• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Randomization of persistent homology and its interaction with Schubert calculus

Research Project

Project/Area Number 20H00119
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Review Section Medium-sized Section 12:Analysis, applied mathematics, and related fields
Research InstitutionKyoto University

Principal Investigator

Hiraoka Yasuaki  京都大学, 高等研究院, 教授 (10432709)

Co-Investigator(Kenkyū-buntansha) 池田 岳  早稲田大学, 理工学術院, 教授 (40309539)
赤木 和人  東北大学, 材料科学高等研究所, 准教授 (50313119)
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)
Project Period (FY) 2020-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥44,850,000 (Direct Cost: ¥34,500,000、Indirect Cost: ¥10,350,000)
Fiscal Year 2023: ¥10,920,000 (Direct Cost: ¥8,400,000、Indirect Cost: ¥2,520,000)
Fiscal Year 2022: ¥10,920,000 (Direct Cost: ¥8,400,000、Indirect Cost: ¥2,520,000)
Fiscal Year 2021: ¥10,920,000 (Direct Cost: ¥8,400,000、Indirect Cost: ¥2,520,000)
Fiscal Year 2020: ¥12,090,000 (Direct Cost: ¥9,300,000、Indirect Cost: ¥2,790,000)
Keywordsパーシステントホモロジー / ランダムトポロジー / 確率論 / シューベルト計算 / マルチパラメータ・パーシステントホモロジー / 非区間表現 / シューベルトカリキュラス / マルチパラメータ・パーシステンス / ランダム化 / 材料科学 / 大数の法則 / 大偏差原理 / 両側シューベルト分解 / シューベルト・カルキュラス
Outline of Research at the Start

マルチパラメータ・パーシステントホモロジーの実用的な分解論を構築することを目標に据えた以下の研究テーマを実施する:(1)パーシステントホモロジーの確率論的安定分解構造の決定,(2)パーシステントホモロジーのシューベルト計算と安定構造の解明,(3)アモルファス炭素材料の階層構造解析.
パーシステントホモロジーの直既約成分に対する確率論的性質を研究し,代数的な分解論を確率論の視点から捉え直す.またシューベルト計算をパーシステントホモロジーに導入し,安定分解構造とシューベルト多様体の交叉理論の関係を調べる.これらの数学的成果をアモルファス炭素材料の解析に適用し,結晶性中距離秩序構造の特徴づけを行う.

Outline of Final Research Achievements

In this project, we conducted several research to deeply understand the mathematical structures of multi-parameter persistence modules based on probability theory and combinatorics. Regarding its applications, we also performed structural analysis of several materials using the methods developed in this project. The basic idea is to focus on the randomness inherent in real data and try to find stable decomposition structures in the sense of probability theory. We clarified the relationship between persistence diagrams and Schubert varieties and found a natural multiplication on persistence diagrams. Furthermore, we succeeded in generalizing the interval approximation, and obtained estimates about the multiplicities of non-intervals under a certain random topology setting.

Academic Significance and Societal Importance of the Research Achievements

ビッグデータ時代の到来を受け、膨大かつ複雑なデータに対する数学的なデータ記述子を開発する必要性が高まっている。この背景の元、本研究ではデータの「かたち」と「うごき」に着目するパーシステントホモロジーの数学研究に主に取り組んだ。特にこのプロジェクトではデータを詳細に調べるパラメータが2つ以上の問題を扱っている。観測データにはノイズがのっていることを想定し、適切な確率論的状況のもとで新規データ記述子を開発することに成功した。

Report

(6 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Annual Research Report
  • 2021 Annual Research Report
  • 2020 Comments on the Screening Results   Annual Research Report
  • Research Products

    (58 results)

All 2024 2023 2022 2021 2020 Other

All Int'l Joint Research (2 results) Journal Article (13 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 13 results,  Open Access: 12 results) Presentation (42 results) (of which Int'l Joint Research: 11 results,  Invited: 38 results) Book (1 results)

  • [Int'l Joint Research] Ohio State University/Virginia Tech University(米国)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Ohio State University/Virginia Tech University(米国)

    • Related Report
      2022 Annual Research Report
  • [Journal Article] Topological data analysis of TEM-based structural features affecting the thermal conductivity of amorphous Ge2024

    • Author(s)
      Wu Yen-Ju、Akagi Kazuto、Goto Masahiro、Xu Yibin
    • Journal Title

      International Journal of Heat and Mass Transfer

      Volume: 221 Pages: 125012-125012

    • DOI

      10.1016/j.ijheatmasstransfer.2023.125012

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Torsion-weighted spanning acycle entropy in cubical lattices and Mahler measures2024

    • Author(s)
      Hiraoka Yasuaki、Shirai Tomoyuki
    • Journal Title

      Journal of Applied and Computational Topology

      Volume: 00 Issue: 6 Pages: 1-31

    • DOI

      10.1007/s41468-024-00163-y

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Closed k-Schur Katalan functions as K-homology Schubert representatives of the affine Grassmannian2024

    • Author(s)
      Ikeda Takeshi、Iwao Shinsuke、Naito Satoshi
    • Journal Title

      Transactions of the American Mathematical Society, Series B

      Volume: 11 Issue: 20 Pages: 667-702

    • DOI

      10.1090/btran/184

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Persistent Homology and Its Application to Chainlets in the Bitcoin Graph2023

    • Author(s)
      Shirai Tomoyuki
    • Journal Title

      JPS Conference Proceedings

      Volume: 40 Pages: 1-14

    • DOI

      10.7566/jpscp.40.011001

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] The multiplicity of a singularity in a vexillary Schubert variety2023

    • Author(s)
      Anderson David、Ikeda Takeshi、Jeon Minyoung、Kawago Ryotaro
    • Journal Title

      Advances in Mathematics

      Volume: 435 Pages: 109366-109366

    • DOI

      10.1016/j.aim.2023.109366

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Zeros of the i.i.d.Gaussian Laurent Series on an Annulus: Weighted Szeg? Kernels and Permanental-Determinantal Point Processes2022

    • Author(s)
      Katori Makoto、Shirai Tomoyuki
    • Journal Title

      Communications in Mathematical Physics

      Volume: - Issue: 3 Pages: 1099-1151

    • DOI

      10.1007/s00220-022-04365-2

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Disordered complex networks: Energy optimal lattices and persistent homology2022

    • Author(s)
      Ghosh Subhroshekhar, Miyoshi Naoto, Shirai Tomoyuki
    • Journal Title

      IEEE Transactions on Information Theory

      Volume: Early Access Issue: 8 Pages: 1-1

    • DOI

      10.1109/tit.2022.3163604

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] A limit theorem for persistence diagrams of random filtered complexes built over marked point processes2022

    • Author(s)
      Tomoyuki Shirai, Kiyotaka Suzaki
    • Journal Title

      Modern Stoch. Theory Appl

      Volume: ー Pages: 1-18

    • DOI

      10.15559/22-vmsta214

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Expected Number of Zeros of Random Power Series with Finitely Dependent Gaussian Coefficients2022

    • Author(s)
      Noda, K., Shirai, T
    • Journal Title

      J Theor Probab

      Volume: ー Issue: 3 Pages: 1534-1554

    • DOI

      10.1007/s10959-022-01203-y

    • Related Report
      2022 Annual Research Report 2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Local universality of determinantal point processes on Riemannian manifolds2022

    • Author(s)
      Makoto Katori, Tomoyuki Shirai
    • Journal Title

      Proceedings of the Japan Academy Series A: Mathematical Sciences

      Volume: 98 Pages: 95-100

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Local universality of determinantal point processes on Riemannian manifolds2022

    • Author(s)
      Makoto Katori, Tomoyuki Shirai
    • Journal Title

      Proc. Japan Acad. Ser. A Math. Sci.

      Volume: 98 Pages: 95-100

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Persistent Homology Analysis for Materials Research and Persistent Homology Software: HomCloud2022

    • Author(s)
      Obayashi Ippei、Nakamura Takenobu、Hiraoka Yasuaki
    • Journal Title

      Journal of the Physical Society of Japan

      Volume: 91 Issue: 9 Pages: 091013-091013

    • DOI

      10.7566/jpsj.91.091013

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Algebraic stability theorem for derived categories of zigzag persistence modules2022

    • Author(s)
      Hiraoka Yasuaki、Ike Yuichi、Yoshiwaki Michio
    • Journal Title

      Journal of Topology and Analysis

      Volume: - Pages: 1-45

    • DOI

      10.1142/s1793525322500091

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Presentation] Zeros of random power series with stationary Gaussian coefficients.2024

    • Author(s)
      Tomoyuki Shirai
    • Organizer
      French Japanese Conference on Probability and Interactions program IHES
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Zeros of Gaussian power series with dependent coefficients.2024

    • Author(s)
      Tomoyuki Shirai
    • Organizer
      iPOD seminar at Mathematics Institute, Leiden University
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] シミュレーションデータのミクロな「形」とマクロなイオン挙動をつなぐ2023

    • Author(s)
      赤木 和人
    • Organizer
      第2回計算イオニクス研究会(第83回固体イオニクス研究会)
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] 幾何学を使ってミクロ構造とマクロ物性をつなぐ2023

    • Author(s)
      赤木 和人
    • Organizer
      日本MRS「水素科学技術連携研究会」
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Enumeration of connected bipartite graphs with given Betti number2023

    • Author(s)
      白井朋之
    • Organizer
      日本数学会年会
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] $\alpha$-行列式点過程のポアソン表現2023

    • Author(s)
      白井朋之
    • Organizer
      東京確率論セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Zeros of random power series with dependent Gaussian coefficients.2023

    • Author(s)
      Tomoyuki Shirai
    • Organizer
      Mini-Symposium with Prof. G. Schehr
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Zeros of Gaussian power series with dependent coefficients.2023

    • Author(s)
      Tomoyuki Shirai
    • Organizer
      The 21st Symposium Stochastic Analysis on Large Scale Interacting Systems RIMS
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Zeros of random power series with stationary Gaussian coefficients.2023

    • Author(s)
      Tomoyuki Shirai
    • Organizer
      Random Interacting Systems, Scaling Limits, and Universality. Workshop in Honour of Takashi Hara. NUS
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] アフィン・グラスマン多様体の同変シューベルト・カルキュラス2023

    • Author(s)
      池田岳
    • Organizer
      「第63回代数学シンポジウム」名古屋大学多元数理科学研究科
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Equivariant homology of the affine Grassmannian of the symplectic group2023

    • Author(s)
      Takeshi Ikeda
    • Organizer
      AlCoVE: an Algebraic Combinatorics Virtual Expedition
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] シンプレクティック群のアフィン・グラスマン多様体のトーラス同変シューベルト類2023

    • Author(s)
      池田岳
    • Organizer
      「変換群の幾何とトポロジー」 RIMS共同研究(公開型)
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] アフィン・グラスマン多様体のシューベルト・カルキュラス - 特に C 型の場合2023

    • Author(s)
      池田岳
    • Organizer
      「幾何学と表現論を巡って」,AGU 表現論セミナー・研究会
    • Related Report
      2022 Annual Research Report
  • [Presentation] Equivariant Schubert Calculus of Affine Grassmanniann of Symplectic Group2023

    • Author(s)
      Takeshi Ikeda
    • Organizer
      Gauge Theory, Moduli Spaces and Representation Theory, Kashiwa 2023
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Equivariant Schubert Calculus of Affine Grassmanniann of Symplectic Group2023

    • Author(s)
      池田岳
    • Organizer
      Gauge Theory, Moduli Spaces and Representation Theory, Kashiwa 2023 In honor of the 60th birthday of Hiraku Nakajima
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] アフィン・グラスマン多様体のシューベルト・カルキュラス 特に C 型の場合2023

    • Author(s)
      池田岳
    • Organizer
      AGU 表現論セミナー・研究会
    • Related Report
      2021 Annual Research Report
  • [Presentation] シミュレーションデータのミクロな「形」とマクロなイオン挙動をつなぐ2023

    • Author(s)
      赤木和人
    • Organizer
      第2回計算イオニクス研究会(第83回固体イオニクス研究会)
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Enumeration of connected bipartite graphs with given Betti number2023

    • Author(s)
      藪奥 哲史, 蓮井 太朗, 白井 朋之
    • Organizer
      日本数学会2023年度年会
    • Related Report
      2021 Annual Research Report
  • [Presentation] パーシステントホモロジー:理論と応用2023

    • Author(s)
      平岡裕章
    • Organizer
      日本数学会2023年度年会企画特別講演
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Zeros of random analytic functions with dependent Gaussian coefficients2022

    • Author(s)
      Tomoyuki Shirai
    • Organizer
      談話会, Kanazawa University
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] $\alpha$行列式点過程のポアソン表現2022

    • Author(s)
      Tomoyuki Shirai
    • Organizer
      Crossroad of Statistical Physics and Probability Theory, Chuo University, Tokyo
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Persistent homology and its applications, program2022

    • Author(s)
      Tomoyuki Shirai
    • Organizer
      Blockchain Kaigi 2022
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Zeros of the i.i.d. Gaussian Laurent series on an annulus2022

    • Author(s)
      Tomoyuki Shirai
    • Organizer
      The Statistical Physics of Continuum Particle Systems with Strong Interactions
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] パーシステントホモロジーとその周辺の話題2022

    • Author(s)
      白井朋之
    • Organizer
      ブロックチェーン研究会, Online
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Zeros of the i.i.d. Gaussian Laurent series on an annulus2022

    • Author(s)
      Tomoyuki Shirai
    • Organizer
      Non-Commutative Probability and Related Topics 2022
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] 行列式点過程の数理とその応用 ---ランダム行列から機械学習へ---2022

    • Author(s)
      白井朋之
    • Organizer
      AIE講演会
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Correlations between zeros of Gaussian polyanalytic functions2022

    • Author(s)
      白井朋之
    • Organizer
      大規模相互作用系の確率解析
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Geometric variants of Schur’s Q-functions2022

    • Author(s)
      池田岳
    • Organizer
      Integrability, combinatorics and representation theory MATRIX/RIMS tandem workshop
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] パーシステントホモロジーとその周辺の話題2022

    • Author(s)
      白井朋之
    • Organizer
      ブロックチェーン研究会
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] パーシステントホモロジーと確率論2022

    • Author(s)
      白井朋之
    • Organizer
      集中講義 特別講義
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Zeros of random analytic functions with dependent Gaussian coefficients2022

    • Author(s)
      白井朋之
    • Organizer
      談話会
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Persistent homology and its applications2022

    • Author(s)
      Tomoyuki Shirai
    • Organizer
      Blockchain Kaigi 2022 (BCK2022)
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Is the persistence diagram really a stable data descriptor?2022

    • Author(s)
      Yasuaki Hiraoka
    • Organizer
      International Conference on "Topology and its Applications to Engineering and Life Science".
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Is the persistence diagram really a stable data descriptor?2022

    • Author(s)
      Yasuaki Hiraoka
    • Organizer
      Computational Persistence 2022
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Girth, magnitude homology, and phase transition of diagonality2022

    • Author(s)
      Yasuaki Hiraoka
    • Organizer
      Conference on Applied, Combinatorial and Toric Topology
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] トポロジカルデータ解析を活用した実材料の性能予測2022

    • Author(s)
      赤木和人
    • Organizer
      日本応用数理学会2022年度年会
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] K理論的シューベルト・カルキュラス2022

    • Author(s)
      池田岳
    • Organizer
      日本数学会202年度年会 函数解析学分科会 特別講演
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] トポロジカルデータ解析を用いた電池材料の構造情報の定量化2021

    • Author(s)
      赤木和人
    • Organizer
      2021年日本表面真空学会学術講演会
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Zeros of random analytic functions with dependent Gaussian coefficients2021

    • Author(s)
      白井朋之
    • Organizer
      確率・ 統計・ 行列ワークショップ 彦根 2021
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] パーシステントホモロジーとその応用2021

    • Author(s)
      白井朋之
    • Organizer
      日本オペレーションズ・リサーチ学会 待ち行列研究部会 第299部会
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] トポロジカルデータ解析による電解質材料の特徴抽出2020

    • Author(s)
      赤木和人
    • Organizer
      電気化学会東北支部 第51回セミコンファレンス
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] アフィン・シューベルト計算と可積分系2020

    • Author(s)
      池田岳
    • Organizer
      早稲田大学整数論セミナー
    • Related Report
      2020 Annual Research Report
  • [Book] 数理科学2023年3月号特集「統計物理の視点で捉える確率論」2023

    • Author(s)
      白井朋之 編
    • Publisher
      サイエンス社
    • Related Report
      2023 Annual Research Report

URL: 

Published: 2020-04-28   Modified: 2025-01-30  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi