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Research on free probability and its applications to probability, combinatorics and representation theories

Research Project

Project/Area Number 15K17549
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Basic analysis
Research InstitutionHokkaido University

Principal Investigator

Hasebe Takahiro  北海道大学, 理学研究院, 准教授 (00633166)

Research Collaborator Yoshinaga Masahiko  
Miyatani Toshinori  
Tsujie Shuhei  
Ueda Yuki  
Asai Nobuhiro  
Sakuma Noriyoshi  
Collins Benoit  
Lehner Franz  
Franz Uwe  
Schleissinger Sebastian  
Arizmendi Octavio  
Huang Hao-Wei  
Wang Jiun-Chau  
Szpojankowski Kamil  
Bozejko Marek  
Ejsmont Wiktor  
Simon Thomas  
Wang Min  
Thorbjornsen Steen  
Skoufranis Paul  
Gu Yinzheng  
Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywordsfree probability / infinite divisibility / Levy processes / unimodal distributions / combinatorics / cumulants / Markov processes / stable distributions / マルコフ過程 / 非可換確率過程 / Loewner chain / コーシー変換 / 単峰性 / 自由安定分布 / 自由ガウス分布 / 自由確率論 / キュムラント / 自由Levy過程 / 自由一般化逆ガウス分布 / 正規分布 / 自由たたみこみ / Coxeter群 / Fock空間 / ウィグナーの半円分布 / 無限分解可能分布 / order polynomial / chromatic polynomial / ランダム行列
Outline of Final Research Achievements

Free probability is a field strongly motivated by mathematics of quantum physics (functional analysis and operator algebras). It focuses on "non-commutative random variables". This theory is constructed in analogy with probability theory, and there are surprising correspondences with probability theory. This research project has investigated various aspects of free probability, including limit theorems in comparison with probability theory, unified theory of cumulants, and applications to asymptotic representation theory of symmetric groups. In addition to finding analogy with probability theory, this project created some feedback to probability theory by finding a new aspect of Markov processes and probability distributions.

Academic Significance and Societal Importance of the Research Achievements

近年の数学においては様々な分野の相互関係が見つかっている.本研究でも分野横断的な側面が強く,確率論,関数解析,組合せ論,複素関数論などの分野を活用して研究が進み,逆にこういった分野への新たな視点を提供することもできた.特にランダム行列やマルコフ過程などのように応用範囲の広い分野に対しても新たな視点が得られた.これによって学術的な交流が活発になり,かつ将来的に成果が社会に還元されるためのポテンシャルとなった.また後継者に研究を伝えていくことで,将来を担う世代の教育にも貢献できる.

Report

(5 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (61 results)

All 2019 2018 2017 2016 2015 Other

All Int'l Joint Research (22 results) Journal Article (12 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 12 results,  Open Access: 5 results,  Acknowledgement Compliant: 2 results) Presentation (25 results) (of which Int'l Joint Research: 13 results,  Invited: 6 results) Remarks (2 results)

  • [Int'l Joint Research] CIMAT(メキシコ)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] University of Lille 1/University of Franche-Comte(フランス)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] University of Wuerzburg(ドイツ)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] National Sun Yat-sen University(台湾)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] York University(カナダ)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] National Sun Yat-sen University(台湾)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Warshawa University/Wroclaw University of Economics(ポーランド)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] University of Lille 1(フランス)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Aarhus University(デンマーク)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] CIMAT(メキシコ)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research]

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Centro de Investigacion en Matematicas(メキシコ)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Wroclaw University/Wroclaw University of Economics(ポーランド)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] University of Lille 1(フランス)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] University of Saskatchewan(カナダ)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] National Sun Yat-sen University(台湾)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research]

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] CIMAT(メキシコ)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] Wroclaw University(ポーランド)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] Aarhus University(デンマーク)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] Graz University of Technology(オーストリア)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] National Sun Yat-sen University(台湾)

    • Related Report
      2015 Research-status Report
  • [Journal Article] The Normal Distribution Is Freely Self-decomposable2019

    • Author(s)
      Takahiro Hasebe, Noriyoshi Sakuma and Steen Thorbjornsen
    • Journal Title

      International Mathematics Research Notices

      Volume: 2019 Issue: 6 Pages: 1758-1787

    • DOI

      10.1093/imrn/rnx171

    • NAID

      120006803958

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Limit theorems in bi-free probability theory2018

    • Author(s)
      T. Hasebe, H.-W. Huang and J.-C. Wang
    • Journal Title

      Probab. Theory Related Fields

      Volume: 172 Issue: 3-4 Pages: 1081-1119

    • DOI

      10.1007/s00440-017-0825-6

    • NAID

      120006769947

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Limit theorems for free Levy processes2018

    • Author(s)
      O. Arizmendi and T. Hasebe
    • Journal Title

      Electron. J. Probab. 23

      Volume: 23 Issue: none Pages: 1-36

    • DOI

      10.1214/18-ejp224

    • NAID

      120006606589

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Free probability for purely discrete eigenvalues of random matrices2018

    • Author(s)
      COLLINS Benoit、HASEBE Takahiro、SAKUMA Noriyoshi
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 70 Issue: 3 Pages: 1111-1150

    • DOI

      10.2969/jmsj/77147714

    • NAID

      130007420328

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Large time unimodality for classical and free Brownian motions with initial distributions2018

    • Author(s)
      T. Hasebe and Y. Ueda
    • Journal Title

      ALEA Lat. Am. J. Probab. Math. Stat.

      Volume: 15 Issue: 1 Pages: 353-374

    • DOI

      10.30757/alea.v15-15

    • NAID

      120006734185

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] 非可換確率論における独立性と無限分解可能分布2018

    • Author(s)
      長谷部高広
    • Journal Title

      数学

      Volume: 70 Pages: 296-320

    • NAID

      130007880386

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Unimodality for free Levy processes2017

    • Author(s)
      T. Hasebe and N. Sakuma
    • Journal Title

      Ann. Inst. Henri Poincare Probab. Stat.

      Volume: 53 Issue: 2 Pages: 916-936

    • DOI

      10.1214/16-aihp742

    • NAID

      120006313449

    • Related Report
      2017 Research-status Report 2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] Noncommutative probability of type D2017

    • Author(s)
      M. Bozejko, W. Ejsmont and T. Hasebe
    • Journal Title

      Internat. J. Math.

      Volume: 28 Issue: 02 Pages: 1750010-1750010

    • DOI

      10.1142/s0129167x17500100

    • NAID

      120006382290

    • Related Report
      2017 Research-status Report 2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] Order quasisymmetric functions distinguish rooted trees2017

    • Author(s)
      T. Hasebe and S. Tsujie
    • Journal Title

      J. Algebraic Combin.

      Volume: 46 Issue: 3-4 Pages: 499-515

    • DOI

      10.1007/s10801-017-0761-7

    • NAID

      120006764191

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Euler characteristic reciprocity for chromatic, flow and order polynomials.2017

    • Author(s)
      Takahiro Hasebe, Toshinori Miyatani, Masahiko Yoshinaga
    • Journal Title

      Journal of Singularities

      Volume: 16 Pages: 212-227

    • DOI

      10.5427/jsing.2017.16k

    • NAID

      120006734193

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Free infinite divisibility for powers of random variables2016

    • Author(s)
      Takahiro Hasebe
    • Journal Title

      ALEA, Lat. Am. J. Probab. Math. Stat.

      Volume: 13 Pages: 309-336

    • NAID

      120006734184

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Radial Bargmann representation for the Fock space of type B2016

    • Author(s)
      Nobuhiro Asai, Marek Bozejko and Takahiro Hasebe
    • Journal Title

      Journal of Mathematical Physics

      Volume: 57 Issue: 2 Pages: 021702-021702

    • DOI

      10.1063/1.4939748

    • NAID

      120006661339

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] Monotone increment processes, classical Markov processes and Loewner chains2019

    • Author(s)
      T. Hasebe
    • Organizer
      New developments in free probability and applications (CRM, Montreal)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On free multiplicative convolution with free normal distributions on the unit circle2019

    • Author(s)
      Y. Ueda
    • Organizer
      Functional analysis seminar (Besancon, France)
    • Related Report
      2018 Annual Research Report
  • [Presentation] Monotone Increment Processes, Classical Markov Processes and Loewner Chains2019

    • Author(s)
      Uwe Franz
    • Organizer
      Functional analysis seminar (Besancon, France)
    • Related Report
      2018 Annual Research Report
  • [Presentation] Monotone Increment Processes, Classical Markov Processes and Loewner Chains2019

    • Author(s)
      Uwe Franz
    • Organizer
      British Mathematical Colloquium, Probability Workshop (Lancaster Univ.)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Monotone Increment processes, Loewner chains and $\triangleright$-homogeneous Markov processes2019

    • Author(s)
      Uwe Franz
    • Organizer
      Oberseminar Freie Wahrscheinlichkeitstheorie (Saarland University)
    • Related Report
      2018 Annual Research Report
  • [Presentation] Recent progress in bi-free harmonic analysis2019

    • Author(s)
      Jiun-Chau Wang
    • Organizer
      The 9th Abstract Harmonic Analysis 2018 (National Sun Yat-sen University, Kaohsiung, Taiwan)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Limits of free Levy processes at small time2018

    • Author(s)
      T. Hasebe
    • Organizer
      18th Workshop: Noncommutative Probability, Operators Algebras, Random Matrices and Related Topics, with Applications (Bedlewo, Poland)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Loewner chains, Markov processes and monotone increment processes2018

    • Author(s)
      T. Hasebe
    • Organizer
      International Workshop on Conformal Dynamics and Loewner Theory (Tohoku Univ.)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Unimodality for classical and free Brownian motions with initial distributions2018

    • Author(s)
      T. Hasebe
    • Organizer
      RIMS共同研究(公開型): 量子場の数理とその周辺 2018 (Kyoto Univ.)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Markov processes and monotone increment processes and Loewner chains2018

    • Author(s)
      T. Hasebe
    • Organizer
      Noncommutative probability and related fields (Hokkaido Univ.)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Free multiplicative convolution with free circular normal distributions and its properties2018

    • Author(s)
      植田優基
    • Organizer
      無限分解可能過程に関連する諸問題(23) (統計数理研究所)
    • Related Report
      2018 Annual Research Report
  • [Presentation] Free infinite divisibility for Generalized Power distributions with Free Poisson term2018

    • Author(s)
      Y. Ueda
    • Organizer
      Seminar of Discrete Harmonic Analysis (SeDAH) (Wroclaw University, Poland)
    • Related Report
      2018 Annual Research Report
  • [Presentation] Unimodality for free convolution with semicircle distributions2018

    • Author(s)
      Y. Ueda
    • Organizer
      18th workshop: Noncommutative Probability, Operator Algebras, Random Matrices and Related Topics, with Applications (Bedlewo, Poland)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Some properties of free stable distributions2018

    • Author(s)
      Thomas Simon
    • Organizer
      Oberseminar Stochastik (University of Munster, Germany)
    • Related Report
      2018 Annual Research Report
  • [Presentation] Free infinite divisibility for powers of random variables2017

    • Author(s)
      T. Hasebe
    • Organizer
      Workshop on interactions between commutative and noncommutative probability (Kyoto University)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Free generalized inverse gaussian distribution2017

    • Author(s)
      T. Hasebe
    • Organizer
      Analysis in quantum information theory (IHP, Paris)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Limit theorems for free Levy processes2017

    • Author(s)
      T. Hasebe
    • Organizer
      非可換確率論とその応用 (お茶の水女子大学)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Free probabilistic analysis of random matrices converging to compact operators2017

    • Author(s)
      Takahiro Hasebe
    • Organizer
      Les Probas du vendredi
    • Place of Presentation
      パリ (フランス)
    • Related Report
      2016 Research-status Report
  • [Presentation] D型Fock空間2017

    • Author(s)
      長谷部高広
    • Organizer
      日本数学会
    • Place of Presentation
      首都大学東京 (東京都、八王子市)
    • Related Report
      2016 Research-status Report
  • [Presentation] コンパクト作用素に収束するランダム行列に対する自由確率論的な解析2016

    • Author(s)
      長谷部高広
    • Organizer
      日本数学会
    • Place of Presentation
      筑波大学第一エリア1E303 (茨城県・つくば市)
    • Year and Date
      2016-03-18
    • Related Report
      2015 Research-status Report
  • [Presentation] Free infinite divisibility for powers of random variables2016

    • Author(s)
      Takahiro Hasebe
    • Organizer
      17th Workshop: Non-commutative Harmonic Analysis
    • Place of Presentation
      Bedlewo (ポーランド)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] コンパクト作用素のように振る舞うランダム行列の研究2016

    • Author(s)
      長谷部高広
    • Organizer
      月曜解析セミナー
    • Place of Presentation
      北海道大学 (北海道、札幌市)
    • Related Report
      2016 Research-status Report
  • [Presentation] Fock space deformed by Coxeter groups2016

    • Author(s)
      Takahiro Hasebe
    • Organizer
      作用素環セミナー
    • Place of Presentation
      東京大学 (東京都)
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Free probabilistic analysis of random matrices converging to compact operators2015

    • Author(s)
      長谷部高広
    • Organizer
      確率論シンポジウム
    • Place of Presentation
      岡山大学大学院自然科学研究棟2階大講義室(岡山県・岡山市)
    • Year and Date
      2015-12-16
    • Related Report
      2015 Research-status Report
  • [Presentation] Analogy between cumulants and Baker-Campbell-Hausdorff formula2015

    • Author(s)
      Takahiro Hasebe
    • Organizer
      Workshop on interactions between commutative and noncommutative probability
    • Place of Presentation
      京都大学理学部3号館110(京都府・京都市)
    • Year and Date
      2015-08-05
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Remarks] Takahiro Hasebe's homepage

    • URL

      https://www.math.sci.hokudai.ac.jp/~thasebe/

    • Related Report
      2018 Annual Research Report
  • [Remarks] Takahiro Hasebe's Homeage

    • URL

      https://www.math.sci.hokudai.ac.jp/~thasebe/

    • Related Report
      2017 Research-status Report

URL: 

Published: 2015-04-16   Modified: 2022-08-19  

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