Research

Publications

• D. Corey, M. Joswig, J. Schanz, M. Wack, M. Weber, Quantum automorphisms of matroids, Journal of Algebra vol. 667, 2025. Link

  Motivated by the vast literature of quantum automorphism groups of graphs, we define and study quantum automorphism groups of matroids. A key feature of quantum groups is that there are many quantizations of a classical group, and this phenomenon manifests in the cryptomorphic characterizations of matroids. Our primary goals are to understand, using theoretical and computational techniques, the relationship between these quantum groups and to find when these quantum groups exhibit quantum symmetry. Finally, we prove a matroidal analog of Lovász's theorem characterizing graph isomorphisms in terms of homomorphism counts.

• L. Schmitz, M. Wack, Finite Gröbner bases for quantum symmetric groups, Journal of Symbolic Computation vol. 136, 2026. Link

  Non-commutative Gröbner bases of two-sided ideals are not necessarily finite. Motivated by this, we provide a closed-form description of a finite and reduced Gröbner bases for the two-sided ideal used in the construction of Wangs quantum symmetric group. In particular, this proves that the word problem for quantum symmetric groups is decidable.

• D. Corey, S. Schmidt, M. Wack, Matroid isomorphism games, (preprint), 2025. Link

  We define and study a collection of matroid isomorphism games corresponding to various axiomatic characterizations of matroids. These are nonlocal games played between two cooperative players. Each game is played on two matroids, and the matroids are isomorphic if and only if the game has a perfect classical winning strategy. We define notions of quantum isomorphism in terms of perfect quantum commuting strategies, and we find a pair of nonisomorphic matroids that are quantum isomorphic. We also give a purely algebraic characterization of quantum isomorphic matroids. Finally, we use this notion of quantum isomorphism to describe a new type of quantum automorphism group of a matroid and derive a sufficient condition for a matroid to have nonclassical quantum automorphism.

• Z. Geiselmann, M. Joswig, L. Kastner, K. Mundinger, S. Pokutta, C. Spiegel, M. Wack, M. Zimmer, 121 Patchworked Curves of Degree Seven, (preprint), 2026. Link

  The 121 real schemes, i.e., ambient isotopy classes, of smooth real plane algebraic curves of degree seven were classified by Viro (1984). By constructing one patchwork of the dilated triangle for each real scheme, we provide an explicit method for constructing polynomials realizing each real scheme. In particular, every real scheme of degree seven can be realized as a T-curve. We also show that the same does not hold for degree eight: there are known maximal real schemes (M-curves) that cannot be realized as a T-curve. This resolves a question of Itenberg and Viro (1996).

• Z. Geiselmann, M. Joswig, L. Kastner, K. Mundinger, S. Pokutta, C. Spiegel, M. Wack, M. Zimmer, Fast Isotopy Computation for T-Curves, (preprint), 2026. Link

  A T-curve of degree d is given by a regular unimodular triangulation of the dilated two simplex together with a sign distribution on its lattice points. By Viro's Patchworking Theorem, this determines the ambient isotopy type (a.k.a. real scheme) of a smooth real plane projective algebraic curve of the same degree. We present a near-quadratic time algorithm for extracting that isotopy type from the triangulation and the signs. Through a GPU-accelerated implementation, this allows one to compute billions of real schemes per second, enabling exhaustive enumeration at scale. This algorithm was essential for our recent construction of all 121 real schemes of degree seven by T-curves.