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New developments on the freeness and topology of hyperplane arrangements, and random walks

Research Project

Project/Area Number 16K13744
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionKyushu University

Principal Investigator

ABE TAKURO  九州大学, マス・フォア・インダストリ研究所, 准教授 (50435971)

Co-Investigator(Kenkyū-buntansha) 白井 朋之  九州大学, マス・フォア・インダストリ研究所, 教授 (70302932)
Research Collaborator Yano Kouji  
Project Period (FY) 2016-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords超平面配置 / 自由配置 / 加除定理 / 旗代数 / 剰余的自由配置 / 剰余旗 / BHR乱歩 / 特性多項式 / 剰余定理 / 乱歩 / 加法定理 / ポアンカレ多項式 / 多数加法・除去・制限定理 / ホモロジー群 / トポロジー / ランダムウォーク
Outline of Final Research Achievements

The aim of this research project is (1) to obtain a good topological understanding of the free hyperplane arrangements, and (2) to relate random walks with free arrangements by using (1). As results for (2), we have only obtained some experimental studies. However, for (1), we obtained better results than expected. Namely, we proved that whether we can apply the addition-deletion theorems for free arrangements or not depends only on the characteristic polynomial, a topological invariant of arrangements. Since characteristic polynomials are also combinatorial invariant, we could show that the addition-deletion theorems are combinatorial, which are very strong results in arrangement theory.

Academic Significance and Societal Importance of the Research Achievements

超平面配置とはベクトル空間中の超平面の有限族である。具体的には、平面中の直線や空間中の平面たちなどが一例となる。これらの様々な性質、例えばこれらが平面や空間をいくつに分割するかなどといった問題を、他の性質と関連付けることが超平面配置の研究である。本研究計画においては、この性質の中でも自由性と呼ばれる極めて良い代数的性質が、新しく超平面を加えたり減らしたりする際にどのように変化するかを考察し、それがトポロジーや組合せ論という、違う側面から完全に統制されていることを証明した。このように、代数的性質が何かで支配されている例は稀であり、超平面配置研究における大きな進展を得たという学術的意義がある。

Report

(4 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report

Research Products

(26 results)

All 2019 2018 2017 2016 Other

All Journal Article (3 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 3 results) Presentation (16 results) (of which Int'l Joint Research: 10 results,  Invited: 12 results) Remarks (7 results)

  • [Journal Article] Deletion theorem and combinatorics of hyperplane arrangements2019

    • Author(s)
      Takuro Abe
    • Journal Title

      Mathematische Annalen

      Volume: 印刷中

    • DOI

      10.1007/s00208-018-1713-9

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Multiple addition, deletion and restriction theorems for hyperplane arrangements2019

    • Author(s)
      Takuro Abe and Hiroaki Terao
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: 印刷中

    • DOI

      10.1090/proc/14592

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Splitting types of bundles of logarithmic vector fields along plane curves2018

    • Author(s)
      Takuro Abe and Alexandru Dimca
    • Journal Title

      International Journal of Mathematics

      Volume: 29

    • DOI

      10.1142/s0129167x18500556

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Combinatorics of the addition-deletion theorems for arrangements2019

    • Author(s)
      Takuro Abe
    • Organizer
      On hyperplane arrangements, configuration spaces and related topics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Combinatorics of the addition-deletion theorems for free arrangements2019

    • Author(s)
      Takuro Abe
    • Organizer
      CIMPA - IMH Research School HYPERPLANE ARRANGEMENTS: RECENT ADVANCES AND OPEN PROBLEMS
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Generators of logarithmic derivation modules of hyperplane arrangements2018

    • Author(s)
      Takuro Abe
    • Organizer
      Arrangements of Hypersurfaces
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Generators for logarithmic derivation modules of hyperplane arrangements2018

    • Author(s)
      Takuro Abe
    • Organizer
      Topology and Geometry: A conference in memory of Stefan Papadima (1953-2018)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 超平面配置の代数・自由性と組み合わせ論2018

    • Author(s)
      阿部拓郎
    • Organizer
      組合せ論サマースクール2018
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] The b_2-equality and free arrangements2018

    • Author(s)
      Takuro Abe
    • Organizer
      New perspectives in hyperplane arrangements
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 超平面配置の対数的ベクトル場と自由性2018

    • Author(s)
      阿部拓郎
    • Organizer
      日本数学会秋季総合分科会特別講演(代数学分科会)
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] 自由超平面配置の加除定理の組み合わせ依存性2018

    • Author(s)
      阿部拓郎
    • Organizer
      数理科学談話会(信州大学)
    • Related Report
      2018 Annual Research Report
  • [Presentation] Poincare polynomials and free arrangements2018

    • Author(s)
      阿部拓郎
    • Organizer
      A walk between hyperplane arrangements, computer algebra and algorithms
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The b_2-inequality and freeness of the restrictions of hyperplane arrangements2017

    • Author(s)
      阿部拓郎
    • Organizer
      Advances in Hyperplane Arrangements
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 自由配置に関する近年の進展とその応用2017

    • Author(s)
      阿部拓郎
    • Organizer
      不変式・超平面配置と平坦構造
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Algebra and combinatorics of hyperplane arrangements2017

    • Author(s)
      阿部拓郎
    • Organizer
      The 15th Japan-Korea Workshop on Algebra and Combinatorics
    • Place of Presentation
      熊本大学
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Divisional flags and freeness of hyperplane arrangements2016

    • Author(s)
      阿部拓郎
    • Organizer
      The Japanese Conference on Combinatorics and its Applications, Mini symposium:Combinatorics of hyperplane arrangements
    • Place of Presentation
      京都大学
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] T. Abe, Divisionally free arrangements of hyperplanes2016

    • Author(s)
      阿部拓郎
    • Organizer
      MS seminar
    • Place of Presentation
      IPMU、東京大学柏キャンパス
    • Related Report
      2016 Research-status Report
  • [Presentation] Recent topics on free arrangements of hyperplanes2016

    • Author(s)
      阿部拓郎
    • Organizer
      Summer Conference on Hyperplane Arrangements(SCHA) in Sapporo
    • Place of Presentation
      北海道大学
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Divisionally free arrangements of hyperplanes2016

    • Author(s)
      阿部拓郎
    • Organizer
      第61回代数学シンポジウム
    • Place of Presentation
      佐賀大学
    • Related Report
      2016 Research-status Report
    • Invited
  • [Remarks] Takuro Abe

    • URL

      https://sites.google.com/site/takuroabemath/home?authuser=0

    • Related Report
      2018 Annual Research Report
  • [Remarks] researchmap

    • URL

      https://researchmap.jp/7000008882

    • Related Report
      2018 Annual Research Report
  • [Remarks] TAKURO ABE

    • URL

      https://sites.google.com/site/takuroabemath/home

    • Related Report
      2017 Research-status Report
  • [Remarks] 九州大学-研究者情報

    • URL

      http://hyoka.ofc.kyushu-u.ac.jp/search/details/K006207/index.html

    • Related Report
      2017 Research-status Report
  • [Remarks] Takuro Abe-Kyushu University

    • URL

      https://kyushu-u.pure.elsevier.com/en/persons/takuro-abe

    • Related Report
      2017 Research-status Report
  • [Remarks] TAKURO ABE

    • URL

      https://www.math.kyoto-u.ac.jp/~abetaku/

    • Related Report
      2016 Research-status Report
  • [Remarks] 九州大学-研究者情報

    • URL

      http://hyoka.ofc.kyushu-u.ac.jp/search/details/K006207/

    • Related Report
      2016 Research-status Report

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Published: 2016-04-21   Modified: 2020-03-30  

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