Homepage of Lukas Kühne
Foto: Universität Bielefeld
Contact Information
Research Interests
My research lies at the crossroads of algebra, combinatorics, algebraic geometry, and algorithms.
My current interests include hyperplane arrangements, matroids and polytopes.
I enjoy using and developing computer algebra tools as part of my research.
Writing
Preprints

Faces of Cosmological Polytopes, with Leonid Monin.

On arrangements of hyperplanes from connected subgraphs, with Michael Cuntz.

On entropic and almost multilinear representability of matroids, with Geva Yashfe.

Enumerating chambers of hyperplane arrangements with symmetry, with Taylor Brysiewicz and Holger Eble.
Published

Computing the nonfree locus of the moduli space of arrangements and Terao's freeness conjecture, with Mohamed Barakat.
To appear in Mathematics of Computation.
An extended abstract appeared in the Oberwolfach workshop report 5/2021 and in the Computeralgebra Rundbrief Ausgabe 68.

On the geometry of flag HilbertPoincaré series for matroids, with Joshua Maglione.
To appear in Algebraic Combinatorics.
An extended abstract appeared in the proceedings of the conference Formal Power Series and Algebraic Combinatorics (FPSAC 2022).

Likelihood Degenerations, with Daniele Agostini, Taylor Brysiewicz, Claudia Fevola, Bernd Sturmfels, and Simon Telen.
To appear in Advances in Mathematics.

Von Staudt Constructions for SkewLinear and Multilinear Matroids, with Rudi Pendavingh and Geva Yashfe.
To appear in Combinatorial Theory.

The Universality of the Resonance Arrangement and its Betti Numbers.
To appear in Combinatorica.
An extended abstract appeared in the proceedings of the conference Formal Power Series and Algebraic Combinatorics (FPSAC 2021).

Representability of Matroids by cArrangements is Undecidable, with Geva Yashfe.
Israel Journal of Mathematics (2022). (arXiv)
An extended abstract appeared in the proceedings of the conference Formal Power Series and Algebraic Combinatorics (FPSAC 2020).

On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture, with Mohamed Barakat, Reimer Behrends, Christopher Jefferson, and Martin Leuner.
SIAM J. Discrete Math., 35(2), (2021) 12011223. (arXiv)

Locally heavy hyperplanes in multiarrangements, with Takuro Abe.
Journal of Pure and Applied Algebra 226 (2022). (arXiv)

Heavy hyperplanes in multiarrangements and their freeness, with Takuro Abe.
Journal of Algebraic Combinatorics 48 (2018), pp. 581–606. (arXiv)
Software
Jointly with Benjamin Schröter, we are currently developing an Oscar module for matroids.
Taylor Brysiewicz, Holger Eble, and I wrote the julia package CountingChambers.jl.
This package counts the chambers of a hyperplane arrangement by computing its characteristic polynomial.
It performs this computation via a modified deletionrestriction algorithm which takes advantage of the arrangement's symmetries.
In joint work with Mohamed Barakat, I wrote the GAP package MatroidGeneration to create an ArangoDB database of all rank 3 matroids up to size 14 whose characteristic polynomial factors over the integer.
These matroids are among others relevant in the study of free or supersolvable hyperplane arrangements.
Feel free to try it and contact us for questions, suggestions or feedback.
Previously, I worked at the Research Insitute for Discrete Mathematics in Bonn to develop and apply tools from combinatorial optimization in a a chip design project in cooperation with IBM.
Travel
A list of conferences where I have been/expect to be.
 Combinatorics, geometry and commutative algebra of hyperplane arrangements, January 2023, Fukuoka, Japan
 Complex and Symplectic Curve Configurations, December 2022, Nantes, France
 Geometry meets Combinatorics in Bielefeld, September 2022, Bielefeld, Germany
 Arrangements in Ticino, June 2022, Locarno, Switzerland
 Combinatorial Coworkspace — a session in algebraic and geometric combinatorics, March 2022, Kleinwalsertal, Austria
 Feynman integrals and nonlinear algebra, March 2022, MPI Leipzig, Germany
 ComputeralgebraTagung der Fachgruppe Computeralgebra, March 2022, Munich, Germany
 Formal Power Series and Algebraic Combinatorics (FPSAC), January 2022, online attendance
 SIAM Conference on Applied Algebraic Geometry at Texas A&M University, August 2021, online attendance
 Semester Program on Combinatorial Algebraic Geometry at ICERM, February  May 2021, Brown University, USA, online attendance
 Workshop on Logarithmic Vector Fields and Freeness of Divisors
and Arrangements at MFO, January 2021, Oberwohlfach, Germany, online attendance
 Workshop on Discrete Geometry at MFO, September 2020, Oberwohlfach, Germany
 Arrangements at Home, August 2020, online attendance
 Formal Power Series and Algebraic Combinatorics (FPSAC), July 2020, online attendance
 Combinatorial Coworkspace — a session in algebraic and geometric combinatorics, March 2020, Kleinwalsertal, Austria
 Algebraic Geometry  Varieties, Polyhedra, Computation, October 2019, FU Berlin, Germany
 Hyperplane Arrangements and Reflection Groups, September 2019, Leibniz Universität Hannover, Germany
 Canadian Discrete and Algorithmic Mathematics Conference, May 2019, Simon Fraser University, Vancouver, Canada
 Arrangements at Western, May 2019, Western University, London (ON), Canada
 Spring School and Workshop on Polytopes, March 2019, Ruhr Universität Bochum, Germany
 Combinatorial Geometries 2018 at CIRM, September 2018, Luminy, France
 Summer School  New Perspectives in Hyperplane Arrangements, September 2018, Ruhr Universität Bochum, Germany
 HighDimensional Combinatorics, April 2018, IIAS Jerusalem, Israel
 Einstein Workshop Discrete Geometry and Topology, March 2018, FU Berlin, Germany
 Geometric and Topological Combinatorics, October 2017, MSRI, USA
Teaching
At Bielefeld University:
 WiSe 2022:
 Lecture course on discrete mathematics for computer science students
 Proseminar on graph theory
 SoSe 2022:
 Proseminar on matroid theory
Since 2022 I advise Leonie Mühlherr as a PhD student.
I gave summer courses for gifted high school students at the German Pupils Academy (Deutsche SchülerAkademie) on linear optimization (2016 and 2018), topological data analysis (2019), and Gödel's incompleteness theorems (2022).
In 2014 and 2015, I held tutorials for the lectures Linear Algebra I and Analysis II in Bonn.
From 2012 to 2014, I gave tutorials in Kaiserslautern for the lectures Fundamentals of Mathematics I and II, Algebraic Structures and Elementary Number Theory.
Other