Updated on 2024/12/04

写真b

 
ABE Takuro
 
*Items subject to periodic update by Rikkyo University (The rest are reprinted from information registered on researchmap.)
Affiliation*
College of Science Department of Mathematics
Graduate School of Science Doctoral Program in Mathematics
Graduate School of Science Master's Program in Mathematics
Title*
Professor
Degree
博士(理学) ( 京都大学 )
Research Theme*
  • ベクトル空間中の超平面の有限集合として定義される超平面配置を、特に可換環論・ベクトル束といった代数・代数幾何的視点を用いて、その代数と幾何学・組み合わせ論との関連を探求しています。また超平面配置はルート系・ワイル群をその起源とするため、それらが関連する様々な数学、例えばHessenberg多様体や、余不変式環などの研究も行っています。

  • Research Interests
  • Algebraic Geometry

  • Root systems

  • Vector bundles

  • Logarithmic vector fields

  • Arrangements of hyperplanes

  • Campus Career*
    • 4 2023 - Present 
      College of Science   Department of Mathematics   Professor
    • 4 2023 - Present 
      Graduate School of Science   Master's Program in Mathematics   Professor
    • 4 2023 - Present 
      Graduate School of Science   Doctoral Program in Mathematics   Professor
     

    Research Areas

    • Natural Science / Algebra  / Arrangements of hyperplanes

    • Natural Science / Algebra  / Algebraic Geometry

    Research History

    • 4 2023 - Present 
      Rikkyo University   Department of Mathematics, Faculty of Science   Professor

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      Country:Japan

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    • 7 2019 - 3 2023 
      Kyushu University   Institute of Mathematics for Industry   Professor

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    • 3 2016 - 6 2019 
      Kyushu University   Institute of Mathematics for Industry   Associate Professor

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    • 10 2010 - 2 2016 
      Kyoto University   Graduate School of Engineering   Lecturer

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    • 10 2008 - 9 2010 
      Kyoto University   Graduade School of Science   Program specific assistant professor (GCOE)

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    • 4 2008 - 9 2008 
      JSPS

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    • 4 2006 - 3 2008 
      Hokkaido University   Graduate School of Science

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    Education

    • 4 2003 - 3 2006 
      Kyoto University

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    • 4 2001 - 3 2003 
      Kyoto University   Graduate School of Science   Division of Mathematics

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    • 4 1997 - 3 2001 
      Kyoto University   Faculty of Science

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    Committee Memberships

    • 4 2018 - 3 2019 
      日本数学会   九州支部評議委員

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      Committee type:Academic society

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    Papers

    • Projective dimensions of hyperplane arrangements Peer-reviewed

      Takuro Abe

      Transactions of the American Mathematical Society377 ( 11 ) 7793 - 7827   11 2024

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      Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.1090/tran/9196

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    • Free reflection multiarrangements and quasi-invariants Peer-reviewed International coauthorship

      Takuro Abe, Naoya Enomoto, Misha Feigin, Masahiko Yoshinaga

      International Mathematics Research Notices   22 10 2024

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      Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.1093/imrn/rnae236

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    • Projective dimension of weakly chordal graphic arrangements Peer-reviewed International coauthorship

      Takuro Abe, Lukas Kühne, Paul Muecksch, Leonie Muehlherr

      Algebraic Combinatorics, accepted   10 2024

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      Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)  

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    • A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups Peer-reviewed

      Takuro Abe, Gerhard Roehrle, Christian Stump, Masahiko Yoshinaga

      Journal of Combinatorial Algebra8 ( 3/4 ) 251 - 278   23 7 2024

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      Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.4171/JCA/94

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    • Vertex-weighted digraphs and freeness of arrangements between Shi and Ish Peer-reviewed

      Takuro Abe, Tan Nhat Tran, Shuhei Tsujie

      European Journal of Combinatorics118   5 2024

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      Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.1016/j.ejc.2024.103920

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    • Generalization of the addition and restriction theorems from free arrangements to the class of projective dimension one Peer-reviewed

      ABE, TAKURO

      Algebraic Combinatorics7 ( 2 ) 413 - 429   29 4 2024

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      Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.5802/alco.340

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    • On some freeness-type properties for line arrangements Peer-reviewed

      Takuro Abe, Denis Ibadula, Anca Măcinic

      ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE25 ( 1 ) 427 - 447   2 2024

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      Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)   Publisher:Scuola Normale Superiore - Edizioni della Normale  

      DOI: 10.2422/2036-2145.202105_038

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    • Addition-deletion theorems for the Solomon-Terao polynomials and B-sequences of hyperplane arrangements Peer-reviewed International journal

      Abe, Takuro

      Mathematische Zeitschrift306 ( 205 )   11 1 2024

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      Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.1007/s00209-023-03426-z

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    • Addition-deletion theorem for free hyperplane arrangements and combinatorics Peer-reviewed

      Takuro Abe

      Journal of Algebra610   1 - 17   19 7 2022

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      Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.1016/j.jalgebra.2022.06.028

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    • Double points of free projective line arrangements Peer-reviewed

      Takuro Abe

      International Mathematical Research Notices2022 ( 3 ) 1811 - 1824   2 2022

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      Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.1093/imrn/rnaa145

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    • Locally heavy hyperplanes in multiarrangements Peer-reviewed

      Takuro Abe, Lukas Kühne

      Journal of Pure and Applied Algebra226 ( 1 ) 106791 - 106791   1 2022

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      Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)   Publisher:Elsevier BV  

      DOI: 10.1016/j.jpaa.2021.106791

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    • Roots of the characteristic polynomials of hyperplane arrangements and their restrictions and localizations Peer-reviewed

      Takuro Abe

      Topology and its Applications (special volume), to appear   12 2021

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      Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.1016/j.topol.2021.107990

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    • A Hyper-surface Arrangement Model of Ranking Distributions Peer-reviewed

      Shizuo Kaji, Akira Horiguchi, Takuro Abe, Yohsuke Watanabe

      Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining   796 - 804   14 8 2021

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      Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:ACM  

      A distribution on the permutations over a fixed finite set is called a ranking distribution. Modelling ranking distributions is one of the major topics in preference learning as such distributions appear as the ranking data produced by many judges. In this paper, we propose a geometric model for ranking distributions. Our idea is to use hyper-surface arrangements in a metric space as the representation space, where each component cut out by hyper-surfaces corresponds to a total ordering, and its volume is proportional to the probability. In this setting, the union of components corresponds to a partial ordering and its probability is also estimated by the volume. Similarly, the probability of a partial ordering conditioned by another partial ordering is estimated by the ratio of volumes. We provide a simple iterative algorithm to fit our model to a given dataset. We show our model can represent the distribution of a real-world dataset faithfully and can be used for prediction and visualisation purposes.

      DOI: 10.1145/3447548.3467253

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    • A Characterization of High Order Freeness for Product Arrangements and Answers to Holm’s Questions Peer-reviewed International journal

      Takuro Abe, Norihiro Nakashima

      Algebras and Representation Theory24 ( 3 ) 585 - 599   6 2021

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      Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.1007/s10468-020-09961-1

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    • Plus-one generated and next to free arrangements of hyperplanes Peer-reviewed

      Takuro Abe

      International Mathematical Research Notices2021 ( 12 ) 9233 - 9261   6 2021

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      Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.1093/imrn/rnz099

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    • Addition–deletion results for the minimal degree of logarithmic derivations of hyperplane arrangements and maximal Tjurina line arrangements

      Takuro Abe, Alexandru Dimca, Gabriel Sticlaru

      Journal of Algebraic Combinatorics   16 11 2020

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      Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)   Publisher:Springer Science and Business Media LLC  

      DOI: 10.1007/s10801-020-00986-9

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      Other Link: http://link.springer.com/article/10.1007/s10801-020-00986-9/fulltext.html

    • On A_1^2 restrictions of Weyl arrangements Peer-reviewed

      Takuro Abe, Hiroaki Terao, Tan Nhat Tran

      Journal of Algebraic Combinatorics   18 9 2020

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:Springer Science and Business Media LLC  

      DOI: 10.1007/s10801-020-00979-8

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      Other Link: http://link.springer.com/article/10.1007/s10801-020-00979-8/fulltext.html

    • Hessenberg varieties and hyperplane arrangements Peer-reviewed

      Takuro Abe, Tatsuya Horiguchi, Mikiya Masuda, Satoshi Murai, Takashi Sato

      Journal für die reine und angewandte Mathematik (Crelles Journal)2020 ( 764 ) 241 - 286   1 7 2020

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:Walter de Gruyter GmbH  

      <title>Abstract</title>Given a semisimple complex linear algebraic group <inline-formula id="j_crelle-2018-0039_ineq_9999_w2aab3b7e1517b1b6b1aab1c14b1b1Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_1056.png" /><tex-math>{ { G } }</tex-math></alternatives></inline-formula> and a lower ideal <italic>I</italic> in positive roots of <italic>G</italic>, three objects arise:
      the ideal arrangement <inline-formula id="j_crelle-2018-0039_ineq_9998_w2aab3b7e1517b1b6b1aab1c14b1b7Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0652.png" /><tex-math>{\mathcal{A}_{I } }</tex-math></alternatives></inline-formula>, the regular nilpotent Hessenberg variety <inline-formula id="j_crelle-2018-0039_ineq_9997_w2aab3b7e1517b1b6b1aab1c14b1b9Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0711.png" /><tex-math>{\operatorname{Hess}(N,I)}</tex-math></alternatives></inline-formula>, and the regular semisimple Hessenberg variety <inline-formula id="j_crelle-2018-0039_ineq_9996_w2aab3b7e1517b1b6b1aab1c14b1c11Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0713.png" /><tex-math>{\operatorname{Hess}(S,I)}</tex-math></alternatives></inline-formula>.
      We show that
      a certain graded ring derived from the logarithmic derivation module of <inline-formula id="j_crelle-2018-0039_ineq_9995_w2aab3b7e1517b1b6b1aab1c14b1c13Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0652.png" /><tex-math>{\mathcal{A}_{I } }</tex-math></alternatives></inline-formula> is isomorphic to
      <inline-formula id="j_crelle-2018-0039_ineq_9994_w2aab3b7e1517b1b6b1aab1c14b1c15Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0431.png" /><tex-math>{H^{*}(\operatorname{Hess}(N,I))}</tex-math></alternatives></inline-formula> and <inline-formula id="j_crelle-2018-0039_ineq_9993_w2aab3b7e1517b1b6b1aab1c14b1c17Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0432.png" /><tex-math>{H^{*}(\operatorname{Hess}(S,I))^{W } }</tex-math></alternatives></inline-formula>,
      the invariants in <inline-formula id="j_crelle-2018-0039_ineq_9992_w2aab3b7e1517b1b6b1aab1c14b1c19Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0433.png" /><tex-math>{H^{*}(\operatorname{Hess}(S,I))}</tex-math></alternatives></inline-formula> under an action of the Weyl group <italic>W</italic> of <italic>G</italic>.
      This isomorphism is shown
      for general Lie type,
      and generalizes Borel’s celebrated theorem showing that the coinvariant algebra of <italic>W</italic> is isomorphic to the cohomology ring of the flag variety <inline-formula id="j_crelle-2018-0039_ineq_9991_w2aab3b7e1517b1b6b1aab1c14b1c27Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0410.png" /><tex-math>{G/B}</tex-math></alternatives></inline-formula>.

      This surprising connection between Hessenberg varieties and hyperplane
      arrangements enables us to produce a number of interesting
      consequences. For instance, the surjectivity of the restriction map
      <inline-formula id="j_crelle-2018-0039_ineq_9990_w2aab3b7e1517b1b6b1aab1c14b2b1Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0428.png" /><tex-math>{H^{*}(G/B)\to H^{*}(\operatorname{Hess}(N,I))}</tex-math></alternatives></inline-formula> announced by Dale
      Peterson
      and an affirmative answer to
      a conjecture of Sommers and Tymoczko are immediate consequences. We also
      give an explicit ring presentation of <inline-formula id="j_crelle-2018-0039_ineq_9989_w2aab3b7e1517b1b6b1aab1c14b2b3Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0431.png" /><tex-math>{H^{*}(\operatorname{Hess}(N,I))}</tex-math></alternatives></inline-formula> in
      types <italic>B</italic>, <italic>C</italic>, and <italic>G</italic>. Such a presentation was already known in type
      <italic>A</italic> and when <inline-formula id="j_crelle-2018-0039_ineq_9988_w2aab3b7e1517b1b6b1aab1c14b2c13Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0711.png" /><tex-math>{\operatorname{Hess}(N,I)}</tex-math></alternatives></inline-formula> is the Peterson variety. Moreover, we find
      the volume polynomial of <inline-formula id="j_crelle-2018-0039_ineq_9987_w2aab3b7e1517b1b6b1aab1c14b2c15Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0711.png" /><tex-math>{\operatorname{Hess}(N,I)}</tex-math></alternatives></inline-formula> and see that the hard
      Lefschetz property and the Hodge–Riemann relations hold for
      <inline-formula id="j_crelle-2018-0039_ineq_9986_w2aab3b7e1517b1b6b1aab1c14b2c17Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0711.png" /><tex-math>{\operatorname{Hess}(N,I)}</tex-math></alternatives></inline-formula>, despite the fact that it is a singular variety in general.

      DOI: 10.1515/crelle-2018-0039

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      Other Link: https://www.degruyter.com/downloadpdf/journals/crll/2020/764/article-p241.xml

    • On complex supersolvable line arrangements Peer-reviewed

      Takuro Abe, Alexandru Dimca

      Journal of Algebra552   38 - 51   6 2020

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      Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.1016/j.jalgebra.2020.02.007

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    • Multiple addition, deletion and restriction theorems for hyperplane arrangements Peer-reviewed

      Takuro Abe, Hiroaki Terao

      Proceedings of the American Mathematical Society147 ( 11 ) 4835 - 4845   11 2019

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      DOI: 10.1090/proc/14592

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    • Solomon-Terao algebra of hyperplane arrangements Peer-reviewed

      Abe, Takuro, Maeno, Toshiaki, Mural, Satoshi, Numata, Yasuhide

      JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN71 ( 4 ) 1027 - 1047   10 2019

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:MATH SOC JAPAN  

      We introduce a new algebra associated with a hyperplane arrangement A, called the Solomon-Terao algebra ST(A, eta), where eta is a homogeneous polynomial. It is shown by Solomon and Terao that ST(A, eta) is Artinian when eta is generic. This algebra can be considered as a generalization of coinvariant algebras in the setting of hyperplane arrangements. The class of Solomon-Terao algebras contains cohomology rings of regular nilpotent Hessenberg varieties. We show that ST(A, eta) is a complete intersection if and only if A is free. We also give a factorization formula of the Hilbert polynomials of ST(A, eta) when A is free, and pose several related questions, problems and conjectures.

      DOI: 10.2969/jmsj/79957995

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    • Deletion theorem and combinatorics of hyperplane arrangements Peer-reviewed

      Takuro Abe

      Mathematische Annalen373 ( 1--2 ) 581 - 595   2 2019

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      Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.1007/s00208-018-1713-9

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    • Heavy hyperplanes in multiarrangements and their freeness Peer-reviewed

      Takuro Abe, Lukas Kuehne

      Journal of Algebraic Combinatorics48 ( 4 ) 581 - 606   12 2018

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      Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.1007/s10801-017-0806-y

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    • Splitting types of bundles of logarithmic vector fields along plane curves Peer-reviewed

      Takuro Abe, Alexandru Dimca

      International Journal of Mathematics29 ( 8 )   2018

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      Language:English   Publishing type:Research paper (scientific journal)  

      DOI: 10.1142/S0129167X18500556

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    • A basis construction of the extended Catalan and Shi arrangements of the type &ITA&IT2 Peer-reviewed

      Takuro Abe, Daisuke Suyama

      JOURNAL OF ALGEBRA493   20 - 35   1 2018

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:ACADEMIC PRESS INC ELSEVIER SCIENCE  

      In [10], Terao proved the freeness of multi-Coxeter arrangements with constant multiplicities by giving an explicit construction of bases. Combining it with algebro-geometric method, Yoshinaga proved the freeness of the extended Catalan or Shi arrangements in [12]. However, there have been no explicit constructions of the bases for the logarithmic derivation modules of the extended Catalan and Shi arrangements. In this paper, we give the first explicit construction of them when the root system is of the type A(2). (C) 2017 Elsevier Inc. All rights reserved.

      DOI: 10.1016/j.jalgebra.2017.09.024

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    • The freeness of Ish arrangements Peer-reviewed

      Takuro Abe, Daisuke Suyama, Shuhei Tsujie

      JOURNAL OF COMBINATORIAL THEORY SERIES A146   169 - 183   2 2017

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:ACADEMIC PRESS INC ELSEVIER SCIENCE  

      The Ish arrangement was introduced by Armstrong to give a new interpretation of the q, t-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free. (C) 2016 Elsevier Inc. All rights reserved.

      DOI: 10.1016/j.jcta.2016.09.008

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    • Restrictions of free arrangements and the division theorem Peer-reviewed

      Takuro Abe

      Springer INdAM Series19   389 - 401   2017

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      Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:Springer International Publishing  

      This is a survey and research note on the modified Orlik conjecture derived from the division theorem introduced in Abe (Invent. Math. 204(1), 317–346, 2016). The division theorem is a generalization of classical addition-deletion theorems for free arrangements. The division theorem can be regarded as a modified converse of the Orlik’s conjecture with a combinatorial condition, i.e., an arrangement is free if the restriction is free and the characteristic polynomial of the restriction divides that of an arrangement. In this article we recall, summarize, pose and re-formulate some of results and problems related to the division theorem based on Abe (Invent. Math. 204(1), 317–346, 2016), and study the modified Orlik’s conjecture with partial answers.

      DOI: 10.1007/978-3-319-58971-8_14

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    • Non-recursive freeness and non-rigidity Peer-reviewed

      T. Abe, M. Cuntz, H. Kawanoue, T. Nozawa

      DISCRETE MATHEMATICS339 ( 5 ) 1430 - 1449   5 2016

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:ELSEVIER SCIENCE BV  

      In the category of free arrangements, inductively and recursively free arrangements are important. In particular, in the former, Terao's open problem asking whether freeness depends only on combinatorics is true. A long standing problem whether all free arrangements are recursively free or not was settled by the second author and Hoge very recently, by giving a free but non-recursively free plane arrangement consisting of 27 planes. In this paper, we construct free but non-recursively free plane arrangements consisting of 13 and 15 planes, and show that the example with 13 planes is the smallest in the sense of the cardinality of planes. In other words, all free plane arrangements consisting of at most 12 planes are recursively free. To show this, we completely classify all free plane arrangements in terms of inductive freeness and three exceptions when the number of planes is at most 12. Several properties of the 15 plane arrangement are proved by computer programs. Also, these two examples solve negatively a problem posed by Yoshinaga on the moduli spaces, (inductive) freeness and, rigidity of free arrangements. (C) 2016 Published by Elsevier B.V.

      DOI: 10.1016/j.disc.2015.12.017

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    • Divisionally free arrangements of hyperplanes Peer-reviewed

      Takuro Abe

      INVENTIONES MATHEMATICAE204 ( 1 ) 317 - 346   4 2016

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:SPRINGER HEIDELBERG  

      Let be the triple of hyperplane arrangements. We show that the freeness of and the division of by imply the freeness of . This "division theorem" improves the famous addition-deletion theorem, and it has several applications, which include a definition of "divisionally free arrangements". It is a strictly larger class of free arrangements than the classical important class of inductively free arrangements. Also, whether an arrangement is divisionally free or not is determined by the combinatorics. Moreover, we show that a lot of recursively free arrangements, to which almost all known free arrangements are belonging, are divisionally free.

      DOI: 10.1007/s00222-015-0615-7

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    • Free filtrations of affine Weyl arrangements and the ideal-Shi arrangements Peer-reviewed

      Takuro Abe, Hiroaki Terao

      JOURNAL OF ALGEBRAIC COMBINATORICS43 ( 1 ) 33 - 44   2 2016

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:SPRINGER  

      In this article, we prove that the ideal-Shi arrangements are free central arrangements of hyperplanes satisfying the dual partition formula. Then, it immediately follows that there exists a saturated free filtration of the cone of any affine Weyl arrangement such that each subarrangement of the filtration satisfies the dual partition formula. This generalizes the main result in Abe et al. (J. Eur. Math. Soc., to appear) which affirmatively settled a conjecture by Sommers and Tymoczko (Trans. Am. Math. Soc. 358:3493-3509, 2006).

      DOI: 10.1007/s10801-015-0624-z

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    • The freeness of ideal subarrangements of Weyl arrangements Peer-reviewed

      T. Abe, M. Barakat, M. Cuntz, T. Hoge, H. Terao

      JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY18 ( 6 ) 1339 - 1348   2016

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:EUROPEAN MATHEMATICAL SOC  

      A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. The proof of the main theorem is classification-free. It heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula.

      DOI: 10.4171/JEMS/615

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    • LOGARITHMIC BUNDLES OF DEFORMED WEYL ARRANGEMENTS OF TYPE A(2) Peer-reviewed

      Takuro Abe, Daniele Faenzi, Jean Valles

      BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE144 ( 4 ) 745 - 761   2016

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:FRENCH MATHEMATICAL SOC  

      We consider deformations of the Weyl arrangement of type A(2), which include the extended Shi and Catalan arrangements. These last ones are well-known to be free. We study their sheaves of logarithmic vector fields in all other cases, and show that they are Steiner bundles. Also, we determine explicitly their unstable lines. As a corollary, some counter-examples to the shift isomorphism problem are given.

      DOI: 10.24033/bsmf.2726

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    • A mathematical problem for security analysis of hash functions and pseudorandom generators Peer-reviewed

      Koji Nuida, Takuro Abe, Shizuo Kaji, Toshiaki Maeno, Yasuhide Numata

      International Journal of Foundations of Computer Science26 ( 2 ) 169 - 194   8 2 2015

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      In this paper, we specify a class of mathematical problems, which we refer to as "Function Density Problems" (FDPs, in short), and point out novel connections of FDPs to the following two cryptographic topics; theoretical security evaluations of keyless hash functions (such as SHA-1), and constructions of provably secure pseudorandom generators (PRGs) with some enhanced security property introduced by Dubrov and Ishai (STOC 2006). Our argument aims at proposing new theoretical frameworks for these topics (especially for the former) based on FDPs, rather than providing some con-crete and practical results on the topics. We also give some examples of mathematical discussions on FDPs, which would be of independent interest from mathematical viewpoints. Finally, we discuss possible directions of future research on other crypto-graphic applications of FDPs and on mathematical studies on FDPs themselves.

      DOI: 10.1142/S0129054115500100

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    • Simple-root bases for Shi arrangements Peer-reviewed

      Takuro Abe, Hiroaki Terao

      JOURNAL OF ALGEBRA422   89 - 104   1 2015

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      In his affirmative answer to the Edelman-Reiner conjecture, Yoshinaga proved that the logarithmic derivation modules of the cones of the extended Shi arrangements are free modules. However, we know very little about the bases themselves except their existence. In this article, we prove the unique existence of two distinguished bases which we call the simple-root basis plus (SRB+) and the simple-root basis minus (SRB-). They are characterized by nice divisibility properties relative to the simple roots. (C) 2014 Elsevier Inc. All rights reserved.

      DOI: 10.1016/j.jalgebra.2014.09.011

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    • Roots of characteristic polynomials and intersection points of line arrangements. Peer-reviewed

      Takuro Abe

      Journal of Singularities8   100 - 117   10 2014

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      DOI: 10.5427/jsing.2014.8h

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    • Free arrangements and coefficients of characteristic polynomials Peer-reviewed

      Takuro Abe, Masahiko Yoshinaga

      Mathematische Zeitschrift275 ( 3-4 ) 911 - 919   12 2013

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      Ziegler showed that the multirestriction of a free arrangement is also free. After Ziegler's work, several results concerning the "reverse direction", i.e., characterizing freeness of an arrangement via that of its multirestriction, have appeared. In this paper, we prove a new characterization of freeness in which the second Betti number of the arrangement plays a crucial role. © 2013 Springer-Verlag Berlin Heidelberg.

      DOI: 10.1007/s00209-013-1165-6

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    • Chambers of 2-affine arrangements and freeness of 3-arrangements Peer-reviewed

      Takuro Abe

      JOURNAL OF ALGEBRAIC COMBINATORICS38 ( 1 ) 65 - 78   8 2013

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      We give a lower bound on the number of chambers divided by a family of lines in the real plane. Also, a sufficient condition for 3-arrangements to be free is given in terms of chambers. To prove them, we give the upper bound on differences of exponents for balanced 2-multiarrangements in terms of the cardinality of hyperplanes.

      DOI: 10.1007/s10801-012-0393-x

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    • Characteristic Polynomials, eta-Complexes, and Freeness of Tame Arrangements Peer-reviewed

      Takuro Abe

      MICHIGAN MATHEMATICAL JOURNAL62 ( 1 ) 117 - 130   2013

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    • Equivariant multiplicities of Coxeter arrangements and invariant bases Peer-reviewed

      Takuro Abe, Hiroaki Terao, Atsushi Wakamiko

      ADVANCES IN MATHEMATICS230 ( 4-6 ) 2364 - 2377   7 2012

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      Let A be an irreducible Coxeter arrangement and W be its Coxeter group. Then W naturally acts on A. A multiplicity in : A -&gt; Z is said to be equivariant when in is constant on each W-orbit of A. in this article, we prove that the multi-derivation module D(A, m) is a free module whenever in is equivariant by explicitly constructing a basis, which generalizes the main theorem of Terao (2002) [12]. The main tool is a primitive derivation and its covariant derivative. Moreover, we show that the W-invariant part D(A, m)(W) for any multiplicity m is a free module over the W-invariant subring. (C) 2012 Elsevier Inc. All rights reserved.

      DOI: 10.1016/j.aim.2012.04.015

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    • Exponents of 2-multiarrangements and multiplicity lattices Peer-reviewed

      Takuro Abe, Yasuhide Numata

      JOURNAL OF ALGEBRAIC COMBINATORICS35 ( 1 ) 1 - 17   2 2012

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      We introduce a concept of multiplicity lattices of 2-multiarrangements, determine the combinatorics and geometry of that lattice, and give a criterion and method to construct a basis for derivation modules effectively.

      DOI: 10.1007/s10801-011-0291-7

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    • ON A CONJECTURE OF ATHANASIADIS RELATED TO FREENESS OF A FAMILY OF HYPERPLANE ARRANGEMENTS Peer-reviewed

      Takuro Abe

      MATHEMATICAL RESEARCH LETTERS19 ( 2 ) 469 - 474   2012

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      We prove a characterization of freeness, conjectured by Athanasiadis, for the family of hyperplane arrangements, which lie between the Coxeter and the Catalan arrangement of the type A(l). One direction was already proved in [2]. Here we prove the other direction

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    • The freeness of Shi-Catalan arrangements Peer-reviewed

      Takuro Abe, Hiroaki Terao

      EUROPEAN JOURNAL OF COMBINATORICS32 ( 8 ) 1191 - 1198   11 2011

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      Let W be a finite Weyl group and A be the corresponding Weyl arrangement. A deformation of A is an affine arrangement which is obtained by adding to each hyperplane H is an element of A several parallel translations of H by the positive root (and its integer multiples) perpendicular to H. We say that a deformation is W-equivariant if the number of parallel hyperplanes of each hyperplane H is an element of A depends only on the W-orbit of H. We prove that the conings of the W-equivariant deformations are free arrangements under a Shi-Catalan condition and give a formula for the number of chambers. This generalizes Yoshinaga&apos;s theorem conjectured by Edelman-Reiner. (C) 2011 Elsevier Ltd. All rights reserved.

      DOI: 10.1016/j.ejc.2011.06.005

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    • Primitive filtrations of the modules of invariant logarithmic forms of Coxeter arrangements Peer-reviewed

      Takuro Abe, Hiroaki Terao

      JOURNAL OF ALGEBRA330 ( 1 ) 251 - 262   3 2011

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      We define primitive derivations for Coxeter arrangements which may not be irreducible. Using those derivations, we introduce the primitive filtrations of the module of invariant logarithmic differential forms for an arbitrary Coxeter arrangement with an arbitrary multiplicity. In particular, when the Coxeter arrangement is irreducible with a constant multiplicity, the primitive filtration was studied in Abe and Terao (2010) [2], which generalizes the Hodge filtration introduced by K. Saito (e.g., Saito, 2003 [6]). (C) 2010 Elsevier Inc. All rights reserved.

      DOI: 10.1016/j.jalgebra.2010.12.010

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    • A primitive derivation and logarithmic differential forms of Coxeter arrangements Peer-reviewed

      Takuro Abe, Hiroaki Terao

      MATHEMATISCHE ZEITSCHRIFT264 ( 4 ) 813 - 828   4 2010

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      Let W be a finite irreducible real reflection group, which is a Coxeter group. We explicitly construct a basis for the module of differential 1-forms with logarithmic poles along the Coxeter arrangement by using a primitive derivation. As a consequence, we extend the Hodge filtration, indexed by nonnegative integers, into a filtration indexed by all integers. This filtration coincides with the filtration by the order of poles. The results are translated into the derivation case.

      DOI: 10.1007/s00209-009-0489-8

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    • Coxeter multiarrangements with quasi-constant multiplicities Peer-reviewed

      Takuro Abe, Masahiko Yoshinaga

      JOURNAL OF ALGEBRA322 ( 8 ) 2839 - 2847   10 2009

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      We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant multiplicity is combinatorially computable. (C) 2009 Elsevier Inc. All rights reserved.

      DOI: 10.1016/j.jalgebra.2009.07.025

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    • Signed-eliminable graphs and free multiplicities on the braid arrangement Peer-reviewed

      Takuro Abe, Koji Nuida, Yasuhide Numata

      JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES80   121 - 134   8 2009

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:OXFORD UNIV PRESS  

      We define specific multiplicities on the braid arrangement by using signed graphs. To consider their freeness, we introduce the notion of signed-eliminable graphs as a generalization of Stanley's classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of certain deformations of the braid arrangement in terms of directed graphs.

      DOI: 10.1112/jlms/jdp019

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    • The Stability of the Family of B2-Type Arrangements Peer-reviewed

      Takuro Abe

      COMMUNICATIONS IN ALGEBRA37 ( 4 ) 1193 - 1215   2009

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      We introduce the family of B2-type arrangements as a generalization of the classical Coxeter arrangement of type B2 and consider the stability and the freeness of it. We show the freeness and (semi)stability are determined by the combinatorics. Moreover, we give a partial answer to the 4-shift problem, which is a conjecture on the combinatorics and geometry induced from the B2-type arrangements.

      DOI: 10.1080/00927870802465969

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    • TOTALLY FREE ARRANGEMENTS OF HYPERPLANES Peer-reviewed

      Takuro Abe, Hiroaki Terao, Masahiko Yoshinaga

      PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY137 ( 4 ) 1405 - 1410   2009

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      A central arrangement A of hyperplanes in an l-dimensional vector space V is said to be totally free if a multiarrangement (A, m) is free for any multiplicity m : A -&gt; Z(&gt; 0). It has been known that A is totally free whenever l &lt;= 2. In this article, we will prove that there does not exist any totally free arrangement other than the obvious ones, that is, a product of one-dimensional arrangements and two-dimensional ones.

      DOI: 10.1090/S0002-9939-08-09755-4

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    • The Euler multiplicity and addition-deletion theorems for multiarrangements Peer-reviewed

      Takuro Abe, Hiroaki Terao, Max Wakefield

      JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES77   335 - 348   4 2008

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:CAMBRIDGE UNIV PRESS  

      The addition-deletion theorems for hyperplane arrangements, which were originally shown by Terao [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980) 293-320.], provide useful ways to construct examples of free arrangements. In this article, we prove addition-deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the Euler multiplicity, of a restricted multiarrangement. We compute the Euler multiplicities in many cases. Then we apply the addition-deletion theorems to various arrangements, including supersolvable arrangements and the Coxeter arrangement of type A(3), to construct free and non-free multiarrangements.

      DOI: 10.1112/jlms/jdm110

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    • Splitting criterion for reflexive sheaves Peer-reviewed

      Takuro Abe, Masahiko Yoshinaga

      PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY136 ( 6 ) 1887 - 1891   2008

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      We give a criterion for a reflexive sheaf to split into a direct sum of line bundles.

      DOI: 10.1090/S0002-9939-08-09367-2

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    • The characteristic polynomial of a multiarrangement Peer-reviewed

      Takuro Abe, Hiroaki Terao, Max Wakefield

      ADVANCES IN MATHEMATICS215 ( 2 ) 825 - 838   11 2007

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:ACADEMIC PRESS INC ELSEVIER SCIENCE  

      Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates 'global' data to 'local' data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free. (c) 2007 Elsevier Inc. All rights reserved.

      DOI: 10.1016/j.aim.2007.04.019

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    • Free and non-free multiplicity on the deleted A(3) arrangement Peer-reviewed

      Takuro Abe

      PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES83 ( 7 ) 99 - 103   10 2007

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      Language:English   Publishing type:Research paper (scientific journal)   Publisher:JAPAN ACAD  

      We give the first complete classification of free and non-free multiplicities on an arrangement, called the deleted A(3) arrangement, which admits both of them.

      DOI: 10.3792/pjaa.83.99

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    • The elementary transformation of vector bundles on regular schemes Peer-reviewed

      Takuro Abe

      TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY359 ( 9 ) 4285 - 4295   2007

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      We give a generalized definition of an elementary transformation of vector bundles on regular schemes by using Maximal Cohen-Macaulay sheaves on divisors. This definition is a natural extension of that given by Maruyama, and has a connection with that given by Sumihiro. By this elementary transformation, we can construct, up to tensoring line bundles, all vector bundles from trivial bundles on nonsingular quasi-projective varieties over an algebraically closed field. Moreover, we give an application of this theory to reflexive sheaves.

      DOI: 10.1090/S0002-9947-07-04161-X

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    • The stability of the family of A<sub>2</sub>-type arrangements Peer-reviewed

      T. Abe

      Journal of Mathematics of Kyoto University46 ( 3 ) 617 - 636   2006

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    Misc.

    • A Mathematical Problem for Security Analysis of Hash Functions and Pseudorandom Generators. Peer-reviewed

      Koji Nuida, Takuro Abe, Shizuo Kaji, Toshiaki Maeno, Yasuhide Numata

      Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)7038 LNCS   144 - 160   2011

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    Presentations

    • Hessenberg variety and Solomon-Terao algebra of hyperplane arrangements Invited

      Takuro Abe

      16 11 2024 

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      Event date: 16 11 2024 - 18 11 2024

      Language:Japanese   Presentation type:Oral presentation (general)  

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    • Cokernel of the Euler restriction map of projective line arrangements Invited International conference

      Takuro Abe

      One-day workshop "Bundles and lattices"  25 8 2024 

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      Event date: 25 8 2024 - 25 8 2024

      Language:English   Presentation type:Oral presentation (general)  

      Venue:Osaka University  

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    • Double points of line arrangements in the complex projective plane Invited International conference

      Takuro Abe

      Suzhou Workshop on Matroid Theory  10 7 2024  Soochow University, University of Science and Technology of China

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      Event date: 8 7 2024 - 12 7 2024

      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Soochow University  

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    • Double Points of Line Arrangements in the Complex Projective Plane International conference

      Takuro Abe

      BICMR Lecture - Algebraic Geometry  14 5 2024  BICMR

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      Event date: 14 5 2024 - 14 5 2024

      Language:English   Presentation type:Public lecture, seminar, tutorial, course, or other speech  

      Venue:University of Beijing  

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    • Multi-Euler derivation

      Abe Takuro

      Multiarrangements of type B_2 and related topics  27 3 2024 

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      Event date: 26 3 2024 - 27 3 2024

      Language:Japanese   Presentation type:Oral presentation (general)  

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    • Regularity of logarithmic modules and Solomon-Terao polynomials of hyperplane arrangements

      Abe, Takuro

      Semianr of Algebra  27 2 2024  University of Sevilla

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      Event date: 27 2 2024 - 27 2 2024

      Language:English   Presentation type:Public lecture, seminar, tutorial, course, or other speech  

      Venue:University of Sevilla  

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    • Multi-Euler derivations Invited

      Takuro Abe

      OBERSEMINAR "ARRANGEMENTS AND SYMMETRIES"  15 1 2024  Ruhr University Bochum

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      Event date: 15 1 2024 - 15 1 2024

      Language:English   Presentation type:Public lecture, seminar, tutorial, course, or other speech  

      Venue:Ruhr University Bochum  

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    • Projective dimensions of weakly chordal graphic arrangements Invited

      Takuro Abe

      JSPS-VAST Bilateral Joint Research Project Workshop "Topology of Singularities and Related Topics"  14 9 2023 

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      Event date: 11 9 2023 - 15 9 2023

      Language:English   Presentation type:Oral presentation (invited, special)  

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    • Algebraic understanding of integer roots of characteristic polynomials of hyperplane arrangements

      Order Seminar  3 7 2023 

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      Event date: 3 7 2023 - 3 7 2023

      Language:English   Presentation type:Public lecture, seminar, tutorial, course, or other speech  

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    • B-sequences of logarithmic modules of hyperplane arrangements Invited International conference

      Takuro Abe

      Characteristic Polynomials of Hyperplane Arrangements and Ehrhart Polynomials of Convex Polytopes  16 2 2023  RIMS

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      Event date: 14 2 2023 - 17 2 2023

      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:RIMS  

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    • B-sequences of the logarithmic modules International conference

      Takuro Abe

      Combinatorics, geometry and commutative algebra of hyperplane arrangements  16 1 2023 

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      Event date: 16 1 2023 - 20 1 2023

      Language:English   Presentation type:Oral presentation (general)  

      Venue:Kyushu University  

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    • Addition-restriction theorems and projective dimensions of logarithmic vector fields of hyperplane arrangements Invited

      Takuro Abe

      Project seminar — Moduli and Friends (online)  24 10 2022 

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      Event date: 24 10 2022 - 24 10 2022

      Language:English   Presentation type:Public lecture, seminar, tutorial, course, or other speech  

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    • Sylvester-Gallai theorem and the splitting type of logarithmic vector fields Invited

      Takuro Abe

      20 10 2022 

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      Event date: 18 10 2022 - 21 10 2022

      Language:Japanese   Presentation type:Oral presentation (invited, special)  

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    • Addition theorem for free arrangements, revisited

      Takuro Abe

      超平面配置の数学とその進展2022  16 2 2022 

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      Event date: 16 2 2022 - 16 2 2022

      Language:Japanese   Presentation type:Oral presentation (general)  

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    • Logarithmic modules of hyperplane arrangements and their freeness Invited

      Takuro Abe

      JSPS-VAST Bilateral Joint Research Project Workshop "Singularities, arrangements, and low-dim. topology"  10 12 2021 

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      Event date: 6 12 2021 - 10 12 2021

      Language:English   Presentation type:Oral presentation (invited, special)  

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    • Double points and free deletions of plane arrangements

      Takuro Abe

      Discrete Geometric Structure Seminar  17 11 2021 

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      Event date: 17 11 2021 - 17 11 2021

      Language:Japanese   Presentation type:Public lecture, seminar, tutorial, course, or other speech  

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    • Projective dimension of logarithmic modules of hyperplane arrangements Invited

      Takuro Abe

      RIMS x OIST Jointly-funded Workshop: "Interactions of New Trends in Algebraic Geometry and Singularities"  11 10 2021 

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      Event date: 11 10 2021 - 15 10 2021

      Language:English   Presentation type:Oral presentation (invited, special)  

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    • Logarithmic vector fields and differential forms revisited Invited International coauthorship International conference

      Takuro Abe

      Oberseminar Arrangements and Symmetries  12 7 2021  Ruhr University Bochum

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      Event date: 12 7 2021 - 12 7 2021

      Language:English   Presentation type:Public lecture, seminar, tutorial, course, or other speech  

      Venue:online  

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    • Projective dimensions and addition-deletion theorems for hyperplane arrangements Invited International conference

      Takuro Abe

      Arrangements at Home, III: Algebraic Aspects  14 8 2020 

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      Event date: 13 8 2020 - 14 8 2020

      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:online  

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    • Double points and freeness of line arrangements in the projective plane Invited International conference

      Takuro Abe

      Afternoon Seminars on Hyperplanes on line  26 6 2020  Shinshu University

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      Event date: 26 6 2020 - 26 6 2020

      Language:English   Presentation type:Oral presentation (general)  

      Venue:online  

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    • Sylvester's problem and logarithmic vector fields of projective line arrangements

      Takuro Abe

      Seminaires de Mathematiques et de leurs Applications  18 2 2020  University of Pau and Pays de l'Adour, Pau, France

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    • 複素数体上の超可解配置の Dirac-Motzkin 予想と対数的ベクト ル場の分裂型 Invited

      阿部 拓郎

      第7回 代数幾何学研究集会-宇部-  11 1 2020  宇部工業高等専門学校

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      Event date: 11 1 2020 - 12 1 2020

      Language:Japanese   Presentation type:Oral presentation (general)  

      Venue:山口県宇部市  

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    • Free arrangements, combinatorics and geometry Invited International conference

      Takuro Abe

      Hyperplane Arrangements and Singularities  6 12 2019  University of Tokyo

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      Event date: 2 12 2019 - 6 12 2019

      Language:English   Presentation type:Oral presentation (general)  

      Venue:University of Tokyo  

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    • Recent developments on free arrangements and combinatorics

      Takuro Abe

      Seminar of Algebra  16 10 2019  Universidad de Sevilla

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      Language:English   Presentation type:Public lecture, seminar, tutorial, course, or other speech  

      Venue:Sevilla  

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    • Free arrangements, restrictions and related topics Invited International conference

      Takuro Abe

      Hyperplane Arrangements and Reflection Groups  27 9 2019  Leibniz Universität Hannover

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Leibniz Universität Hannover, Germany  

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    • Solomon-Terao algebra of hyperplane arrangements and singularities Invited International conference

      Takuro Abe

      Special Session on Geometry and Topology of Singularities  15 9 2019  University of Wisconsin-Madison

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:University of Wisconsin-Madison, Madison  

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    • Around free arrangements and Solomon-Terao algebras Invited

      Takuro Abe

      Hyperplane Arrangements in Wakkanai  23 8 2019 

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    • Recent topics on free arrangements Invited International conference

      Takuro Abe

      Arrangements at Western  23 5 2019  Western University

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:London, Canada  

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    • Combinatorics of the addition-deletion theorems for free arrangements International conference

      Takuro Abe

      CIMPA - IMH Research School HYPERPLANE ARRANGEMENTS: RECENT ADVANCES AND OPEN PROBLEMS  21 3 2019  VAST and CIMPA

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:Hanoi  

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    • Combinatorics of the addition-deletion theorems for arrangements Invited International conference

      Takuro Abe

      On hyperplane arrangements, configuration spaces and related topics  21 2 2019  Hokkaido University

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:北海道大学  

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    • Logarithmic vector fields and freeness of hyperplane arrangements (three times talks) Invited International conference

      Takuro Abe

      Free divisors and Hyperplane arrangements  18 12 2018  University Paris Diderot, Paris, France

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:University Paris Diderot  

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    • Hessenberg varieties and hyperplane arrangements Invited International conference

      Takuro Abe

      Hessenberg varieties 2018 in Osaka  13 12 2018  Osaka City University

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:Osaka City University  

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    • Combinatorial dependency of the addition-deletion theorems for hyperplane arrangements Invited

      Takuro Abe

      29 11 2018 

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      Language:Japanese   Presentation type:Public lecture, seminar, tutorial, course, or other speech  

      Venue:Shinshu University  

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    • Hessenbergs and hyperplane arrangements part II Invited International conference

      Takuro Abe

      Hessenberg Varieties in Combinatorics, Geometry and Representation Theory  25 10 2018  BIRS, Banff, Canada

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:BIRS, Banff, Canada  

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    • Logarithmic vector fields and freeness of hyperplane arrangements Invited

      Takuro Abe

      日本数学会秋季総合分科会  27 9 2018  MSJ

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:Okayama University  

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    • The b_2-equality and free arrangements Invited International conference

      Takuro Abe

      New perspectives in hyperplane arrangements  14 9 2018  Ruhr University Bochum, Bochum, Germany.

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Ruhr University Bochum, Bochum, Germany.  

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    • 超平面配置の代数・自由性と組み合わせ論 Invited

      Takuro Abe

      組合せ論サマースクール2018  9 8 2018 

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

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    • Free arrangements of hyperplanes and applications (three times talks) Invited International conference

      Takuro Abe

      Workshop on Algebraic Geometry  26 6 2018  Academy of Mathematics and Systems Science, Beijing, China

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Beijing, China  

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    • Generators for logarithmic derivation modules of hyperplane arrangements Invited International conference

      Takuro Abe

      Topology and Geometry: A conference in memory of Stefan Papadima (1953-2018)  29 5 2018  Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:Bucharest, Romania  

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    • Generators of logarithmic derivation modules of hyperplane arrangements Invited International conference

      Takuro Abe

      Arrangements of Hypersurfaces  21 4 2018  AMS and Northeastern University, Boston, USA

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Northeastern University, Boston, USA  

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    • Poincare polynomials and free arrangements Invited International conference

      Takuro Abe

      A walk between hyperplane arrangements, computer algebra and algorithms  2 2 2018  Hokkaido University

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:Sapporo, Japan  

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    • Solomon-Terao algebra of hyperplane arrangements Invited International conference

      Takuro Abe

      Hyperplane arrangement and representation stability  19 1 2018  MFO, Oberwolfach, Germany

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Oberwolfach, Germany  

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    • Solomon-Terao algebra of hyperplane arrangements Invited International conference

      Takuro Abe

      Toric Topology 2017 in Osaka  11 12 2017  Osaka City University

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      Language:English   Presentation type:Oral presentation (general)  

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    • 自由配置に関する近年の進展とその応用 Invited

      Takuro Abe

      10 11 2017  Hokkaido University

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      Language:Japanese   Presentation type:Oral presentation (general)  

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    • Hyperplane arrangements and Hessenberg varieties Invited International conference

      Takuro Abe

      The 5th Franco-Japan-Vietnamese Symposium on Singularities  2 11 2017  University of Kagoshima

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      Language:English   Presentation type:Oral presentation (general)  

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    • Freeness and combinatorics of hyperplane arrangements

      Takuro Abe

      Seminaires de Mathematiques et de leurs Applications  21 9 2017  University of Pau and Pays de l'Adour, Pau, France

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:University of Pau and Pays de l'Adour, Pau, France  

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    • Free arrangements and vector bundles International conference

      Takuro Abe

      VIIIe rencontre Pau-Zaragoza d'Algebra et Geometrie  14 9 2017  University of Pau and Pays de l'Adour, Pau, France

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:University of Pau and Pays de l'Adour, Pau, France  

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    • The $b_2$-inequality and freeness of the restrictions of hyperplane arrangements Invited International conference

      Takuro Abe

      Advances in Hyperplane Arrangements  16 8 2017  University of Bremen, Bremen, Germany

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:University of Bremen, Bremen, Germany  

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    • Hyperplane arrangements and Hessenberg varieties Invited International conference

      Takuro Abe

      Advances in Arrangement Theory  26 7 2017  CMS, SMC

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:Centre Mont-Royal, Montreal, Canada  

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    • Hyperplane arrangements and Hessenberg varieties Invited International conference

      Takuro Abe

      Arrangements and beyond  7 6 2017  Centro di Ricerca Matematica Ennio De Giorgi, Pisa, Italy

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:Centro di Ricerca Matematica Ennio De Giorgi, Pisa, Italy  

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    • Hyperplane arrangments, Solomon-Terao algebras and applications to Hessenberg varieties Invited

      Takuro Abe

      26 5 2017  RIMS

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:Kyoto  

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    • Algebra and geometry of Solomon-Terao's formula Invited International conference

      Takuro Abe

      Hyperplane Arrangements and related topics  14 2 2017  Hokkaido University

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      Language:English   Presentation type:Oral presentation (general)  

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    • Algebra and combinatorics of hyperplane arrangements Invited International conference

      Takuro Abe

      The 15th Japan-Korea Workshop on Algebra and Combinatorics  10 2 2017  Kumamoto University

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      Language:English   Presentation type:Oral presentation (general)  

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    • Divisionally free arrangements of hyperplanes Invited

      Takuro Abe

      第61回代数学シンポジウム  9 9 2016 

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

      Venue:Saga University  

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    • Recent topics on free arrangements of hyperplanes Invited International conference

      Takuro Abe

      Summer Conference on Hyperplane Arrangements(SCHA) in Sapporo  11 8 2016  Hokkaido University

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:Sapporo, Japan  

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    • Freeness and flags of hyperplane arrangements Invited International conference

      Takuro Abe

      Special Session on Topology and Combinatorics of Arrangements (in honor of Mike Falk)  3 2016  AMS Sectional Meeting, at Stony Brook University, USA.

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      Language:English   Presentation type:Oral presentation (invited, special)  

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    • Divisionally free arrangements of hyperplanes Invited

      Takuro Abe

      26 11 2015  Fukuoka University

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      Language:Japanese   Presentation type:Oral presentation (general)  

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    • Divisional freeness and the second Betti number of hyperplane arrangements Invited International conference

      Takuro Abe

      Hyperplane arrangements and reflection groups  11 8 2015  Leibniz Universität Hannover

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:Hannover, Germany  

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    • Divisionally free arrangements of hyperplanes(講演) Invited International conference

      Takuro Abe

      Differential and combinatorial aspects of singularities  6 8 2015  Technische Universitat Kaiserslautern

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Kaiserslautern, Germany  

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    • Division free theorem for line arrangements and divisionally free arrangements of hyperplanes Invited International conference

      Takuro Abe

      Arrangements of plane curves and related problems  17 3 2015  Tokyo Metropolitan University

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:Tokyo Metropolitan University  

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    • Division and localization of characteristic polynomials of hyperplane arrangements" Invited International conference

      Takuro Abe

      Combinatorics and Algebraic Topology of Configurations  19 2 2015  Centro De Giorgi

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:Pisa, Italy  

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    • 直線配置の数え上げ幾何とベッチ数 Invited

      阿部 拓郎

      信州大学数理科学談話会  6 2 2015 

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      Language:Japanese   Presentation type:Oral presentation (general)  

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    • Intersection points, chambers and the freeness of line arrangements

      Takuro Abe

      Algebraic and Complex geometry seminar at Osaka University  19 12 2014 

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      Language:Japanese   Presentation type:Oral presentation (general)  

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    • Freeness and Geometry of line arrangements Invited International conference

      Takuro Abe

      Hokkaido University - University of Bremen Joint Symposium Special session:Arrangements of Hyperplanes  3 12 2014  University of Bremen and Hokkaido University

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Bremen, Germany  

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    • The freeness of ideal subarrangements of Weyl arrangements

      Takuro Abe

      Seminarie Geometrie et Systemes Dynamiques  28 11 2014  Université de Bourgogne

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      Language:English   Presentation type:Oral presentation (general)  

      Venue:Dijon, France  

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    • Algebra and Combinatorics of line arrangements 1 and 2

      Takuro Abe

      組合せ論と可換代数サマーセミナー2014  24 9 2014 

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      Language:Japanese   Presentation type:Oral presentation (general)  

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    • Stability of line arrangements in the projective plane International conference

      Takuro Abe

      The 1st Workshop of JSPS-MAE Sakura Program ``Geometry and Combinatorics of Hyperplane Arrangements and Related Problems''  2 9 2014  Masahiko Yoshinaga (Hokkaido University)

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:Hokkaido University  

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    • Recent developments on algebra of line arrangements Invited International conference

      Takuro Abe

      The 2nd Franco-Japanese-Vietnamese Symposium on Singularities  27 8 2014  CNRS-JSPS-VAST

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:University of Hokkaido  

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    • Intersection points and Betti numbers of line arrangements Invited International conference

      Takuro Abe

      Special Session ``D-modules and Singularities'' at the First Joint International Meeting RSME-SCM-SEMA-SIMAI-UMI  1 7 2014  RSME-SCM-SEMA-SIMAI-UMI

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      Language:English   Presentation type:Oral presentation (invited, special)  

      Venue:The University of the Basque Country  

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    • Intersection points and Betti numbers of line arrangements

      Japan Conference on Graph Theory and Combinatorics  20 5 2014  KAWARABAYASHI Large Graph project and Nihon University

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      Language:English   Presentation type:Other  

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    • Intersection points and Betti numbers of the complements of line arrangements Invited International conference

      Takuro Abe

      Branched Coverings, Degenerations, and Related Topics 2014  10 3 2014  広島大学

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      Language:English   Presentation type:Oral presentation (general)  

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    • The freeness of ideal subarrangements of Weyl arrangements Invited International conference

      Takuro Abe

      New perspectives in hyperplane and reflection arrangements  10 2 2014  Ruhr University Bochum

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      Language:English   Presentation type:Oral presentation (general)  

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    • 平面直線の交点と補空間のベッチ数

      第5回 代数曲面ワークショップ at 秋葉原  26 10 2013  首都大学東京

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

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    • 平面中の直線配置の交点数と補空間のベッチ数

      情報セキュリティと関連する離散数学  9 8 2013  信州大学

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      Language:Japanese   Presentation type:Other  

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    • Roots of characteristic polynomials and intersection points of line arrangements International conference

      Takuro Abe

      Geometry and Topology of Arrangements of Hypersurfaces  29 6 2013  American Mathematical Society, Romanian Mathematical Society

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      Language:English   Presentation type:Oral presentation (invited, special)  

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    • 超平面配置の代数と組み合わせ論

      大阪組み合わせ論セミナー  15 6 2013  大阪市立大学

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      Language:Japanese   Presentation type:Other  

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    • Height bases for Weyl arrangements Invited International conference

      Takuro Abe

      Hyperplane Arrangements: Combinatorial and Geometric Aspects  21 2 2013  Ruhr University Bochum, Germany

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      Language:English   Presentation type:Oral presentation (invited, special)  

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    • Height basis for Weyl arrangements

      Workshop on Free Divisors and Differential Equations  8 11 2012  Tokyo University of Agriculture and Technology

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      Language:English   Presentation type:Oral presentation (invited, special)  

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    • 関数密度問題とハッシュ関数の安全性評価

      情報セキュリティ基盤の数理構造と安全性解析  5 9 2012  九州大学マス・フォア・インダストリ研究所

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

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    • Digraphs and free deformations between Coxeter and Catalan arrangements of the type A

      代数幾何の関連する諸分野  8 8 2012  北海道大学

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

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    • 超平面配置の代数と幾何

      関数方程式サマーセミナー  2 8 2011 

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      Language:Japanese   Presentation type:Oral presentation (invited, special)  

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    • Chambers and freeness of line arrangements

      Workshop on free divisors  31 5 2011  University of Warwick

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      Language:English   Presentation type:Oral presentation (invited, special)  

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    • Free Coxeter arrangements and applications

      The 2nd Kyushu University-POSTECH Joint Workshop --Algebraic Geometry and Related Topics--  18 3 2011  九州大学

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      Language:English   Presentation type:Oral presentation (invited, special)  

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    • Exponents and freeness of 3-arrangements

      特異点と多様体の幾何学  16 9 2010  山形大学

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      Language:English   Presentation type:Oral presentation (invited, special)  

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    • 原始微分とコクセター型多重配置

      RIMS研究集会``表現論と組合わせ論''  27 8 2009  北海道大学

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      Language:English   Presentation type:Oral presentation (invited, special)  

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    • A logarithmic module of a generalized multiarrangement

      The 6th HU and SNU Symposium on Mathematics --Geometry and Topology-- as a part of The 11th HU-SNU Joint Symposium  8 11 2008  北海道大学

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      Language:English   Presentation type:Oral presentation (invited, special)  

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    • Coxeter multiarrangements with quasi-constant multiplicities

      Geometry and Analysis on Complex Varieties  21 11 2007  東京農業工業大学

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      Language:English   Presentation type:Oral presentation (invited, special)  

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    • The addition-deletion theorems for multiarrangements and graphic deformation of the braid arrangement

      CIMPA summer school on arrangements and local systems  18 6 2007  CIMPA, Galatasaray University

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      Language:English   Presentation type:Oral presentation (invited, special)  

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    • Construction of rank two reflexive sheaves from finite fields

      Geometry and Analysis on Complex Algebraic Geometry  11 12 2006  RIMS

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      Language:English   Presentation type:Oral presentation (invited, special)  

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    • The stability of A_2 and B_2-type arrangements Invited International conference

      Takuro Abe

      Recent Developments in Arrangements and Configulation Spaces  10 8 2006  MSRI, Berkeley

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      Language:English   Presentation type:Oral presentation (invited, special)  

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    Teaching Experience

    • 10 2024 - 3 2025 
      群論入門 ( 立教大学 )

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    • 4 2024 - 9 2024 
      代数学I(環論) ( 立教大学 )

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    • 10 2023 - 3 2024 
      Advanced Linear ALgebra ( Rikkyo University )

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    • 10 2023 - 3 2024 
      Algebra II ( Rikkyo University )

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    • 4 2023 - 9 2023 
      Linear Algebra 2 ( Rikkyo University )

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    • 10 2018 - 3 2019 
      Ordinary Differential equations ( Kyushu University )

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    •  
      代数幾何学大意 ( 九州大学 )

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    •  
      数理モデル概論 ( 九州大学 )

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    •  
      複素関数論 ( 九州大学 )

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    •  
      工業数学F1(複素関数論) ( 京都大学 )

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    •  
      微分積分学続論B(常微分方程式) ( 京都大学 )

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    •  
      微分積分学続論A(ベクトル解析) ( 京都大学 )

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    •  
      微分積分学 ( 京都大学 )

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    •  
      線形代数学続論 ( 京都大学 )

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    •  
      線形代数学 ( 京都大学 )

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    Professional Memberships

    •  
      THE MATHEMATICAL SOCIETY OF JAPAN

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    Research Projects

    • Fusion of enumerative and algebraic geometry, and exploration of quasi-geomtric invariants

      Takuro Abe

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      6 2023 - 3 2029

      Grant number:23K17298

      Authorship:Principal investigator 

      Grant amount:\26000000 ( Direct Cost: \20000000 、 Indirect Cost:\6000000 )

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    • Discrete structures related to hyperplane arrangements, generalization, deepening, and applications

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

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      4 2023 - 3 2028

      Grant number:23H00081

      Grant amount:\46540000 ( Direct Cost: \35800000 、 Indirect Cost:\10740000 )

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    • Research of logarithmic vector fields of hyperplane arrangements

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

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      4 2021 - 3 2026

      Grant number:21H00975

      Grant amount:\17030000 ( Direct Cost: \13100000 、 Indirect Cost:\3930000 )

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    • Fusion of algebra, geometry and combinatorics based on the roots of Poincare polynomials of hyperplane arrangements

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Challenging Research (Exploratory) 

      Takuro Abe

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      7 2020 - 3 2023

      Grant number:20K20880

      Authorship:Principal investigator 

      Grant amount:\6500000 ( Direct Cost: \5000000 、 Indirect Cost:\1500000 )

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    • Explicit study of hyperplane arrangements and related stratified spaces via discrete structures

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

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      4 2018 - 3 2023

      Grant number:18H01115

      Grant amount:\16770000 ( Direct Cost: \12900000 、 Indirect Cost:\3870000 )

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    • D加群を用いた超平面配置とSolomon-寺尾複体の革新的研究

      日本学術振興会  国際共同研究加速基金(国際共同研究強化(A)) 

      阿部 拓郎

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      4 2019 - 3 2022

      Authorship:Principal investigator  Grant type:Competitive

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    • 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

      日本学術振興会  科研費(基盤研究B) 

      阿部 拓郎

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      4 2016 - 3 2021

      Authorship:Principal investigator  Grant type:Competitive

      Grant amount:\17680000 ( Direct Cost: \13600000 、 Indirect Cost:\4080000 )

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    • 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

      日本学術振興会  科研費(挑戦的萌芽研究) 

      阿部 拓郎

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      4 2016 - 3 2018

      Authorship:Principal investigator  Grant type:Competitive

      Grant amount:\3510000 ( Direct Cost: \2700000 、 Indirect Cost:\810000 )

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    • Panoptic Study of Hyperplane arrangements

      Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

      Terao Hiroaki, ABE Takuro, KAMIYA Hidehiko, KOHNO Toshitake, TAKEMURA Akimichi, TERAOSOMA Tomohide, MATSUMOTO Kohji, Cuntz Michael, Hoge Torsten, NAKASHIMA Norihiro, Roehrle Gerhard, Schenck Hal

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      4 2012 - 3 2017

      Grant number:24244001

      Grant amount:\35620000 ( Direct Cost: \27400000 、 Indirect Cost:\8220000 )

      The following three essential themes were selected to grasp the vastness of the research of hyperplane arrangements. The obtained results are as follows:
      (A) An explicit formula for a basis of the derivation module of the Shi arrangements coming from the Weyl arrangement of type A was obtained. In the formula, the Bernoulli polynomials naturally appear. We proved that any ideal subarrangement is free and that the height distribution and the exponents are dual to each other. (B) Hypergeometric integrals are regarded as a period map on the moduli space of arrangements of hyperplanes and can be described in terms of local system cohomology. In this context, a new method was obtained to compute the cohomology groups of the Aomoto complex. (C) The multiple addition theorem was proved. Consequently we learned that the class of hyperplane arrangements in which the so-called Terao conjecture holds true is quite large.

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    • 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

      日本学術振興会  科研費(若手研究(B)) 

      阿部 拓郎

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      4 2012 - 3 2016

      Authorship:Principal investigator  Grant type:Competitive

      Grant amount:\4420000 ( Direct Cost: \3400000 、 Indirect Cost:\1020000 )

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    • 多重配置の自由性の解析と関連する幾何学の創出

      日本学術振興会  科研費(若手研究(B)) 

      阿部 拓郎

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      4 2009 - 3 2012

      Authorship:Principal investigator  Grant type:Competitive

      Grant amount:\3640000 ( Direct Cost: \2800000 、 Indirect Cost:\840000 )

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    • 多重配置の自由性の解析と関連する幾何学の創出

      日本学術振興会  科研費(特別研究員奨励費) 

      阿部 拓郎

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      4 2008 - 3 2009

      Authorship:Principal investigator  Grant type:Competitive

      Grant amount:\800000 ( Direct Cost: \800000 )

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    • 超平面配置上の自由重複度の研究

      日本学術振興会  科学研究費助成事業 

      阿部 拓郎

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      2008 - 2008

      Grant number:20740001

      Grant amount:\1170000 ( Direct Cost: \900000 、 Indirect Cost:\270000 )

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    Academic Activities

    • An editor of ``Commentarii Mathematici Universitatis Sancti Pauli''

      Rikkyo University  2023 - Present

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      Type:Other 

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    • Editor of `` Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial''

      2021 - Present

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      Type:Other 

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    • FPSAC 2024 Program Committee

      Planning, management, etc.

      Ruhr-Universität Bochum  ( Ruhr-Universität Bochum ) 22 7 2024 - 26 7 2024

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      Type:Academic society, research group, etc. 

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    • ICIAM2023 Local Scientific Program Organizer

      Planning, management, etc.

      ICIAM  20 8 2023 - 25 8 2023

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      Type:Academic society, research group, etc. 

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    Social Contribution

    • おもしろサイエンスワールド2024

      Advisor, Organizing member

      立教大学理学部  おもしろサイエンスワールド2024  8 8 2024 - 8 8 2024

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      Type:Science cafe

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    • 「わける」と「数える」がつなぐ数学

      Lecturer

      情報・システム研究機構統計数理研究所、明治大学先端数理科学インスティテュート、九州大学マス・フォア・インダストリ研究所、京都大学数理解析研究所、大阪公立大学数学研究所(主管)  数学・数理科学5研究拠点合同市民講演会「はじける数学!ブレークスルーって何だろう?  大阪公立大学  6 11 2022 - 6 11 2022

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      Audience: Junior students, High school students, General

      Type:Lecture

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    • 超平面配置の数学

      Appearance

      九州大学  2018年度九州大学公開講座「現代数学入門」  12 8 2018 - 12 8 2018

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      Type:Lecture

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    • 平面直線を用いた二次元植木算

      Lecturer

      九州大学  平成29年度九州大学オープンキャンパス数学科紹介プログラム, 企画講演  6 8 2017 - 6 8 2017

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      Type:University open house

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