研究者紹介システム

ESCOLAR Gaw Emerson
エスカラ ガウ エマソン
大学院人間発達環境学研究科 人間環境学専攻
助教
数学関係
Last Updated :2022/01/10

研究者情報

所属

  • 【主配置】

    大学院人間発達環境学研究科 人間環境学専攻
  • 【配置】

    国際人間科学部 環境共生学科, 発達科学部 人間環境学科

学位

  • 博士(数理学), 九州大学

研究活動

研究キーワード

  • アルゴリズム
  • 表現論
  • 位相的データ解析

研究分野

  • 自然科学一般 / 応用数学、統計数学 / 応用トポロジー
  • 自然科学一般 / 数学基礎 / 応用トポロジー

論文

  • Mickaël Buchet, Emerson G. Escolar

    A recent work by Lesnick and Wright proposed a visualisation of $2$D
    persistence modules by using their restrictions onto lines, giving a family of
    $1$D persistence modules. We give a constructive proof that any $1$D
    persistence module with finite support can be found as a restriction of some
    indecomposable $2$D persistence module with finite support. As consequences of
    our construction, we are able to exhibit indecomposable $2$D persistence
    modules whose support has holes as well as an indecomposable $2$D persistence
    module containing all $1$D persistence modules with finite support as line
    restrictions. Finally, we also show that any finite-rectangle-decomposable $n$D
    persistence module can be found as a restriction of some indecomposable
    $(n+1)$D persistence module.

    2020年, J. Appl. Comput. Topol., 4 (3), 387 - 424

    [査読有り]

    研究論文(学術雑誌)

  • Asashiba Hideto, Escolar Emerson G, Hiraoka Yasuaki, Takeuchi Hiroshi

    2019年01月, JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS, 36 (1), 97 - 130

    [査読有り]

  • Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension.

    Mickaël Buchet, Emerson G. Escolar

    Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018年, 34th International Symposium on Computational Geometry, SoCG 2018, June 11-14, 2018, Budapest, Hungary, 15:1-15:13

    [査読有り]

  • Yasuaki Hiraoka, Takenobu Nakamura, Akihiko Hirata, Emerson G. Escolar, Kaname Matsue, Yasumasa Nishiura

    This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is expressed as 2D histograms. Then, specific distributions such as curves and islands in the PDs identify meaningful shape characteristics of the atomic configuration. Although the method can be applied to a wide variety of disordered systems, it is applied here to silica glass, the Lennard-Jones system, and Cu-Zr metallic glass as standard examples of continuous random network and random packing structures. In silica glass, the method classified the atomic rings as short-range and medium-range orders and unveiled hierarchical ring structures among them. These detailed geometric characterizations clarified a real space origin of the first sharp diffraction peak and also indicated that PDs contain information on elastic response. Even in the Lennard-Jones system and Cu-Zr metallic glass, the hierarchical structures in the atomic configurations were derived in a similar way using PDs, although the glass structures and properties substantially differ from silica glass. These results suggest that the PDs provide a unified method that extracts greater depth of geometric information in amorphous solids than conventional methods.

    NATL ACAD SCIENCES, 2016年06月, PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, 113 (26), 7035 - 7040, 英語

    [査読有り]

    研究論文(学術雑誌)

  • Emerson G. Escolar, Yasuaki Hiraoka

    In this work, we discuss the problem of finding optimal cycles for homology groups of simplicial complexes and for persistent homology of filtrations. We review the linear programming formulation of the optimal homologous cycle problem and its extension to allow for multiple cycles. By inserting these linear programming problems into the persistent homology algorithm, we are able to compute an optimal cycle, that has been optimized at birth, for every persistent interval in the persistent diagram.

    SPRINGER-VERLAG TOKYO, 2016年, OPTIMIZATION IN THE REAL WORLD: TOWARD SOLVING REAL-WORLD OPTIMIZATION PROBLEMS, 13, 79 - 96, 英語

    [査読有り]

    研究論文(国際会議プロシーディングス)

  • Emerson G. Escolar, Yasuaki Hiraoka

    This is a summary paper of Escolar and Hiraoka (Persistence modules on commutative ladders of finite type. Discrete Comput Geom 55, 100-157 (2016)) which presents an extension of persistence modules as representations on quivers with nontrivial relations. In particular, the mathematical and algorithmic results in that paper enable us to detect robust and common topological structures of two geometric objects. In this paper, we only deal with a special type of persistence modules defined on the so-called commutative triple ladder for the sake of simplicity. We aim to explain the essence of Auslander-Reiten theory in connection with persistence modules.

    SPRINGER JAPAN, 2016年, MATHEMATICAL CHALLENGES IN A NEW PHASE OF MATERIALS SCIENCE, 166, 69 - 82, 英語

    [査読有り]

    研究論文(国際会議プロシーディングス)

  • Emerson G. Escolar, Yasuaki Hiraoka

    We study persistence modules defined on commutative ladders. This class of persistence modules frequently appears in topological data analysis, and the theory and algorithm proposed in this paper can be applied to these practical problems. A new algebraic framework deals with persistence modules as representations on associative algebras and the Auslander-Reiten theory is applied to develop the theoretical and algorithmic foundations. In particular, we prove that the commutative ladders of length less than 5 are representation-finite and explicitly show their Auslander-Reiten quivers. Furthermore, a generalization of persistence diagrams is introduced by using Auslander-Reiten quivers. We provide an algorithm for computing persistence diagrams for the commutative ladders of length 3 by using the structure of Auslander-Reiten quivers.

    SPRINGER, 2016年01月, DISCRETE & COMPUTATIONAL GEOMETRY, 55 (1), 100 - 157, 英語

    [査読有り]

    研究論文(学術雑誌)

  • Takenobu Nakamura, Yasuaki Hiraoka, Akihiko Hirata, Emerson G. Escolar, Yasumasa Nishiura

    The characterization of the medium-range (MRO) order in amorphous materials and its relation to the short-range order is discussed. A new topological approach to extract a hierarchical structure of amorphous materials is presented, which is robust against small perturbations and allows us to distinguish it from periodic or random configurations. This method is called the persistence diagram (PD) and introduces scales to many-body atomic structures to facilitate size and shape characterization. We first illustrate the representation of perfect crystalline and random structures in PDs. Then, the MRO in amorphous silica is characterized using the appropriate PD. The PD approach compresses the size of the data set significantly, to much smaller geometrical summaries, and has considerable potential for application to a wide range of materials, including complex molecular liquids, granular materials, and metallic glasses.

    IOP PUBLISHING LTD, 2015年07月, NANOTECHNOLOGY, 26 (30), 304001, 英語

    [査読有り]

    研究論文(学術雑誌)

  • Emerson G. Escolar, Yasuaki Hiraoka

    2014年, A Mathematical Approach to Research Problems of Science and Technology - Theoretical Basis and Developments in Mathematical Modeling, 101 - 118

  • Emerson Escolar, Yasuaki Hiraoka

    2014年, Journal of the Indonesian Mathematical Society, 20, 47 - 75, 英語

    [査読有り]

  • Emerson G. Escolar, Yasuaki Hiraoka

    Persistence modules on commutative ladders naturally arise in topological data analysis. It is known that all isomorphism classes of indecomposable modules, which are the counterparts to persistence intervals in the standard setting of persistent homology, can be derived for persistence modules on commutative ladders of finite type. Furthermore, the concept of persistence diagrams can be naturally generalized as functions defined on the Auslander-Reiten quivers of commutative ladders. A previous paper [4] presents an algorithm to compute persistence diagrams by inductively applying echelon form reductions to a given persistence module. In this work, we show that discrete Morse reduction can be generalized to this setting. Given a quiver complex double-struck X, we show that its persistence module H q(double-struck X) is isomorphic to the persistence module H q(double-struck A) of its Morse quiver complex double-struck A. With this preprocessing step, we reduce the computation time by computing H q(double-struck A) instead, since double-struck A is generally smaller in size. We also provide an algorithm to obtain such Morse quiver complexes. © 2014 Springer-Verlag.

    Springer Verlag, 2014年, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 8592, 144 - 151, 英語

    研究論文(国際会議プロシーディングス)

  • Emerson G. Escolar, Killian Meehan, Michio Yoshiwaki

    Springer Science and Business Media LLC, 2021年10月23日, Applicable Algebra in Engineering, Communication and Computing

    研究論文(学術雑誌)

MISC

  • The Whole in the Parts: Putting $n$D Persistence Modules Inside Indecomposable $(n + 1)$D Ones

    Mickaël Buchet, Emerson G. Escolar

    Multidimensional persistence has been proposed to study the persistence of topological features in data indexed by multiple parameters. In this work, we further explore its algebraic complications from the point of view of higher dimensional indecomposable persistence modules containing lower dimensional ones as hyperplane restrictions. Our previous work constructively showed that any finite rectangle-decomposable $n$D persistence module is the hyperplane restriction of some indecomposable $(n+1)$D persistence module, as a corollary of the result for $n=1$. Here, we extend this by dropping the requirement of rectangle-decomposability. Furthermore, in the case that the underlying field is countable, we construct an indecomposable $(n+1)$D persistence module containing all $n$D persistence modules, up to isomorphism, as hyperplane restrictions. Finally, in the case $n=1$, we present a minimal construction that improves our previous construction.

    2020年12月04日

  • On Approximation of $2$D Persistence Modules by Interval-decomposables

    Hideto Asashiba, Emerson G. Escolar, Ken Nakashima, Michio Yoshiwaki

    In this work, we propose a new invariant for $2$D persistence modules called
    the compressed multiplicity and show that it generalizes the notions of the
    dimension vector and the rank invariant. In addition, we propose an
    "interval-decomposable approximation" $\delta^{\ast}(M)$ of a $2$D persistence
    module $M$. In the case that $M$ is interval-decomposable, we show that
    $\delta^{\ast}(M) = M$. Furthermore, even for representations $M$ not
    necessarily interval-decomposable, $\delta^{\ast}(M)$ preserves the dimension
    vector and the rank invariant of $M$.

    2019年11月05日

    機関テクニカルレポート,技術報告書,プレプリント等

  • Mapping Firms' Locations in Technological Space: A Topological Analysis of Patent Statistics

    Emerson G. Escolar, Yasuaki Hiraoka, Mitsuru Igami, Yasin Ozcan

    Where do firms innovate? Mapping their locations in technological space is
    difficult, because it is high dimensional and unstructured. We address this
    issue by using a method in computational topology called the Mapper algorithm,
    which combines local clustering with global reconstruction. We apply this
    method to a panel of 333 major firms' patent portfolios in 1976--2005 across
    430 technological areas. Results suggest the Mapper graph captures salient
    patterns in firms' patenting histories, and our measures of their uniqueness
    (the type and length of "flares") are correlated with firms' financial
    performances in a statistically and economically significant manner.

    2019年09月01日

    機関テクニカルレポート,技術報告書,プレプリント等

  • On Interval Decomposability of 2D Persistence Modules

    Hideto Asashiba, Mickaël Buchet, Emerson G. Escolar, Ken Nakashima, Michio Yoshiwaki

    In persistent homology of filtrations, the indecomposable decompositions
    provide the persistence diagrams. In multidimensional persistence it is known
    to be impossible to classify all indecomposable modules. One direction is to
    consider the subclass of interval-decomposable persistence modules, which are
    direct sums of interval representations. We introduce the definition of
    pre-interval representations, a more algebraic definition, and study the
    relationships between pre-interval, interval, and indecomposable thin
    representations. We show that over the `equioriented' commutative $2$D grid,
    these concepts are equivalent. Moreover, we provide an algorithm for
    determining whether or not an $n$D persistence module is
    interval/pre-interval/thin-decomposable, under certain finiteness conditions
    and without explicitly computing decompositions.

    2018年12月13日

    機関テクニカルレポート,技術報告書,プレプリント等

  • 8aAP-9 パーシステントホモロジーを用いたアモルファスシリカ構造の表現II(8aAP 領域11,領域12合同 ガラス合同2,領域11(物性基礎論・統計力学・流体物理・応用数学・社会経済物理))

    中村 壮伸, 平岡 裕章, 平田 秋彦, Escolar Emerson, 松江 要, 西浦 廉政

    一般社団法人日本物理学会, 2014年08月22日, 日本物理学会講演概要集, 69 (2), 128 - 128, 日本語

  • 中村 壮伸, 平岡 裕章, 平田 秋彦, Escolar E., 松江 要, 西浦 廉政

    一般社団法人日本物理学会, 2015年, 日本物理学会講演概要集, 70, 2947 - 2947, 日本語

講演・口頭発表等

  • 区間表現による2Dパーシステント表現の近似

    中島 健, 浅芝 秀人, Emerson G. Escolar, 吉脇 理雄

    日本応用数理学会 2021年 研究部会連合発表会, 2021年03月04日

    口頭発表(一般)

  • The Whole in the Parts: Putting nD Persistence Modules Inside Indecomposable (n+1)D Ones

    M. Buchet, E.G. Escolar

    日本応用数理学会 2021年 研究部会連合発表会, 2021年03月04日, 日本語, online

    口頭発表(一般)

  • Introduction to Topological Data Analysis: Ideas and Applications

    Emerson G. Escolar

    departmental colloquium talk (2021 in-house training program), 2021年01月15日, Department of Mathematics and Computer Science, College of Science, University of the Philippines Baguio (online)

    公開講演,セミナー,チュートリアル,講習,講義等

  • Mapping Firms' Locations in Technological Space: A Topological Analysis of Patent Statistics

    Emerson G. Escolar, 平岡裕章, 伊神満, Yasin Ozcan

    [2nd AIP Open Seminar] Talks by Topological Data Analysis Team, 2020年11月18日, Online

    公開講演,セミナー,チュートリアル,講習,講義等

  • Mapperを用いた企業の技術戦略の位相的データ解析

    Emerson G. Escolar, 平岡裕章, 伊神満, Yasin Ozcan

    日本応用数学会2020年度年会, 正会員主催OS:位相的データ解析, 2020年09月09日, online, zoom

    口頭発表(一般)

  • Mapperを用いた企業の技術戦略の位相的データ解析

    Emerson G. Escolar, 平岡裕章, 伊神満, Yasin Ozcan

    TDA for Applications - Tutorial & Workshop, 2020年06月19日, online

    公開講演,セミナー,チュートリアル,講習,講義等

  • Mapping firms’ locations in technological space: a topological analysis of patent statistics

    Emerson G. Escolar, 平岡裕章, 伊神満, Yasin Ozcan

    Kyoto University Applied Mathematics Seminar, 2020年05月26日, Kyoto University, Kyoto

    公開講演,セミナー,チュートリアル,講習,講義等

  • Mapping firms’ locations in technological space: A topological analysis of patent statistics

    Emerson G. Escolar, 平岡裕章, 伊神満, Yasin Ozcan

    日本数学会 2020年度年会, 2020年03月18日, (会は中止だが、発表は成立)

    口頭発表(一般)

  • 企業の技術戦略の位相的データ解析

    Emerson G. Escolar, 平岡裕章, 伊神満, Yasin Ozcan

    2019年度応用数学合同研究集会, 2019年12月12日, 龍谷大学

    口頭発表(一般)

  • Every 1D Persistence Module is a Restriction of Some Indecomposable 2D Persistence Module

    M. Buchet, E.G. Escolar

    ICIAM 2019 Thematic Minisymposium: "Geometry and Topology in Data Analysis", 2019年07月16日, Valencia, Spain, スペイン

    口頭発表(一般)

  • Every 1D Persistence Module is a Restriction of Some Indecomposable 2D Persistence Module

    M. Buchet, E.G. Escolar

    Workshop "Computational Applications of Quiver Representations: TDA and QPA", 2019年05月03日, Bielefeld University, Bielefeld, Germany, ドイツ連邦共和国, 国際会議

    口頭発表(一般)

  • Every 1D Persistence Module is a Restriction of Some Indecomposable 2D Persistence Module

    M. Buchet, E.G. Escolar

    MSJ Spring Meeting 2019, 2019年03月17日, 日本語, Tokyo Institute of Technology

    口頭発表(一般)

  • Every 1D Persistence Module is a Restriction of Some Indecomposable 2D Persistence Module

    M. Buchet, E.G. Escolar

    Workshop on Applied Topology 2019, 2019年01月08日, Kyoto

    口頭発表(一般)

  • Realization of Indecomposable Persistence Modules of Large Dimension

    M. Buchet, E.G. Escolar

    Applied Geometry & Topology 2018, 2018年07月24日, Kyoto University, Kyoto

    口頭発表(一般)

  • Computing Indecomposable Decompositions of Persistence Modules

    guest lecture at Institute of Geometry, 2018年06月21日, TU Graz, Graz, Austria

    口頭発表(一般)

  • Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension

    M. Buchet, E.G. Escolar

    34th International Symposium on Computational Geometry (SoCG 2018), 2018年06月11日, Budapest, Hungary, ハンガリー共和国, 国際会議

    口頭発表(一般)

  • Vietoris-Rips Realization of Indecomposable Persistence Modules of Arbitrarily Large Dimension

    M. Buchet, E.G. Escolar

    MSJ Spring Meeting 2018, 2018年03月18日, The University of Tokyo

    口頭発表(一般)

  • Representation Theory and Persistent Homology in TDA

    RIKEN-AIP Math Group Joint Seminar, Izuyama Kensyuu Center

    口頭発表(一般)

  • Vietoris-Rips Realization of Indecomposable Persistence Modules of Arbitrarily Large Dimension

    M. Buchet, E.G. Escolar

    2017年度応用数学合同研究集会, 2017年12月14日, 龍谷大学瀬田キャンパス

    口頭発表(一般)

  • Vietoris-Rips Realization of Indecomposable Persistence Modules of Large Dimension

    M. Buchet, E.G. Escolar

    Applied Algebraic Topology 2017, Hokkaido University, Sapporo, Japan, 国際会議

    口頭発表(一般)

  • An Introduction to Quiver Representation Theory for Topological Data Analysis

    E.G. Escolar

    A3 foresight workshop: Modeling and Simulation of Hierarchical and Heterogeneous Flow Systems with Applications to Materials Science III, 2016年11月14日, Tohoku Forum for Creativity, Tohoku University

    口頭発表(一般)

  • Persistence of Common Topological Features via Commutative Ladder Quivers

    E.G. Escolar, Y. Hiraoka

    EASIAM 2016, 2016年06月21日, University of Macau, Macau, China

    口頭発表(一般)

  • Introduction to representation theory for topological data analysis

    Emerson Gaw ESCOLAR

    A3 foresight winter school on Mathematics on Materials Science: Topological Data Analysis and Dynamics, 2016年02月16日, AIMR, Tohoku University

    口頭発表(一般)

  • Matrix Method for Persistence Modules on Commutative Ladders of Finite Type

    Minisymposium on Topological Data Analysis and Dynamics, 8th International Congress on Industrial and Applied Mathematics, 2015年08月14日, Beijing, China

    口頭発表(一般)

  • Morse Reduction for Persistence Modules on Commutative Ladders of Finite Type

    E.G. Escolar, Y. Hiraoka

    2014年度応用数学合同研究集会, 2014年12月18日, 龍谷大学

    口頭発表(一般)

  • Computing Persistence Modules of Quiver Complexes

    E.G. Escolar, Y. Hiraoka

    Short presentation (student session), Mathematics of Fluid Dynamics and Material Science, 1st Joint Conference of A3 Foresight Program, 2014年11月22日, International Convention Center Jeju, Korea, 国際会議

    口頭発表(一般)

  • Optimal Cycles in Homology via Linear Programming

    E.G. Escolar, Y. Hiraoka

    IMI Workshop on Optimization in the Real World, 2014年10月15日, Kyushu University

    口頭発表(一般)

  • Computing Persistence Modules on Commutative Ladders of Finite Type

    E.G. Escolar, Y. Hiraoka

    MSJ Autumn Meeting 2014, 2014年09月27日, Hiroshima University

    口頭発表(一般)

  • Computing Persistence Modules on Commutative Ladder Quivers of Finite Type

    E.G. Escolar, Y. Hiraoka

    The 4th International Congress on Mathematical Software. Session: Software for Computational Topology, Hanyang University, Seoul, Korea, 国際会議

    口頭発表(一般)

  • Computing Persistence Modules on Commutative Ladder Quivers of Finite Type

    E.G. Escolar, Y. Hiraoka

    Workshop on Topological Data Analysis, 京都大学, 国際会議

    口頭発表(一般)

  • Optimal Cycles of Homology Groups

    Intersection of Pure Mathematics and Applied Mathematics IV, 2014年02月18日, 九州大学

    口頭発表(一般)

  • Morse Reduction for Zigzag Complexes

    E.G. Escolar, Y. Hiraoka

    AKOOS-PNU International Conference 2014, Busan National University, Busan, South Korea, 国際会議

    口頭発表(一般)

  • Morse Reduction for Zigzag Complexes

    E.G. Escolar, Y. Hiraoka

    The 9th East Asia SIAM Conference, The 2nd Conference on Industrial and Applied Mathematics (EASIAM-CIAM 2013), Institut Teknologi Bandung, Bandung, West Java, Indonesia, 国際会議

    口頭発表(一般)

  • Introduction to Topological Data Analysis

    Emerson G. Escolar

    Ateneo de Manila University, School of Science & Engineering, Department of Mathematics, Mathematics Research Seminar Series, 2021年10月08日, 英語

    [招待有り]

    公開講演,セミナー,チュートリアル,講習,講義等

  • パーシステンス加群の区間分解可能性、区間近似、及びその計算

    Emerson G. Escolar

    IMI共同利用研究:位相的データ解析の理論と応用, 日本語

    [招待有り]

  • Mapping Firms' Locations in Technological Space: A Topological Analysis of Patent Statistics

    Emerson G. Escolar

    ILJU POSTECH MINDS Workshop on Topological Data Analysis and Machine Learning, 2021年07月09日, 英語

    [招待有り]

所属学協会

  • 日本応用数理学会

  • 日本数学会

Works(作品等)

  • OptiPersLP: optimal cycles in persistence via linear programming

    Emerson Gaw ESCOLAR

  • gyoza: commutative ladder persistence by solving matrix problems

    Emerson Gaw ESCOLAR

  • pmgap

    Emerson Gaw ESCOLAR

    computations for Persistence Modules using GAP (開発中)