UC SPARK - University of Canterbury - New Zealand

Associate Professor Gunter Franz Steinke

Mathematics and Statistics

Fields of Research

  • Automorphism groups of geometries
  • Circle geometries and generalized quadrangles
  • Finite Laguerre planes and related geometries
  • Ovals in finite projective planes
  • Topological geometry

Researcher Summary

My research interests are in topological and finite geometry and their connections to groups. I am the leading expert in the theory of topological circle planes and for Laguerre and Minkowski planes. I pioneered the cut-and-paste method for the construction of topological geometries beginning with the first examples of 2-dimensional projective planes whose collineation groups consist only of the identity. In the area of Laguerre planes I discovered the first non-classical models of 4-dimensional Laguerre planes. This brought to an end a 20-year long search and opened the door for further investigations of such planes and related geometries, such as 6-dimensional generalized quadrangles. I further introduced elation Laguerre planes, a particularly nice and well-behaved class of Laguerre planes that have the potential to play a role in the theory of Lagurre planes that is analogous to the one of translation planes in the theory of projective planes. In the area of Minkowski planes I introduced a standard representation of 4-dimensional Minkowski planes which significantly reduced the verification of the various defining topological properties of 4-dimensional Minkowski planes and I constructed the first examples of 4-dimensional non-classical Minkowski planes thus concluding a 30-year long search for such planes.

Subject Area: Disciplines

Future Research

  • Investigate finite elation Laguerre planes (long term)
  • Classify 2-dimensional Minkowski planes with respect to Klein-Kroll types and provide models for each type and characterisations for some types (short to mid term)
  • Classify 3-dimensional compact generalized quadrangles whose automorphism groups are at least 6-dimensionall (long term)
  • Classify all 2-dimensional Minkowski planes whose automorphism groups are 3-dimensional (long term)
  • Investigate new flat topological geometries (mid term)
  • Investigate 6-dimensional compact generalized quadrangles (long term)
  • Classify 2-dimensional Laguerre planes with respect to Kleinewillinghöfer types and provide models for each type and characterisations for some types (short to mid term)
  • Classify all 2-dimensional Laguerre planes whose automorphism groups are 4-dimensional (long term)
  • Further develop the theory of 4-dimensional Laguerre and Minkowski planes (long term)

Key Methodologies

  • Computer searches for finite geometries
  • Lie transformation groups
  • Orbit structure of group actions
  • Representation theory of Lie and finite groups
  • Topology of manifolds

Affiliations

Research/Scholarly/Creative Works

(Displaying research/scholarly/creative work for last six years)
Journal Articles
  • Steinke GF. (2016) On kleinewillingh¨ofer types of finite laguerre planes with respect to homotheties. Australasian Journal of Combinatorics 66(3): 425-435.
  • Steinke GF. and Stroppel MJ. (2015) Simple groups acting two-transitively on the set of generators of a finite elation Laguerre plane. Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 56(1): 285-298. http://dx.doi.org/10.1007/s13366-013-0169-z.
  • Loewen R. and Steinke GF. (2014) The circle space of a spherical circle plane. Bulletin of the Belgian Mathematical Society - Simon Stevin 21(2): 351-364.
  • Steinke GF. and Stroppel MJ. (2013) Finite elation Laguerre planes admitting a two-transitive group on their set of generators. Innovations in Incidence Geometry 13: 207-223. http://www.iig.ugent.be/online/volume-13.php.
Conference Contributions - Other
  • Steinke GF. (2016) Central automorphisms of finite Laguerre planes. Queenstown, New Zealand: Symmetries and Covers of Discrete Objects (SCDO), 14-19 Feb 2016
  • Steinke GF. (2016) The search for 2-dimensional Laguerre planes of Kleinewillinghofer type II.A.2. Victoria University of Wellington: NZMS Colloquium 2016, 5-8 Dec 2016
  • Steinke G. (2015) Collineations of finite 2-affine planes. University of Queensland, Brisbane, Australia: 39th Australasian Conference on Combinatorial Mathematics and Combinatorial Computing (39ACCMCC), 7-11 Dec 2015
  • Steinke GF. (2015) Slice and match - how to modify 2-dimensional geometries. Christchurch, New Zealand: New Zealand Mathematics Colloquium 2015, 1-3 Dec 2015
  • Steinke GF. (2014) Finite Minkowski planes of Klein type 20. Wellington, New Zealand: 38th Australasian Conference on Combinatorial Mathematics and Combinatorial Computing, 1-5 Dec 2014
  • Steinke G. (2013) The topology of 3-interpolating systems on the 2-sphere. Tauranga, New Zealand: New Zealand Mathematical Society Colloquium (NZMS), 3-5 Dec 2013
  • Steinke G. and Stroppel M. (2012) 2-transitive finite elation Laguerre planes are miquelian. Palmerston North, New Zealand: New Zealand Mathematical Society Colloquium (NZMS), 4-6 Dec 2012
  • Steinke GF. (2012) 2-transitivity finite circle geometries. Queenstown, New Zealand: Symmetries of discrete objects (SODO2012), 13-17 Feb 2012
  • Steinke G. (2011) Homotheties in Minkowski planes. Monash University, Melbourne, Australia: 35th Australasian Conference on Combinatorial Mathematics and Combinatorial Computing (35ACCMCC), 5-9 Dec 2011