret bases for the analysis
of Hilbert and Quot schemes. Again the coordinate dependence
played a crucial role.
 A fairly recent topic for me are complexity questions in
commutative algebra.
[40,
86]
contain some
general results on bounds for Gröbner bases using techniques from
involutive bases. In
[44]
we provided the first
complexity results for involutive bases.

My work has been funded by various grants mainly from Studienstiftung des
Deutschen Volkes (German National Scholarship Foundation), from Deutsche Forschungsgemeinschaft
(DFG) (German Science Foundation) and from the European Commission.
At national level, I was from 2009 to 2016 a member of the steering
group of the Special Priority
Programme Algorithmic and Experimental Methods in Algebra, Geometry
and Number Theory (SPP 1489). Within this programme, I conducted
jointly with my
colleague Andreas Weber (Universität Bonn) the project
Bifurcations and Singularities of Algebraic Differential
Equations (BISADE). Currently, I am participating with my group
in the project Symbolic
Methods for Biological Networks (SYMBIONT) which may be considered
as a successor of BISADE. It is a GermanFrench project with participants
in Aachen, Bonn, Lille, Montpellier, Nancy and Paris (a poster describing
the project is available here). In addition, I have received various grants for
international collaborations with groups in Dubna, Genoa, Plymouth,
Tbilissi and Torino, for the organisation of conferences like CASC and ACA
and for guest scientists from Russia, Iran and Italy.
At European level, I participated in an INTAS project with the title
Involutive Systems of Algebraic and Differential Equations (INTAS
991112) in which groups from Mannheim, Karlsruhe, Greifswald, Bangor,
Catania, Dubna, Moscow and Novosibirsk collaborated between 2000 and 2002.
In 2005/6 I headed the Heidelberg team of a European research project
called Global Integrability of Field Theories (GIFT)
within the NESTAdventure programme of the European Commission. The
other teams were located in Karlsruhe, Grenoble, Toulouse, Amsterdam and
Lancaster. Between 2016 and 2018, my group participated in the European
project (FETCSA) Satisfiability Checking and Symbolic Computation
with the other teams located in Aachen, Bath, Coventry, Genoa, Linz, Nancy,
Oxford and Trento.

The following list contains students who have written or are writing theses
under my supervision.
 Undergraduate Theses (Bachelor, Master, Diploma, etc.)
 Nico Burmeister: Qualitative Analysis of Continuous and
Deterministic Models of Population Dynamics; teacher thesis project,
Institut für Mathematik, Universität Kassel
 Thomas Izgin: The Involutive GVW Algorithm and the Computation of
Pommaret Bases; master thesis, Institut für Mathematik,
Universität Kassel 2020
 Filip Skrentny: The Gröbner Walk;
bachelor thesis, Institut für Mathematik, Universität
Kassel 2019
 Marco Horn: Theory and Application of ReedSolomon Codes;
teacher thesis project, Institut für Mathematik, Universität
Kassel 2018
 Marvin Brandenstein: On Characterstic Chains and their
Applications; bachelor thesis, Institut für Mathematik,
Universität Kassel 2018
 Alice Moallemy: The Computation of the Radical of Polynomial
Ideals; bachelor thesis, Institut für Mathematik,
Universität Kassel 2018
 Markus Fülle: Gröbner Bases in Affine Monoid
Algebras; bachelor thesis, Institut für Mathematik,
Universität Kassel 2018
 Thomas Izgin: The Computation of Gröbner Bases and Syzygies
with the GVW Algorithm; bachelor thesis, Institut für
Mathematik, Universität Kassel 2017
 Julian Körting: Algorithmic Factorisation of Univariate
Polynomials over the Integers and over the Rationals; teacher
thesis, Institut für Mathematik, Universität Kassel 2017
 Elishan Braun: Numerical Analysis and Visualisation of Implicit
Ordinary Differential Equations; master thesis, Institut für
Mathematik, Universität Kassel 2017
 Maxim Urich: Geometric Theory of Singularity Induced
Bifurcations; master thesis, Institut für Mathematik,
Universität Kassel 2016
 Matthias Orth: Constructive Theory of Inverse Systems; master
thesis, Institut für Mathematik, Universität Kassel 2016
 Frederick Mücker: Effective Computations with Module
Homomorphisms in the CoCoALib; diploma thesis, Institut
für Mathematik, Universität Kassel 2015
 Sven Nummer: On the Deterministic Modeling of Epidemies and
Endemies after Kermack and McKendrick; teacher thesis, Institut
für Mathematik, Universität Kassel 2015
 Matthias Orth: Stable Ideals and Genericity; bachelor thesis,
Institut für Mathematik, Universität Kassel 2014
 Julius Rahaus: Involutive Bases in Clifford Algebras; bachelor
thesis, Institut für Mathematik, Universität Kassel 2014
 Pierre Pytlik: Effective Genericity for Polynomial Ideals;
diploma thesis, Institut für Mathematik, Universität Kassel
2014
 Maxim Urich: Geometric Completion of Differential Algebraic
Equations; bachelor thesis, Institut für Mathematik,
Universität Kassel 2013
 Mario Albert: Computing Minimal Free Resolutions of Polynomial Ideals
with Pommaret Bases; master thesis, Institut für
Mathematik, Universität Kassel 2013
 Elishan Braun: Discrete Gradient Methods for Hamiltonian Systems
with Constraints; bachelor thesis, Institut für
Mathematik, Universität Kassel 2013
 Sebastian Schütz: Combinatorics of Hilbert Functions;
diploma thesis, Institut für Mathematik, Universität Kassel
2012
 Mario Albert: Janet Bases in CoCoA; bachelor thesis, Institut
für Mathematik, Universität Kassel 2011
 Mehdi Sahbi: Pommaret Bases and the Computation of the Koszul Homology
in the Monomial Case; diploma thesis, Fakultät für
Informatik, Universität Karlsruhe 2007
 Mehdi Sahbi: The Effective Determination of DeltaRegular Coordinates
for Polynomial Ideals; study thesis, Fakultät für
Informatik, Universität Karlsruhe 2006
 Wolfgang Globke: An ObjectOriented Programming Environment for
Differential Geometric Computations in MuPAD; study thesis,
Fakultät für Informatik, Universität Karlsruhe 2006
 Marcus Hausdorf: GeometricAlgebraic Completion of General Systems of
Differential Equations; diploma thesis, Fakultät für
Informatik, Universität Karlsruhe 2000
 Marcus Hausdorf: A General Symmetry Package for Differential Equations
in MuPAD; study thesis, Fakultät für Informatik,
Universität Karlsruhe 1999
 Pavel Lukowicz: Applications of Computeralgebra on Problems in the
Statistical Physics of Neural Networks; diploma thesis, Fakultät
für Informatik, Universität Karlsruhe 1999
 Christoph Zenger: GröbnerBasen for Differential Forms;
diploma thesis, Fakultät für Informatik, Universität
Karlsruhe 1992
 Joachim Schü: Implementation of the CartanKuranishi Theorem in
Axiom; diploma thesis, Fakultät für Informatik,
Universität Karlsruhe 1992
 Ph.D. Theses
 Matthias Orth: Syzygies over NonCommutative and Quotient
Polynomial Rings thesis project, Universität Kassel
 Maxim Urich: Symmetries of Differential Equations via Vessiot
Theory thesis project, Universität Kassel
 Mario Albert: Involutive Bases, Resolutions and Hilbert Schemes,
Universität Kassel and University of Torino 2017
 Matthias Fetzer: Free Resolutions from Involutive Bases,
Universität Kassel 2016
 Michael Schweinfurter: Deterministic Genericity and the Computation
of Homological Invariants, Universität Kassel 2016
 Hassan Errami: Semialgebraic Algorithms for Symbolic Analysis of
Complex Reaction Networks, Universität Kassel and
Universität Bonn 2013
 Matthias Seiß: Root Parametrised Differential Equations for
Groups of Lie Type, Universität Kassel and Universität
Heidelberg 2011
 Eduardo Saenz de Cabezon: Combinatorial Koszul Homology: Computations
and Applications; Departamento de Matematicas y Computacion,
Universidad de La Rioja, Logrono (Spain) 2008
 Dirk Fesser: On Vessiot's Theory of Partial Differential
Equations; Fachbereich Mathematik, Universität Kassel 2008
In addition, I have been referee or member of the jury for the following
theses:
 H. Schatz: Automatic Computation of Continued Fraction
Representations as Solutions of Explicit Differential Equations,
Ph.D. thesis, Universität Kassel 2020
 M. Scheicher: Topics in Multidimensional Behavioural Algebraic
Systems Theory, habilitation thesis, Universität Innsbruck
2019
 G. Regensburger: Algebraic and algorithmic approaches to analysis:
Integrodifferential equations, positive steady states, and
wavelets, habilitation thesis, Universität Linz 2019
 K. Fischer: Identification of Special Functions, given by
Rodrigues formulas, Ph.D. thesis, Universität Kassel 2016
 V. Levandovskyy: Computer Algebraic Analysis, habilitation
thesis, RWTH Aachen 2015
 A. Lakhal: Elimination in Ore Algebras, Ph.D. thesis,
Universität Kassel 2014
 J. Tautges: Reconstruction of Human Motions Based on LowDimensional
Control Signals, Ph.D. thesis, Universität Bonn 2012
 E.O. AbdelRahman: Algorithmic Contributions to the Qualitative
Analysis of Autonomous Parametric Dynamical Systems, Ph.D. thesis,
Universität Bonn 2011
 E. Nana Chiadjeu: Algorithmic Computation of Formal Fourier
Series, Ph.D. thesis, Universität Kassel 2010
 T. Wichmann: Symbolic Reduction Methods for NonLinear DAE
Systems, Ph.D. thesis, Universität Kaiserslautern 2004
 T. Arponen: Numerical Solution and Structural Analysis of
DifferentialAlgebraic Equations, Ph.D. thesis, Helsinki
University of Technology 2001
