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 and bifurcation questions). More details can be
found in my publications.

My work has been funded by various grants mainly from Studienstiftung des
Deutschen Volkes (German National Scholarship Foundation) and from Deutsche Forschungsgemeinschaft
(DFG) (German Science Foundation). In addition, 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. Currently, I am a member of the steering group of the Special
Priority Programme Algorithmic and Experimental Methods in Algebra, Geometry
and Number Theory (SPP 1489) which was invoked by DFG in 2009.
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 in a
European project
(FETCSA) Satisfiability Checking and Symbolic Computation
with the other teams located in Aachen, Bath, Coventry, Genoa, Linz, Nancy,
Oxford, Trento.

The following list contains students who have written or are writing theses
under my supervision.
 Undergraduate Theses (Bachelor, Master, Diploma, etc.)
 Thomas Izgin: The Computation of Gröbner Bases and Syzygies
with the GVW Algorithm; bachelor thesis project, Institut für
Mathematik, Universität Kassel
 Julian Körting: Algorithmic Factorisation of Univariate
Polynomials over the Integers and over the Rationals; teacher
thesis project, Institut für Mathematik, Universität
Kassel
 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:
 Maxim Urich:
 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:
 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
