Frank Klinker

Research interests

My research interest is on the various relations between mathematics on the one hand and natural sciences, engeneering and economics on the other hand.

One focus of my work is on a particular relation between differential geometry and physics: geometric supersymmetry. This expression describes the various connections between differential geometry and supersymmetry. Undoubtedly, two important mathematical fields in this context are representation theory and spin geometry.

In this context, I am generally interested in (pseudo)-Riemannian geometry, geometric analysis, combinatorics, geometric algebras, geometry of supermanifolds, deformation of geometric structures, Yang-Mills theory, supergravity, string theory, and deformation quantization.

Moreover, I like to apply mathematics to problems and questions that arise from practical applications; for example electrical engineering, automation engeneering, finance and markets, or economics in general. So feel free to contact me if you are in need of advice for your mathematical problem.

Cooperations with Industry and Business

  • C. Eggert, ThyssenKrupp Rothe Erde GmbH, Dortmund, Germany (Geometric analysis of AC circuits)
  • R. Gäer, Schniewindt GmbH & Co. KG., Neuenrade, Germany (Geometric analysis of AC circuits)
  • W. Kranz, Kranz Software Engineering, Dortmund, Germany (Applied data smoothing)
  • R. Moorbrink, A. Trog, Franz Högemann GmbH, Garrel, Germany (Optimization of production processes)
  • S. Pohlschröder, Pohlschröder Property Consulting, Dortmund, Germany (Calculation and optimization of trend signals)
  • C. Reineke, ehem. Deutsche Telekom AG, Dortmund, Germany (Applied algebraic problems, geometric analysis of AC circuits)
  • Rentokil GmbH, Regionalfiliale Dortmund, Germany (Static calculations in warehouse logistics)



  • On the characterization of the degree of interpolation polynomials in terms of certain combinatorical matrices. Arab. J. Math. (online 2018). Joint work with C. Reineke (Journal or arXiv)
  • On the regularity of matrices with uniform polynomial entries. São Paulo J. Math. Sci. (2018, online 2017). Joint work with C. Reineke (Journal or arXiv)
  • The general non-symmetric, unbalanced star circuit: On the geometrization of problems in electrical measurement. Math. Semesterber. 64 (2017) no. 1, 25-39. Joint work with R. Gäer, Schniewindt GmbH & Co. KG, and C. Eggert, ThyssenKrupp Rothe Erde GmbH (Journal or arXiv)
  • An explicit description of SL(2,C) in terms of SO+(3,1) and vice versa. Int. Electron. J. Geom. 8 (2015) no. 1, 94-104 (Journal/pdf or arXiv)
  • Eleven-dimensional symmetric supergravity backgrounds, their geometric superalgebras, and a common reduction. Bulg. J. Phys. 41 (2014) no. 2, 130-141 (Journal or arXiv)
  • A family of non-restricted D=11 geometric supersymmetries. J. Geom. Phys. 86 (2014) 534-553 (Journal or arXiv)
  • Connections on Cahen-Wallach spaces. Adv. Appl. Clifford Algebr. 24 (2014) no. 3, 737-768 (Journal or arXiv)
  • Ein optimiertes Glättungsverfahren motiviert durch eine technische Fragestellung (An optimized smoothing method motivated by a technical problem). Math. Semesterber. 59 (2012) no. 1, 29-55, German. Joint work with G. Skoruppa, in cooperation with W. Kranz, Kranz Software Engineering (Journal or Preprint)
  • Generalized duality for k-forms. J. Geom. Phys. 61 (2011) no. 12, 2293-2308 (Journal or arXiv)
  • Exponential Moving Average versus Moving Exponential Average. Math. Semesterber. 58 (2011) no. 1, 97-107. In cooperation with S. Pohlschröder (Journal or Preprint)
  • Polynomial poly-vector fields. Int. Electron. J. Geom. 2 (2009) no. 1, 55-73 (Journal/pdf or arXiv)
  • SUSY structures on deformed supermanifolds. Differential Geom. Appl. 26 (2008) no. 5, 566-582 (Journal or arXiv)
  • The decomposition of the spinor bundle of Grassmann manifolds. J. Math. Phys. 48 (2007) no. 11, 113511, 26 pp (Journal or arXiv)
  • The torsion of spinor connections and related structures. SIGMA, Symmetry Integrability Geom. Methods Appl. 2 (2006) Paper 077, 28 pp (Journal or arXiv)
  • Supersymmetric Killing structures. Comm. Math. Phys., 255 (2005) no. 2, 419-467 (Journal or Preprint)

Preprints / Non-refereed Papers

  • Die Keplersche Fassregel und Numerische Quadraturverfahren (The Keplersche Fassregel and numerical quadrature). 2010, 28 pp (pdf, German); A preprint of sections 1+2 has been published in Mathematik-Seminar des Freistaates Sachsen, Heft Sayda, Leipzig, 2001.
  • Quadratic Poisson structures in dimension four. Appendix to "Polynomial poly-vector fields". 2008, 22 pp. (pdf)
  • The spinor bundle of Riemannian products. arXiv:math.DG/0212058, 2002, 7 pp (arXiv)


  • Supersymmetric extensions of solvable Lorentzian symmetric spaces. Habilitation thesis, TU Dortmund University, 2013, 112 pp (Abstract/Summary).
  • Supersymmetric Killing structures. Dissertation, Leipzig University, 2003, 116 pp (DNB entry)
  • Zusammenhänge auf Prinzipalbündeln und Instantonen auf S⁴ (Connections on Principal Bundles and Instantons on S⁴). Diploma thesis, University of Osnabrück, 2000, 104 pp (pdf, German)

In Preparation / Projects

  • Ist das Glas halbvoll oder halbleer: Eine analytische Betrachtung. (Joint work with C. Reineke)
  • On the classification of homogeneous supergravity backgrounds.
  • Properties of small eigenvalues of the Dirac operator on quaternionic Kähler manifolds and their eigenspaces.
  • The geometric treatment of supersymmetry on homogeneous supermanifolds.

Lecture Notes

  • Grundlagen der Kartographie I: Geometrische Abbildungen (Elementary Cartography: Geometric Maps). Lecture notes, 2017, 25 pp, (pdf, German)
  • Grundlagen der Analysis: Ein dreisemestriger Kurs (Elementary Analysis: A Three-Term Course). Lecture notes, 2016/2018, 334 pp, (pdf, German)
  • Differentialgeometrie I: Kurven und Flächen (Differential Geometry I: Curves and Surfaces). Lecture notes, 2014/2016, 130 pp. (pdf, German)
  • Mathematischer Vorkurs (Preliminary course in mathematics). Lecture notes, 2012, 105 pp. (pdf, German)
  • Lineare Algebra auf Räumen mit anti-kommutierenden Koeffizienten (Linear algebra on spaces with anti-commuting coefficients). 2001, 29 pp. (pdf, German)

Short Notes

  • Spurfreie Matrizen als Kommutatoren (Trace free matrices as commutators). 2011, 9 pp. (pdf, German)
  • Die Gaußsche Gerade (The Gauß line). 2011, 4 pp. (pdf, German)
  • Ein effizientes Verfahren zur Berechnung einer Basis von Summe und Schnitt zweier Vektorräume (An efficient way to calculate a basis of the sum and intersection of two vector spaces). 2008, 4 pp. (pdf, German)
  • The Kontsevitch quantization formula. Talk given at a joint seminar of the groups Differential Geometry and Theoretical Physics III, TU Dortmund, 2004, 14 pp. (pdf)
  • The signs. Supplement to my Dissertation, 2003, 4 pp. (pdf)

Organized Conferences

March 16-19, 2015. TU Dortmund University: International Conference Geometric and Algebraic Methods in Mathematical Physics (GAMMP 2015)