Lukas Kühne

Photo: Universität Bielefeld

About

I am an assistant professor (Juniorprofessor) at Bielefeld University.

Contact

Email: lukas[insert a dot here]kuehne@math.uni-bielefeld.de
Office: V4 - 201
Telephone: +49 521 106 5028

Fakultät für Mathematik
Universität Bielefeld
Postfach 100131
33501 Bielefeld, Germany

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

  1. Dynamical systems on some elliptic modular surfaces via operators on line arrangements
    with Xavier Roulleau.
  2. Line arrangements with many triple points
    with Tomasz Szemberg and Halszka Tutaj-Gasińska.
  3. Regular polygons, line operators, and elliptic modular surfaces as realization spaces of matroids
    with Xavier Roulleau.
  4. Matroids in OSCAR
    with Daniel Corey and Benjamin Schröter.
  5. Projective dimension of weakly chordal graphic arrangements
    with Takuro Abe, Paul Mücksch, and Leonie Mühlherr.
  6. On entropic and almost multilinear representability of matroids
    with Geva Yashfe.

Published or accepted for publication

  1. Faces of Cosmological Polytopes
    with Leonid Monin.
    To appear in Annales de l’Institut Henri Poincaré D.
  2. On arrangements of hyperplanes from connected subgraphs
    with Michael Cuntz.
    To appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze.
  3. Computing Characteristic Polynomials of Hyperplane Arrangements with Symmetries
    with Taylor Brysiewicz and Holger Eble.
    Discrete & Computational Geometry. (arXiv)
  4. Computing the nonfree locus of the moduli space of arrangements and Terao's freeness conjecture
    with Mohamed Barakat.
    Mathematics of Computation, 92 (2023), pp. 1431-1452. (arXiv)
    An extended abstract appeared in the Oberwolfach workshop report 5/2021 and in the Computeralgebra Rundbrief Ausgabe 68.
  5. On the geometry of flag Hilbert-Poincaré series for matroids
    with Joshua Maglione.
    Algebraic Combinatorics, 6(3), (2023), pp. 623-638. (arXiv)
    An extended abstract appeared in the proceedings of the conference Formal Power Series and Algebraic Combinatorics (FPSAC 2022).
  6. Likelihood Degenerations
    with Daniele Agostini, Taylor Brysiewicz, Claudia Fevola, Bernd Sturmfels, and Simon Telen.
    Advances in Mathematics, 414 (2023), Paper No. 108863. (arXiv)
  7. Von Staudt Constructions for Skew-Linear and Multilinear Matroids
    with Rudi Pendavingh and Geva Yashfe.
    Combinatorial Theory, 3(1) (2023), Paper No. 16. (arXiv)
  8. 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).
  9. Representability of Matroids by c-Arrangements 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).
  10. Locally heavy hyperplanes in multiarrangements
    with Takuro Abe.
    Journal of Pure and Applied Algebra, 226(1), (2022) Paper No. 106791. (arXiv)
  11. 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. 1201-1223. (arXiv)
  12. 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 deletion-restriction 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:

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

Selected grants

Events (co-)organized

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 K-Thoery 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
  • Computeralgebra-Tagung 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
  • High-Dimensional Combinatorics, April 2018, IIAS Jerusalem
  • Geometric and Topological Combinatorics, October 2017, MSRI

Short Biography

  • since 2021: Juniorprofessor (without tenure-track) for Combinatorial Algebra, Universität Bielefeld
  • 2020-2021: Postdoctoral researcher, Max-Planck-Institute for Mathematics in the Sciences, Leipzig
  • 2017-2020: PhD, Hebrew University of Jerusalem
  • 2014-2017: Master, Universität Bonn
  • 2011-2014: Bachelor, TU Kaiserslautern

Other

Miscellaneous programs I participated in: International Summer Science Institute, Deutsche Schülerakademie, Bundeswettbewerb Mathematik and Früheinstieg ins Mathematikstudium.