Research

Math.

1. Non-very generic arrangement in low dimension, arxiv.org/abs/2202.04794 ,  https://doi.org/10.2748/tmj.20230922, (2) 77 (3), 357-373, (2025), e-ISSN (2nd Ser.): 2186-585X, (with Settepanella. S)

Abstract: The discriminantal arrangement B(n,k,A) has been introduced by Manin and Schectman in 1989 and it consists of all non-generic translates of a generic arrangement A of n hyperplanes in a k-dimensional space. It is known that its combinatorics depends on the original arrangement A which, following Bayer and Brandt [3], is called very generic if the intersection lattice of the induced discriminantal arrangement has maximum cardinality, non- very generic otherwise. While a complete description of the combinatorics of B(n,k,A) when A is very generic is known (see [2]), very few is known in the non-very generic case. Even to provide examples of non very generic arrangements proved to be a non-trivial task (see [17]). In this paper, we characterize, classify and provide examples of non-very generic arrangements in low dimension.

2. Homogeneous quandles with abelian inner automorphism groups, https://arxiv.org/abs/2403.07383, https://doi.org/10.1016/j.jalgebra.2024.09.004, Journal of Algebra, Volume 663, 2025, Pages 150-170, ISSN 0021-8693, (with Sugawara. S)

Abstract: In this paper, we give a characterization for homogeneous quandles with commutative inner automorphism groups. In particular, it is shown that such a quandle is expressed as an abelian extension of a trivial quandle. Our construction is a generalization of the recent work by Furuki and Tamaru, which gives the construction of disconnected flat quandles.

Chem.

1. Reproducing the Reaction Route Map on the Shape Space from Its Quotient by the Complete Nuclear Permutation-Inversion Group.  Hiroshi Teramoto, Takuya Saito, Masamitsu Aoki, Burai Murayama, Masato Kobayashi, Takenobu Nakamura, Tetsuya Taketsugu. Journal of Chemical Theory and Computation 19(17) 5886-5896 2023/08/29 https://doi.org/10.1021/acs.jctc.3c00500, https://arxiv.org/abs/2305.08072

2. Characterizing Reaction Route Map of Realistic Molecular Reactions Based on Weight Rank Clique Filtration of Persistent Homology.  Burai Murayama, Masato Kobayashi, Masamitsu Aoki, Suguru Ishibashi, Takuya Saito, Takenobu Nakamura, Hiroshi Teramoto, Tetsuya Taketsugu. Journal of Chemical Theory and Computation 19(15) 5007-5023 2023/07/03  https://doi.org/10.1021/acs.jctc.2c01204, https://arxiv.org/abs/2211.15067

1. Arithmetic non-very generic arrangements, https://arxiv.org/abs/2506.23124, (with Das. P, Settepanella. S)

Abstract: A discriminantal hyperplane arrangement B(n,k,A) is constructed from a given (generic) hyperplane arrangement A, which is classified as either very generic or non-very generic depending on the combinatorial structure of B(n,k,A). In particular, A is considered non-very generic if the intersection lattice of B(n,k,A) contains at least one non-very generic intersection -- that is, an intersection that fails to satisfy a specific rank condition established by Athanasiadis in [1]. In this paper, we present arithmetic criteria characterizing non-very generic intersections in discriminantal arrangements and we complete and correct a previous result by Libgober and the third author concerning rank-two intersections in such arrangements.

2. Degeneration in discriminantal arrangements, https://arxiv.org/abs/2404.18835

Abstract: Discriminantal arrangements are hyperplane arrangements, which are generalized braid ones. They are constructed from given hyperplane arrangements, but their combinatorics are not invariant under combinatorial equivalence. However, it is known that the combinatorics of the discriminantal arrangement are constant on a Zariski open set of the space of hyperplane arrangements. In the present paper, we introduce non-very generic varieties in the space of hyperplane arrangements to classify discriminantal arrangements and show that the Zariski open set is the complement of non-very generic varieties. We study their basic properties and construction and provide examples, including infinite families of non-very generic varieties. In particular, the construction we call degeneration is a powerful tool for constructing non-very generic varieties. As an application, we provide lists of non-very generic varieties for spaces of small line arrangements.

3. A categorification for the characteristic polynomial of matroids, https://arxiv.org/abs/2402.09851, (with Yamagata. S)

Abstract: In the present paper, we provide a cohomology group as a categorification of the characteristic polynomial of matroids. The construction depends on the “quasi-representation” of a matroid. For a certain choice of the quasi-representation, we show that our cohomology theory gives a generalization of the chromatic cohomology introduced by L. Helme-Guizon and Y. Rong, and also the characteristic cohomology introduced by Z. Dancso and A. Licata.

1. A categorification of polynomial invariants of matroids, South African-Japanese Discrete Homotopy Meeting 2025, University of Cape Town, 2025/09/10

2. *重み付き半順序集合の不変量について, 離散数学とその応用研究集会2025, 広島YMCA 3号館, 2025/08/19

3. *Towards the categorification of polynomial invariants in matroid theory I, ホモトピー沖縄2025, 沖縄船員会館, 2025/06/28

4. *Invariants for signed matroids, 第41回代数的組合せ論シンポジウム, 早稲田大学, 2025/06/20

5. *Varieties associated with combinatorics of discriminantal arrangements, Arrangement Days in NITech 2025 ,名古屋工業大学, 2025/06/11

6. *Colored Tutte Polynomials and Jacobi Polynomials, Waseda Workshop on Discrete Mathematics and Related Topics 2025, 早稲田大学, 2025/03/17.

7. 超平面配置の一般の位置の程度を表す多様体の余次元について, 第29回代数学若手研究会, 大阪大学, 2025/03/13.

8. マトロイドの階数関数の非可換化, 第21回数学総合若手研究集会, 北海道大学, 2025/03/03.  （テクニカルレポート）

9. 重さ枚挙多項式の畳み込み公式,  第21回組合せ論若手研究集会, 慶應義塾大学, 2025/02/17.

10. *マトロイドの多項式不変量の圏化,  Magnitude 2024,  九州大学, 2025/01/09.

11. Recent developments in the combinatorics of discriminantal arrangements, 日本数学会2024年度秋季総合分科会, 大阪大学, 2024/09/06.

12. 極小な非常に一般的でない直線配置, 離散数学とその応用研究集会2024, 山形大学, 2024/08/20.

13. *マトロイドの特性多項式の圏化, 誤り訂正符号と超平面配置に関わる多項式不変量, 九州大学, 2024/06/27.

14. *内部自己同型群が可換な等質カンドルについて, 第40回代数的組合せ論シンポジウム, 山梨大学, 2024/06/19.（報告集）

15. ある non-very generic な直線配置の構成法, 第6回数理新人セミナー, 九州大学及びオンライン, 2023/02/20. （報告集）

16. *内部自己同型群が可換な等質カンドルについて, 研究集会「カンドルと対称空間」,  大阪公立大学,  2022/12/09. （報告集）

17. *Discriminantal arrangements, Derived matroids, and their combinatorics, Workshop on Combinatorial topics in Shinshu 2022, 信州大学, 2022/07/25

18. 対称群による判別的配置 B(6, 3, A), B(6, 2, A) の分類, 第18回数学総合若手研究集会, 北海道大学, 2022/03/03. （テクニカルレポート）

19. 判別的配置の組合せ型のなす半順序集合, 超平面配置の数学とその進展2022, オンライン, 2022/02/15.

(*: invited

1. Arithmetic conditions for non-very generic arrangements, Topology of Arrangements with an Eye to Applications, University of Pisa, September 1-September 5, 2025.

2. Classification of discriminantal arrangement B(6, k, A) by using the symmetric group of degree 6, Arrangements in Ticino, SUPSI, June 27- July 1, 2022.