九州大学 研究者情報
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基本情報 研究活動 教育活動 社会活動
阿部 拓郎(あべ たくろう) データ更新日:2019.11.08



主な研究テーマ
超平面配置の代数・幾何・組み合わせ論
代数幾何学
キーワード:超平面配置、自由配置、ルート系
2003.03.
研究業績
主要原著論文
1. Takuro Abe, Toshiaki Maeno, Satoshi Murai and Yasuhide Numata, Solomon-Terao algebra of hyperplane arrangements
, Journal of the Matiemathcal Society of Japan, 71, 4, 1027-1047, 2019.10, [URL].
2. Takuro Abe, Hiroaki Terao, Multiple addition, deletion and restriction theorems for hyperplane arrangements, Proceedings of the American Mathematical Society, 10.1090/proc/14592, 147, 11, 4835-4845, 2019.11, [URL], In the study of free arrangements, the most useful result to construct/ check free arrangements is the addition-deletion theorem in [J. Fac. Sci. Univ. Tokyo 27 (1980), 293-320]. Recently, the multiple version of the addition theorem was proved in [J. Eur. Math. Soc. 18 (2016), 1339-1348], called the multiple addition theorem (MAT), to prove the ideal-free theorem. The aim of this article is to give the deletion version of MAT, the multiple deletion theorem (MDT). Also, we can generalize MAT to get MAT2 from the viewpoint of our new proof. Moreover, we introduce the restriction version, a multiple restriction theorem (MRT). Applications of MAT2, including the combinatorial freeness of the extended Catalan arrangements, are given..
3. 阿部拓郎, Deletion theorem and combinatorics of hyperplane arrangements, Mathematische Annalen, 10.1007/s00208-018-1713-9, 373, 1-2, 581-595, 2019.02, [URL], 自由配置から一枚超平面を抜いたものが再び自由になるかを判定する寺尾の除去定理の成立条件が完全に組み合わせ論で記述されることを示した。.
4. 阿部拓郎, Lukas Kühne, Heavy hyperplanes in multiarrangements and their freeness, Journal of Algebraic Combinatorics, https://doi.org/10.1007/s10801-017-0806-y, 48, 4, 581-606, 2018.12.
5. 阿部拓郎, Alexandru Dimca, Splitting types of bundles of logarithmic vector fields along plane curves, International Journal of Mathematics, https://doi.org/10.1142/S0129167X18500556, 29, 8, 1-20, 2018.10.
6. 阿部 拓郎, 陶山大輔, A basis construction of the extended Catalan and Shi arrangements of the type A2, Journal of Algebra, doi.org/10.1016/j.jalgebra.2017.09.024, 493, 20-35, 2018.01.
7. 阿部拓郎, Restrictions of free arrangements and the division theorem, Proceedings of the Intensive Period "Perspectives in Lie Theory",
Springer INdAM Series ; 19
, https://doi.org/10.1007/978-3-319-58971-8_14, 389-401, 2017.12.
8. 阿部 拓郎, Mohamed Barakat, Michael Cuntz, Torsten Hoge, 寺尾宏明, The freeness of ideal subarrangements of Weyl arrangements, Journal of the Eurpoean Mathematical Society, 10.4171/JEMS/615 , 18, 6, 1339-1348, 2016.06, ワイル配置の部分配置であるイデアル配置の自由性を証明した。その結果として、Shapiro-Steinberg-Kostant-Macdonaldによる、ルート系の指数と正ルートの高さ分布の双対分割の一致を、イデアルまで拡張することに成功した。.
9. 阿部 拓郎, Divisionally free arrangements of hyperplanes, Inventiones mathematicae, 10.1007/s00222-015-0615-7, 204, 1, 317-346, 2016.04, 自由配置に関する剰余定理を証明し、剰余的自由配置を定式化することで自由配置の組合せ論的側面を明確にした。.
10. 縫田光司, 阿部 拓郎, 鍛冶静雄, 前野俊昭, 沼田泰英, A mathematical problem for security analysis of hash functions and pseudorandom generators, International Journal of Foundations of Computer Science, 10.1142/s0129054115500100, 26, 169, 2015.02.
11. 阿部 拓郎, Roots of characteristic polynomials and intersection points of line arrangements, Journal of Singularities, 8, 100-117, 2014.11.
12. 阿部 拓郎, Chambers of 2-affine arrangements and freeness of 3-arrangements, Journal of Algebraic Combinatorics, 10.1007/s10801-012-0393-x, 38, 1, 65-78, 2013.08.
13. 阿部 拓郎, 吉永正彦, Free arrangements and coefficients of characteristic polynomials, Mathematische Zeitschrift , 10.1007/s00209-013-1165-6, 275, 3, 911-919, 2013.04.
14. 阿部 拓郎, Characteristic polynomials, η-complexes and freeness of tame arrangements, Michigan Mathematical Journal, 62, 1, 117-130, 2013.01.
15. 阿部 拓郎, 寺尾宏明, 若神子篤史, Equivariant multiplicities of Coxeter arrangements and invariant bases, Advances in Mathematics, 10.1016/j.aim.2012.04.015, 230, 2364-2377, 2012.07.
16. 阿部 拓郎, On the conjecture of Athanasiadis related to freeness of a family of hyperplane arrangements, Mathematical Research Letters, 19, 2, 469-474, 2012.04.
17. 阿部 拓郎, 沼田泰英, Exponents of 2-multiarrangements and multiplicity lattices, Journal of Algebraic Combinatorics, 10.1007/s10801-011-0291-7, 35, 1, 1-17, 2012.02.
18. 阿部 拓郎, 寺尾宏明, A primitive derivation and logarithmic differential forms of Coxeter arrangements, Mathematische Zeitschrift, 264, 4, 813-828, 2010.04.
19. 阿部 拓郎, 吉永正彦, Coxeter multiarrangements with quasi-constant multiplicities, Journal of Algebra, 10.1016/j.jalgebra.2009.07.025, 322, 8, 2839-2847, 2009.10.
20. 阿部 拓郎, 縫田光司, 沼田泰英, Signed-eliminable graphs and free multiplicities on the braid arrangement, Journal of the London Mathematical Society, 10.1112/jlms/jdp019, 80, 1, 121-134, 2009.08.
21. 阿部 拓郎, 吉永正彦, Splitting criterion for reflexive sheaves, Proceedings of the American Mathematical Society, 136, 6, 1887-1891, 2008.06.
22. 阿部 拓郎, 寺尾宏明, Max Wakefield, The Euler multiplicity and addition-deletion theorems for multiarrangements, Journal of the London Mathematical Society, 10.1112/jlms/jdm110, 77, 2, 335-348, 2008.02.
23. 阿部 拓郎, 寺尾宏明, Max Wakefield, The characteristic polynomial of a multiarrangement, Advances in Mathematics, 10.1016/j.aim.2007.04.019, 215, 2, 825-838, 2007.10.
24. 阿部 拓郎, The Elementary transformation of vector bundles on regular schemes, Transactions of the American Mathematical Society, 359, 9, 4285-4295, 2007.09.
25. 阿部 拓郎, The stability of the family of A_2-type arrangements, Journal of Mathematics Kyoto University , 46, 3, 617-635, 2006.07.
主要学会発表等
1. 阿部拓郎, Recent topics on free arrangements, Arrangements at Western, 2019.05, [URL].
2. 阿部拓郎, Combinatorics of the addition-deletion theorems for free arrangements, CIMPA - IMH Research School HYPERPLANE ARRANGEMENTS: RECENT ADVANCES AND OPEN PROBLEMS, 2019.03, [URL].
3. 阿部拓郎, Combinatorics of the addition-deletion theorems for arrangements, On hyperplane arrangements, configuration spaces and related topics, 2019.02, [URL].
4. 阿部拓郎, Logarithmic vector fields and freeness of hyperplane arrangements, Free divisors and Hyperplane arrangements, 2018.12, [URL].
5. 阿部拓郎, Hessenberg varieties and hyperplane arrangements, Hessenberg varieties 2018 in Osaka, 2018.12, [URL].
6. 阿部拓郎, Hessenbergs and hyperplane arrangements part II, Hessenberg Varieties in Combinatorics, Geometry and Representation Theory, 2018.10, [URL].
7. 阿部拓郎, 超平面配置の対数的ベクトル場と自由性, 日本数学会秋季総合分科会, 2018.09, [URL], 超平面配置の対数的ベクトル場とその自由性は、超平面配置研究の中でも中心的な役割を果たしている。
講演者は近年自由正研究において大きな進展を数多くもたらした。それらを、代数・幾何の観点を中心に概説する。.
8. 阿部拓郎, The b_2-equality and free arrangements, New perspectives in hyperplane arrangements, 2018.09, [URL].
9. 阿部拓郎, 超平面配置の代数・自由性と組み合わせ論, 組合せ論サマースクール2018, 2018.08.
10. 阿部拓郎, Free arrangements of hyperplanes and applications, Workshop on Algebraic Geometry, 2018.06.
11. Takuro Abe, Generators for logarithmic derivation modules of hyperplane arrangements, Topology and Geometry: A conference in memory of Stefan Papadima (1953-2018), 2018.05, [URL], The most famous nice generators for logarithmic derivation modules of arrangements are
free bases when the arrangement is free. Dimca and Sticlaru introduced a nearly free
plane curves, which has also a nice set of generators. We study more on a set of nice
generators..
12. Takuro Abe, Generators of logarithmic derivation modules of hyperplane arrangements, Arrangements of Hypersurfaces, 2018.04, [URL], Logarithmic derivation modules are one of the most important objects to study in of hyperplane arrangements and
hypersurfaces. In particular, the freeness of them have been intensively studied. But in general they are not free. By
Dimca and Sticlaru, the near freeness of plane curves and cubic surfaces are introduced, which is close to the freeness
from the viewpoint of the number of generators. We study several properties of free and nearly free curves from algebro-
geometric points of view, and consider the higher dimensional version of them. This talk contains a joint work with Alex
Dimca..
13. 阿部拓郎, Poincare polynomials and free arrangements, A walk between hyperplane arrangements, computer algebra and algorithms, 2018.02, [URL].
14. 阿部拓郎, Solomon-Terao algebra of hyperplane arrangements, Topology of arrangements and representation stability, 2018.01, [URL].
15. 阿部拓郎, Solomon-Terao algebra of hyperplane arrangements, Toric Topology 2017 in Osaka, 2017.12, [URL].
16. 阿部拓郎, 自由配置に関する近年の進展とその応用, 不変式・超平面配置と平坦構造 , 2017.11, [URL].
17. 阿部 拓郎, Hyperplane arrangements and Hessenberg varieties, The 5th Franco-Japan-Vietnamese Symposium on Singularities, 2017.11, [URL].
18. 阿部 拓郎, Free arrangements and vector bundles, VIIIe rencontre Pau-Zaragoza d'Algebra et Geometrie, 2017.09.
19. 阿部 拓郎, The b2-inequality and freeness of the restrictions of hyperplane arrangements, Advances in Hyperplane Arrangements, 2017.08, [URL].
20. 阿部 拓郎, Hyperplane arrangements and Hessenberg varieties, Advances in Arrangement Theory, Mathematical Congress of the Americas, 2017.07, [URL], イデアル配置から構成されるSolomon-寺尾代数と、同じイデアルから構成される正則冪零Hessenberg多様体のコホモロジー環が同型となることを示した。.
21. 阿部 拓郎, Hyperplane arrangements and Hessenberg varieties, Arrangements and beyond, 2017.06, [URL], Hessenberg varieties were introduced by De Mari, Procesi and Shayman as a generalization of flag varieties. Recently, for the regular nilpotent and regular semisimple cases, their topologies are intensively studied, and related to combinatorial and (geometric) representational aspects. However, the algebraic structure of their cohomology groups have been unknown except for the case of type A. Recalling the fact that their cohomology rings are isomorphic to the coinvariant algebras, and Kyoji Saito's original proof of the freeness of Weyl arrangements by using basic invariants, we give a presentation of the cohomology group of a regular nilpotent Hessenberg variety by using a logarithmic derivation module of certain hyperplane arrangements (ideal arrangements) coming from the Hessenberg variety. Also, several properties of cohomology groups like complete intersection, hard Lefschetz properties and Hodge-Riemann relations are shown..
22. 阿部 拓郎, Hyperplane arrangments, Solomon-Terao algebras and applications to Hessenberg varieties, 変換群を核とする代数的位相幾何学, 2017.05, [URL].
23. 阿部 拓郎, Algebra and geometry of Solomon-Terao's formula, Hyperplane Arrangements and related topics, 2017.02.
24. 阿部 拓郎, Algebra and combinatorics of hyperplane arrangements, The 15th Japan-Korea Workshop on Algebra and Combinatorics, 2017.02.
25. 阿部 拓郎, Hyperplane arrangements and Hessenberg varieties, Oberseminar, 2016.12.
26. 阿部 拓郎, Divisionally free arrangements of hyperplanes, 第61回代数学シンポジウム, 2016.09, [URL].
27. 阿部 拓郎, Recent developments on algebra of line arrangements, The 2nd Franco-Japanese-Vietnamese Symposium on Singularities, 2015.08, [URL].
28. 阿部 拓郎, Division and localization of characteristic polynomials of hyperplane arrangements, Combinatorics and Algebraic Topology of Configurations, 2015.02, [URL].
29. 阿部 拓郎, Division free theorem for line arrangements and divisionally free arrangements of hyperplanes, Arrangements of plane curves and related problems, 2015.03, [URL].
30. 阿部 拓郎, Divisional freeness and the second Betti number of hyperplane arrangements, Hyperplane arrangements and reflection groups, 2015.08, [URL].
31. 阿部 拓郎, Divisional freeness and the second Betti number of hyperplane arrangements, Differential and combinatorial aspects of singularities, 2015.08, [URL].
32. 阿部 拓郎, Recent topics on free arrangements of hyperplanes, Summer Conference on Hyperplane Arrangements(SCHA) in Sapporo, 2016.08, [URL], Freeness has been one of the central topics among the theory of hyperplane arrangements. There are several important results like Terao's addition-deletion and factorization therems, the moduli theoretic approach by Yusvinsky and so on. In particular, Yoshinaga's criterion on freeness by using mutliarrangemens gave a breakthrough in this study area, and there have been a lot of new approaches on freeness problem.
In this talk, we discuss several recent results including freeness criterion, multiple addition theorems and divisional freeness. Also, we discuss the new algebraic class of line arramgenents and plane curves in the projective plane called near freeness by Dimca and Sticlaru. Moreover, we pose some problems which appeared in this ten years..
33. 阿部 拓郎, Divisional flags and freeness of hyperplane arrangements, The JapaneseConference on Combinatorics and its Applications, Mini symposium:Combinatorics of hyperplane arrangements, 2016.05, [URL].
34. 阿部 拓郎, Some remarks on nearly free arrangements of lines in the projective plane, Workshop on Hyperplane Arrangements and Singularity Theory, 2016.03.
35. 阿部 拓郎, Freeness and flags of hyperplane arrangements, Special Session on Topology and Combinatorics of Arrangements (in honor of Mike Falk), AMS Sectional Meeting, 2016.03.
学会活動
所属学会名
日本数学会
学会大会・会議・シンポジウム等における役割
2018.09.17~2018.09.19, 量子情報社会に向けた数理的アプローチ, 世話人.
2018.07.25~2018.07.31, Study Group Workshop 2018, 世話人.
2018.06.20~2018.06.21, ベクトル束の分裂・構成・安定性, 世話人.
2018.06.11~2018.06.15, Matroids, reflection groups and free hyperplane arrangements, 世話人.
2018.02.05~2018.02.07, 代数的手法による数理暗号解析, 世話人.
2017.02.13~2017.02.14, Hyperplane Arrangements and related topics, 世話人.
2017.02.06~2017.02.08, 代数幾何学と暗号数理の展開, 世話人.
2016.08.08~2016.08.12, Summer Conference on Hyperplane Arrangements(SCHA) in Sapporo, 世話人.
2016.05.24~2016.05.24, Combinatorics of hyperplane arrangements, 世話人代表.
学術論文等の審査
年度 外国語雑誌査読論文数 日本語雑誌査読論文数 国際会議録査読論文数 国内会議録査読論文数 合計
2017年度
2016年度
2015年度
その他の研究活動
海外渡航状況, 海外での教育研究歴
Université de Pau et des Pays de l'Adour, France, 2017.09~2017.09.
Université de Pau et des Pays de l'Adour, France, 2013.09~2013.09.
外国人研究者等の受入れ状況
2015.10~2016.02, 1ヶ月以上, 京都大学大学院工学研究科, Spain, 日本学術振興会.
研究資金
科学研究費補助金の採択状況(文部科学省、日本学術振興会)
2019年度~2021年度, 国際共同研究強化(A), 代表, D加群を用いた超平面配置とSolomon-寺尾複体の革新的研究.
2016年度~2020年度, 基盤研究(B), 代表, 超平面配置の余不変式環論の創成とその表現論・幾何学の新展開.
2016年度~2018年度, 挑戦的萌芽研究, 代表, 超平面配置の自由性・トポロジーとランダムウォークの新潮流.
2012年度~2015年度, 若手研究(B), 代表, 超平面配置の自由性の多角的解析と関連する幾何学の研究.
2009年度~2011年度, 若手研究(B), 代表, 多重配置の自由性の解析と関連する幾何学の創出.

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