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Panoptic Study of Hyperplane arrangements

Research Project

Project/Area Number 24244001
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionHokkaido University

Principal Investigator

Terao Hiroaki  北海道大学, 理学(系)研究科(研究院), 名誉教授 (90119058)

Co-Investigator(Kenkyū-buntansha) 吉永 正彦  北海道大学, 理学(系)研究科(研究院), 准教授 (90467647)
Co-Investigator(Renkei-kenkyūsha) ABE Takuro  九州大学, マス・フォア・インダストリ研究所, 准教授 (50435971)
KAMIYA Hidehiko  大阪経済大学, 経済学部, 教授 (50300687)
KOHNO Toshitake  東京大学, 数理科学研究科, 教授 (80144111)
TAKEMURA Akimichi  滋賀大学, データサイエンス教育研究センター, 教授 (10171670)
TERAOSOMA Tomohide  東京大学, 大学院数理科学研究科, 教授 (50192654)
MATSUMOTO Kohji  名古屋大学, 多元数理科学研究科, 教授 (60192754)
Research Collaborator Cuntz Michael  ハノーファー大学, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, 教授
Hoge Torsten  ルール大学ボーフム, 数学部門, 講師
NAKASHIMA Norihiro  東京電機大学, 情報環境学部, 助教 (90732115)
Roehrle Gerhard  ルール大学ボーフム, 数学部門, 教授
Schenck Hal  イリノイ大学, 数学部門, 教授
Project Period (FY) 2012-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥35,620,000 (Direct Cost: ¥27,400,000、Indirect Cost: ¥8,220,000)
Fiscal Year 2016: ¥6,500,000 (Direct Cost: ¥5,000,000、Indirect Cost: ¥1,500,000)
Fiscal Year 2015: ¥5,850,000 (Direct Cost: ¥4,500,000、Indirect Cost: ¥1,350,000)
Fiscal Year 2014: ¥7,150,000 (Direct Cost: ¥5,500,000、Indirect Cost: ¥1,650,000)
Fiscal Year 2013: ¥7,020,000 (Direct Cost: ¥5,400,000、Indirect Cost: ¥1,620,000)
Fiscal Year 2012: ¥9,100,000 (Direct Cost: ¥7,000,000、Indirect Cost: ¥2,100,000)
Keywords代数的組合せ論 / 超平面配置 / 代数学 / 自由配置 / 鏡映群 / ワイル配置
Outline of Final Research Achievements

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.

Report

(6 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • 2013 Annual Research Report
  • 2012 Annual Research Report
  • Research Products

    (51 results)

All 2016 2015 2014 2013 2012 Other

All Journal Article (15 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 14 results,  Open Access: 4 results,  Acknowledgement Compliant: 2 results) Presentation (27 results) (of which Int'l Joint Research: 6 results,  Invited: 26 results) Book (3 results) Remarks (5 results) Funded Workshop (1 results)

  • [Journal Article] Canonical systems of basic invariants for unitary reflection groups.2016

    • Author(s)
      Norihiro Nakashima, Hiroaki Terao, Shuhei Tsujie
    • Journal Title

      Canadian Mathematical Bulletin

      Volume: 59(3) Issue: 3 Pages: 617-623

    • DOI

      10.4153/cmb-2016-031-7

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Logarithmic vector fields for curve configurations in $P^2$ with quasihomogeneous singularities.2016

    • Author(s)
      Hal Schenck, Hiroaki Terao, Masahiko Yoshinaga
    • Journal Title

      Math. Res. Lett.

      Volume: 印刷中

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Worpitzky partitions for root systems and characteristic quasi-polynomials.2016

    • Author(s)
      Masahiko Yoshinaga
    • Journal Title

      Tohoku Math. J.

      Volume: 印刷中

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Free filtrations of affine Weyl arrangements and the ideal-Shi2016

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

      Journal of Algebraic Combinatorics

      Volume: 43 Issue: 1 Pages: 33-44

    • DOI

      10.1007/s10801-015-0624-z

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Degeneration of Orlik-Solomon algebras and Milnor fibers of complex line arrangements2016

    • Author(s)
      Pauline Bailet, Masahiko Yoshinaga
    • Journal Title

      Geometriae Dedicata

      Volume: 未定 Issue: 1 Pages: 49-56

    • DOI

      10.1007/s10711-014-0027-7

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Resonant bands, Aomoto complex, and real 4-nets2015

    • Author(s)
      Michele Torielli, Masahiko Yoshinaga
    • Journal Title

      Journal of Singularities

      Volume: 11 Pages: 33-51

    • DOI

      10.5427/jsing.2015.11b

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Simple-root bases for Shi arrangements2015

    • Author(s)
      T. Abe, H. Terao
    • Journal Title

      Journal of Algebra

      Volume: 402 Pages: 89-104

    • DOI

      10.1016/j.jalgebra.2014.09.011

    • NAID

      120005527841

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] The freeness of ideal subarrangements of Weyl arrangements.2015

    • Author(s)
      Takuro Abe, Mohamed Barakat, Michael Cuntz, Torsten Hoge, Hiroaki Terao
    • Journal Title

      Journal of the European Mathematical Society(European Mathematical Society)

      Volume: 印刷中

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] 論説「超平面配置に関する最近の話題」.2014

    • Author(s)
      寺尾宏明、吉永正彦
    • Journal Title

      数学(日本数学会編集、岩波書店発売)

      Volume: 66 Pages: 157-169

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The freeness of ideal subarrangements of Weyl arrangements2014

    • Author(s)
      Takuro Abe, Mohamed Barakat, Michael Cuntz, Torsten Hoge, Hiroaki Terao
    • Journal Title

      DMTCS Proceedings of FPSAC 2014 (26th International Conference on Formal Power Series and Algebraic Combinatorics)

      Volume: 印刷中

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Weyl 配置のイデアル部分配置の自由性2013

    • Author(s)
      寺尾 宏明
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 1870 Pages: 115-126

    • Related Report
      2013 Annual Research Report
  • [Journal Article] The Shi arrangement of the type $D_\ell$2012

    • Author(s)
      R. Gao, D. Pei, H. Terao
    • Journal Title

      Proceedings of the Japan Academy, Ser. A Mathematical Sciences

      Volume: 88 Issue: 3 Pages: 41-45

    • DOI

      10.3792/pjaa.88.41

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Equivariant multiplicities of Coxeter arrangements and invariant bases2012

    • Author(s)
      T. Abe, H. Terao, A. Wakamiko
    • Journal Title

      Advances in Mathematics

      Volume: 230 Issue: 4-6 Pages: 2364-2377

    • DOI

      10.1016/j.aim.2012.04.015

    • NAID

      120005946712

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Shi arrangements and the Bernoulli polynomials2012

    • Author(s)
      D. Suyama, H. Terao
    • Journal Title

      Bull. London Math. Soc.

      Volume: 44 Issue: 3 Pages: 563-570

    • DOI

      10.1112/blms/bdr118

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Arrangements stable under the Coxeter groups2012

    • Author(s)
      Hidehiko Kamiya, Akimichi Takemura, Hiroaki Terao
    • Journal Title

      ``Configuration Spaces: Geometry, Topology and Combinatorics,'' Proceedings of a special period at the De Giorgi Center, Scuola Normale in Pisa, May - June, 2010, Edizioni Della Normale, Pisa

      Volume: なし Pages: 327-354

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Presentation] Subarrangements and restrictions of Weyl arrangements.2016

    • Author(s)
      Hiroaki Terao
    • Organizer
      the Special Session on "Topology and Combinatorics of Arrangements (in honor of Mike Falk)" at an AMS Spring Eastern Sectional Meeting
    • Place of Presentation
      ストーニーブルック(アメリカ合衆国)
    • Year and Date
      2016-03-19
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Multiple addition theorem on arrangements of hyperplanes and a proof of the Shapiro-Steinberg-Kostant-Macdonald dual-partition formula.2015

    • Author(s)
      Hiroaki Terao
    • Organizer
      Braids, Configuration Spaces, and Quantum Topology
    • Place of Presentation
      東京大学(東京都・目黒区)
    • Year and Date
      2015-09-08
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Before and After the Ideal-free Theorem.2015

    • Author(s)
      Hiroaki Terao
    • Organizer
      Hyperplane arrangements and reflection groups
    • Place of Presentation
      ハノーファー(ドイツ)
    • Year and Date
      2015-08-11
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On Parabolic Subarrangements and Restrictions of Weyl arrangements.2015

    • Author(s)
      Hiroaki Terao
    • Organizer
      Differential and combinatorial aspects of singularities
    • Place of Presentation
      カイザースラウテルン(ドイツ)
    • Year and Date
      2015-08-07
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Ideal Free Theorem and Saturated Free Filtrations os Affine Weyl Arrangements.2015

    • Author(s)
      Hiroaki Terao
    • Organizer
      "Computational Geometric Topology in Arrangement Theory"
    • Place of Presentation
      プロビデンス(アメリカ合衆国)
    • Year and Date
      2015-07-10
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Ideal Free Theorem and Saturated Free Filtrations of Affine Weyl Arrangements.2015

    • Author(s)
      Hiroaki Terao
    • Organizer
      Algebraic structures in combinatorics and geometry
    • Place of Presentation
      RIMS京都大学(京都府・京都市)
    • Year and Date
      2015-06-11
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Arrangements of hyperplanes and applications.2015

    • Author(s)
      Hiroaki Terao
    • Organizer
      KAIST自然科学大学数学科講演会
    • Place of Presentation
      KAIST(韓国)
    • Year and Date
      2015-03-13
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Ideal Free Theorem and Saturated Free Filtrations of Affine Weyl Arrangements.2015

    • Author(s)
      Hiroaki Terao
    • Organizer
      The 10th Kagoshima Algebra-Analysis-Geometry Seminar
    • Place of Presentation
      鹿児島大学(鹿児島市)
    • Year and Date
      2015-02-17
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Free filtrations of affine arrangements of infinitely many hyperplanes.2014

    • Author(s)
      Hiroaki Terao
    • Organizer
      Hokkaido University - University of Bremen Joint Seminar "Arrangements of Hyperplanes"
    • Place of Presentation
      ブレーメン大学(ドイツ)
    • Year and Date
      2014-12-01
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Multiple addition theorem on arrangements of hyperplanes and a proof of the Shapiro-Steinberg-Kostant-Macdonald dual-partition formula.2014

    • Author(s)
      Hiroaki Terao
    • Organizer
      The 1st Workshop of JSPS-MAE Sakura Program "Geometry and Combinatorics of Hyperplane Arrangements and Related Problems"
    • Place of Presentation
      北海道大学(札幌市)
    • Year and Date
      2014-09-03
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Ideals of the roots posets and a new proof of the dual-partition formula by Shapiro-Steinberg-Kostant-Macdonlald.2014

    • Author(s)
      Hiroaki Terao
    • Organizer
      The 2nd Franco-Japanese-Vietnamese Symposium on Singularities
    • Place of Presentation
      北海道大学(札幌市)
    • Year and Date
      2014-08-26
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Chambers of Arrangements and Arrow's Impossibility Theorem.2014

    • Author(s)
      Hiroaki Terao
    • Organizer
      Recent developments on geometrical and algebraic methods in Economics
    • Place of Presentation
      北海道大学(札幌市)
    • Year and Date
      2014-08-22
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On ideal subarrangements of Weyl arrangements.2014

    • Author(s)
      Hiroaki Terao
    • Organizer
      The 13th International Workshop in Real and Complex Singularities
    • Place of Presentation
      Instituto de Ciencias Matematicas e de Computacao Universidade de Sao Paulo(ブラジル)
    • Year and Date
      2014-07-28
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On ideal subarrangements of Weyl arrangements.2014

    • Author(s)
      Hiroaki Terao
    • Organizer
      FPSAC '14 (The 26th International Conference on Formal Power Series and Algebraic Combinatorics)
    • Place of Presentation
      DePaul University(米国)
    • Year and Date
      2014-06-30
    • Related Report
      2014 Annual Research Report
  • [Presentation] Finite Reflection Groups, Free Arrangements, Ideal Subarrangements of Weyl arrangements2013

    • Author(s)
      Hiroaki Terao
    • Organizer
      CIMPA-UNESCO-MESR-MINECO-VIETNAM research school on "Geometry and Topology of Singular Varieties. Theory and Applications
    • Place of Presentation
      the Institute of Mathematics, Vietnam Academy of Science and Technology(ヴェトナム)
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] The freeness of ideal subarrangements of Weyl arrangements

    • Author(s)
      Hiroaki Terao
    • Organizer
      The 6th Pacific RIM Conference on Mathematics 2013
    • Place of Presentation
      札幌コンベンションセンター(札幌市)
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Multiple addition/deletion theorems and the existence of free paths in the affine Weyl arrangements

    • Author(s)
      Hiroaki Terao
    • Organizer
      Combinatorics Seminar
    • Place of Presentation
      名古屋大学(名古屋市)
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] On ideal subarrangements of Weyl arrangements

    • Author(s)
      Hiroaki Terao
    • Organizer
      Hyperplane arrangements and characteristic classes
    • Place of Presentation
      京都大学数理解析研究所(京都市)
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] On ideal subarrangements of Weyl arrangements

    • Author(s)
      Hiroaki Terao
    • Organizer
      Japanese Turkish Joint Geometry Meeting
    • Place of Presentation
      Galatasaray University(トルコ)
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] The freeness of ideal-Shi arrangements and free paths in affine Weyl arrangements (and a question)

    • Author(s)
      Hiroaki Terao
    • Organizer
      New perspectives in hyperplane and reflection arrangements
    • Place of Presentation
      Ruhr-Universitat Bochum(ドイツ)
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Height basis of Weyl arrangements

    • Author(s)
      Hiroaki Terao
    • Organizer
      "Arrangements in Pyrenees" School on hyperplane arrangements and related topics
    • Place of Presentation
      Universite de Pau(フランス)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Primitive derivations, Shi arrangements, Bernoulli polynomials and the height-free conjecture

    • Author(s)
      Hiroaki Terao
    • Organizer
      FPSAC '12 (The 24th International Conference on Formal Power Series and Algebraic Combinatorics)
    • Place of Presentation
      名古屋大学(名古屋市)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Primitive derivations, Shi arrangements and Bernoulli polynomials

    • Author(s)
      Hiroaki Terao
    • Organizer
      2012 Shanghai Conference on Algebraic Combinatorics
    • Place of Presentation
      Shanghai Jiao Tong University(中華人民共和国)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Primitive derivations, Shi arrangements and Bernoulli polynomials

    • Author(s)
      Hiroaki Terao
    • Organizer
      colloquium
    • Place of Presentation
      Northeastern Normal University(中華人民共和国)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Logarithmic derivation modules of Shi arrangements associated with root systems

    • Author(s)
      Hiroaki Terao
    • Organizer
      Combinatorial Representation Theory and Related Topics
    • Place of Presentation
      京都大学数理解析研究所(京都市)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Logarithmic derivation modules of Shi arrangements associated with root systems

    • Author(s)
      Hiroaki Terao
    • Organizer
      Workshop on "Free Divisors and Differential Equations"
    • Place of Presentation
      東京農工大学(小金井市)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Logarithmic derivations modules of Shi arrangements associated with root systems

    • Author(s)
      Hiroaki Terao
    • Organizer
      Conference on "Hyperplane Arrangements: Combinatorial and Geometric Aspects"
    • Place of Presentation
      Ruhr-Universitat Bochum(ドイツ)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Book] DMTCS Proceedings, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)2016

    • Author(s)
      Takuro Abe, Mohamed Barakat, Michael Cuntz, Torsten Hoge, Hiroaki Terao
    • Total Pages
      956
    • Publisher
      DMTCS
    • Related Report
      2015 Annual Research Report
  • [Book] 周期と実数の$0$-認識問題: Kontsevich-Zagierの予想. 問題・予想・原理の数学22016

    • Author(s)
      吉永正彦
    • Total Pages
      193
    • Publisher
      数学書房
    • Related Report
      2015 Annual Research Report
  • [Book] 不変量と対称性: 現代数学のこころ2013

    • Author(s)
      今井 淳・寺尾宏明・中村博昭
    • Total Pages
      292
    • Publisher
      筑摩書房
    • Related Report
      2012 Annual Research Report
  • [Remarks] 寺尾研究室HP

    • URL

      http://www.math.sci.hokudai.ac.jp/~terao/

    • Related Report
      2016 Annual Research Report
  • [Remarks] 北海道大学国際本部寺尾宏明ホームページ

    • URL

      http://www.math.sci.hokudai.ac.jp/~terao/

    • Related Report
      2015 Annual Research Report
  • [Remarks] 北海道大学大学院理学研究院数学部門寺尾宏明ホームページ

    • URL

      http://www.math.sci.hokudai.ac.jp/~terao/

    • Related Report
      2014 Annual Research Report
  • [Remarks] Hiroaki Terao's Page

    • URL

      http://www.math.sci.hokudai.ac.jp/~terao/

    • Related Report
      2013 Annual Research Report
  • [Remarks] Hiroaki Terao's Page

    • URL

      http://www.math.sci.hokudai.ac.jp/~terao/

    • Related Report
      2012 Annual Research Report
  • [Funded Workshop] Summer Conference on Hyperplane Arrangements(SCHA) in Sapporo2016

    • Place of Presentation
      北海道大学学術交流会館
    • Year and Date
      2016-08-08
    • Related Report
      2016 Annual Research Report

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Published: 2013-05-15   Modified: 2019-07-29  

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