Associate Professor Gunter Franz Steinke
Mathematics and Statistics
Phone: +64 3 364 2987 ext. 92426
Office: Erskine 603
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 cutandpaste method for the construction of topological geometries beginning with the first examples of 2dimensional projective planes whose collineation groups consist only of the identity. In the area of Laguerre planes I discovered the first nonclassical models of 4dimensional Laguerre planes. This brought to an end a 20year long search and opened the door for further investigations of such planes and related geometries, such as 6dimensional generalized quadrangles. I further introduced elation Laguerre planes, a particularly nice and wellbehaved 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 4dimensional Minkowski planes which significantly reduced the verification of the various defining topological properties of 4dimensional Minkowski planes and I constructed the first examples of 4dimensional nonclassical Minkowski planes thus concluding a 30year long search for such planes.
Subject Area: Disciplines
Future Research

Investigate finite elation Laguerre planes (long term)

Classify 2dimensional Minkowski planes with respect to KleinKroll types and provide models for each type and characterisations for some types (short to mid term)

Classify 3dimensional compact generalized quadrangles whose automorphism groups are at least 6dimensionall (long term)

Classify all 2dimensional Minkowski planes whose automorphism groups are 3dimensional (long term)

Investigate new flat topological geometries (mid term)

Investigate 6dimensional compact generalized quadrangles (long term)

Classify 2dimensional Laguerre planes with respect to Kleinewillinghöfer types and provide models for each type and characterisations for some types (short to mid term)

Classify all 2dimensional Laguerre planes whose automorphism groups are 4dimensional (long term)

Further develop the theory of 4dimensional 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. and Stroppel MJ. (2017) On elation Laguerre planes with a twotransitive orbit on the set of generators. Stuttgarter Mathematische Berichte 2017009: 18.

Steinke GF. (2016) On kleinewillingh¨ofer types of finite laguerre planes with respect to homotheties. Australasian Journal of Combinatorics 66(3): 425435.

Steinke GF. and Stroppel MJ. (2015) Simple groups acting twotransitively on the set of generators of a finite elation Laguerre plane. Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 56(1): 285298. http://dx.doi.org/10.1007/s133660130169z.

Loewen R. and Steinke GF. (2014) The circle space of a spherical circle plane. Bulletin of the Belgian Mathematical Society  Simon Stevin 21(2): 351364.

Steinke GF. and Stroppel MJ. (2013) Finite elation Laguerre planes admitting a twotransitive group on their set of generators. Innovations in Incidence Geometry 13: 207223. http://www.iig.ugent.be/online/volume13.php.
Conference Contributions  Other

Steinke GF. (2018) Projective planes, Laguerre planes and generalized quadrangles that admit large groups of automorphisms. University of Auckland: Groups and Geometry 2018, 2326 Jan 2018

Steinke GF. (2017) Almost 2transitive finite elation Laguerre planes. Monash University, Melbourne, Australia: 5th International Combinatorics Conference (5ICC), 49 Dec 2017

Steinke GF. (2016) Central automorphisms of finite Laguerre planes. Queenstown, New Zealand: Symmetries and Covers of Discrete Objects (SCDO), 1419 Feb 2016

Steinke GF. (2016) The search for 2dimensional Laguerre planes of Kleinewillinghofer type II.A.2. Victoria University of Wellington: NZMS Colloquium 2016, 58 Dec 2016

Steinke G. (2015) Collineations of finite 2affine planes. University of Queensland, Brisbane, Australia: 39th Australasian Conference on Combinatorial Mathematics and Combinatorial Computing (39ACCMCC), 711 Dec 2015

Steinke GF. (2015) Slice and match  how to modify 2dimensional geometries. Christchurch, New Zealand: New Zealand Mathematics Colloquium 2015, 13 Dec 2015

Steinke GF. (2014) Finite Minkowski planes of Klein type 20. Wellington, New Zealand: 38th Australasian Conference on Combinatorial Mathematics and Combinatorial Computing, 15 Dec 2014

Steinke G. (2013) The topology of 3interpolating systems on the 2sphere. Tauranga, New Zealand: New Zealand Mathematical Society Colloquium (NZMS), 35 Dec 2013

Steinke G. and Stroppel M. (2012) 2transitive finite elation Laguerre planes are miquelian. Palmerston North, New Zealand: New Zealand Mathematical Society Colloquium (NZMS), 46 Dec 2012

Steinke GF. (2012) 2transitivity finite circle geometries. Queenstown, New Zealand: Symmetries of discrete objects (SODO2012), 1317 Feb 2012