Lukas Kühne
Photo: Universität Bielefeld
About
Contact
Research
My research lies at the crossroads of combinatorics, algebra and geometry.
My current interests include hyperplane arrangements, matroids and polytopes.
I enjoy using and developing experimental and computational tools as part of my research.
Writing
Preprints

Dynamical systems on some elliptic modular surfaces via operators on line arrangements
with Xavier Roulleau.

Line arrangements with many triple points
with Tomasz Szemberg and Halszka TutajGasińska.

Regular polygons, line operators, and elliptic modular surfaces as realization spaces of matroids
with Xavier Roulleau.

Matroids in OSCAR
with Daniel Corey and Benjamin Schröter.

Projective dimension of weakly chordal graphic arrangements
with Takuro Abe, Paul Mücksch, and Leonie Mühlherr.

On entropic and almost multilinear representability of matroids
with Geva Yashfe.
Published or accepted for publication

Faces of Cosmological Polytopes
with Leonid Monin.
To appear in Annales de l’Institut Henri Poincaré D.

On arrangements of hyperplanes from connected subgraphs
with Michael Cuntz.
To appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze.

Computing Characteristic Polynomials of Hyperplane Arrangements with Symmetries
with Taylor Brysiewicz and Holger Eble.
Discrete & Computational Geometry. (arXiv)

Computing the nonfree locus of the moduli space of arrangements and Terao's freeness conjecture
with Mohamed Barakat.
Mathematics of Computation, 92 (2023), pp. 14311452. (arXiv)
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.
Algebraic Combinatorics, 6(3), (2023), pp. 623638.
(arXiv)
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.
Advances in Mathematics, 414 (2023), Paper No. 108863.
(arXiv)

Von Staudt Constructions for SkewLinear and Multilinear Matroids
with Rudi Pendavingh and Geva Yashfe.
Combinatorial Theory, 3(1) (2023), Paper No. 16. (arXiv)

The Universality of the Resonance Arrangement and its Betti Numbers.
Combinatorica (2023). (arXiv)
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, 252, (2022), pp. 95–147. (arXiv)
An extended abstract appeared in the proceedings of the conference Formal Power Series and Algebraic Combinatorics (FPSAC 2020).

Locally heavy hyperplanes in multiarrangements
with Takuro Abe.
Journal of Pure and Applied Algebra, 226(1), (2022) Paper No. 106791. (arXiv)

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 Journal of Discrete Mathematics, 35(2), (2021) pp. 12011223. (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.
Teaching
At Bielefeld University:
 Winter term 2023:
 Lecture course Linear Algebra 2
 Summer terum 2023:
 Lecture course Linear Algebra 1
 Winter term 2022:
 Lecture course on discrete mathematics for computer science students
 Proseminar on graph theory
 Summer term 2022:
 Proseminar on matroid theory
I gave summer courses for 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).
PhD students
 Sebastian Degen, 2023  present, joint with KaiUwe Schmidt, funded by DFG
 Leonie Mühlherr, 2022  present
Selected grants
Events (co)organized
 MFO Miniworkshop Alcoved Polytopes in Physics and Optimization, Oberwolfach, 27 March 2025, with N. Early, L. Monin and J. Yu
 MaRDI Barcamp on Research Data Management in Mathematics, ZiF in Bielefeld, 4 July 2023, with C. Köhler and T. Krause
 Summer school for high school students, Bielefeld, 1214 June 2023, with C. AlfesNeumann
 Geometry meets Combinatorics in Bielefeld, 59 September 2022, with M. JuhnkeKubitzke, R. Sanyal, and C. Voll
 FOMUS: Foundations of Mathematics: Univalent Foundations and Set Theory, ZiF in Bielefeld, 1823 July 2016, with B. Grabmayr, D. Kant, D. Sarikaya, and M. Viehstädt
Travel
A list of conferences where I have been/expect to be.
 Alcoved Polytopes in Physics and Optimization, March 2025, MFO
 Member during the Special Year on Algebraic and Geometric Combinatorics, September—December 2024, IAS Princeton
 Combinatorics, Resurgence and Algebraic Geometry in Quantum Field Theory, August 2024, MPI Bonn
 Arrangements, Matroids and Logarithmic Vector Fields, June 2024, MFO
 Combinatorial Coworkspace — a session in algebraic and geometric combinatorics, March 2024, Kleinwalsertal
 Hyperplane Arrangements 2023, December 2023, Rikkyo University, Tokyo
 Moduli and Friends, September 2023, IMAR, Bucharest
 Hodge and KThoery meets Combinatorics, September 2023, Bernoulli Center, EPFL
 GMZ 60, May 2023, FU Berlin
 Combinatorics, geometry and commutative algebra of hyperplane arrangements, January 2023, Kyushu University, Fukuoka
 Complex and Symplectic Curve Configurations, December 2022, University of Nantes
 Geometry meets Combinatorics in Bielefeld, September 2022, Bielefeld Univesity
 Arrangements in Ticino, June 2022, Locarno
 Combinatorial Coworkspace — a session in algebraic and geometric combinatorics, March 2022, Kleinwalsertal
 Feynman integrals and nonlinear algebra, March 2022, MPI Leipzig
 ComputeralgebraTagung der Fachgruppe Computeralgebra, March 2022, Munich
 Formal Power Series and Algebraic Combinatorics (FPSAC), January 2022, online
 SIAM Conference on Applied Algebraic Geometry at Texas A&M University, August 2021, online
 Semester Program on Combinatorial Algebraic Geometry, February  May 2021, ICERM, online
 Workshop on Logarithmic Vector Fields and Freeness of Divisors
and Arrangements, January 2021, MFO, online
 Workshop on Discrete Geometry at MFO, September 2020, MFO
 Arrangements at Home, August 2020, online
 Formal Power Series and Algebraic Combinatorics (FPSAC), July 2020, online
 Combinatorial Coworkspace — a session in algebraic and geometric combinatorics, March 2020, Kleinwalsertal
 Algebraic Geometry  Varieties, Polyhedra, Computation, October 2019, FU Berlin
 Hyperplane Arrangements and Reflection Groups, September 2019, Leibniz Universität Hannover
 Canadian Discrete and Algorithmic Mathematics Conference, May 2019, Simon Fraser University, Vancouver
 Arrangements at Western, May 2019, Western University, London (ON)
 Spring School and Workshop on Polytopes, March 2019, Ruhr Universität Bochum
 Combinatorial Geometries 2018 at CIRM, September 2018, CIRM
 Summer School  New Perspectives in Hyperplane Arrangements, September 2018, Ruhr Universität Bochum
 HighDimensional Combinatorics, April 2018, IIAS Jerusalem
 Geometric and Topological Combinatorics, October 2017, MSRI
Short Biography
 since 2021: Juniorprofessor (without tenuretrack) for Combinatorial Algebra, Universität Bielefeld
 20202021: Postdoctoral researcher, MaxPlanckInstitute for Mathematics in the Sciences, Leipzig
 20172020: PhD, Hebrew University of Jerusalem
 20142017: Master, Universität Bonn
 20112014: Bachelor, TU Kaiserslautern
Other