Research Interests

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 bio-chemical 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 second-order 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 low-dimensional 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 in-depth 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.
Research Funding

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 German-French 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 99-1112) 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 NEST-Adventure 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 (FET-CSA) Satisfiability Checking and Symbolic Computation with the other teams located in Aachen, Bath, Coventry, Genoa, Linz, Nancy, Oxford and Trento.

Students

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 Reed-Solomon 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 Delta-Regular Coordinates for Polynomial Ideals; study thesis, Fakultät für Informatik, Universität Karlsruhe 2006
    • Wolfgang Globke: An Object-Oriented Programming Environment for Differential Geometric Computations in MuPAD; study thesis, Fakultät für Informatik, Universität Karlsruhe 2006
    • Marcus Hausdorf: Geometric-Algebraic 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: Gröbner-Basen for Differential Forms; diploma thesis, Fakultät für Informatik, Universität Karlsruhe 1992
    • Joachim Schü: Implementation of the Cartan-Kuranishi Theorem in Axiom; diploma thesis, Fakultät für Informatik, Universität Karlsruhe 1992
  • Ph.D. Theses
    • Matthias Orth: Syzygies over Non-Commutative 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: Semi-algebraic 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: Integro-differential 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 Low-Dimensional Control Signals, Ph.D. thesis, Universität Bonn 2012
  • E.O. Abdel-Rahman: 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 Non-Linear DAE Systems, Ph.D. thesis, Universität Kaiserslautern 2004
  • T. Arponen: Numerical Solution and Structural Analysis of Differential-Algebraic Equations, Ph.D. thesis, Helsinki University of Technology 2001
Academic Heritage

With the help of The Mathematics Genealogy Project, it has become quite easy to investigate one's own academic heritage. I was quite surprised to detect in my heritage besides three Nobel laureate (two in physics, one in medicine) the reformator Philipp Melanchthon. Large parts of my heritage can be seen in this PDF file.

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