Izeki, Hiroyasu

写真a

Affiliation

Faculty of Science and Technology, Department of Mathematics (Yagami)

Position

Professor

Career 【 Display / hide

  • 1992.04
    -
    1995.03

    東京都立大学理学部助手

  • 1992.04
    -
    1995.03

    東京都立大学理学部助手

  • 1992.04
    -
    1995.03

    東京都立大学理学部助手

  • 1995.04
    -
    1997.03

    名古屋大学大学院多元数理科学研究科助教授

  • 1995.04
    -
    1997.03

    名古屋大学大学院多元数理科学研究科助教授

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Academic Background 【 Display / hide

  • 1989.03

    Keio University, Faculty of Science and Engineering, 数理科学科・数学専攻

  • 1989.03

    Keio University, Faculty of Science and Engineering, 数理科学科・数学専攻

    University, Graduated

  • 1991.03

    Keio University, Graduate School, Division of Science and Engineering, 数理科学専攻

  • 1991.03

    Keio University, Graduate School, Division of Science and Engineering, 数理科学専攻

    Graduate School, Completed, Master's course

Academic Degrees 【 Display / hide

  • 博士(理学), Keio University, Dissertation, 1994.12

 

Research Areas 【 Display / hide

  • Natural Science / Geometry

  • Natural Science / Geometry

Research Keywords 【 Display / hide

  • rigidity

  • rigidity

  • fixed-point property

  • fixed-point property

  • finitely generated group

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Research Themes 【 Display / hide

  • rigidity and fixed-point property of discrete groups, 

    2003
    -
    Present

 

Books 【 Display / hide

  • Yamabe Problem

    Osamu Kobayashi, Kazuo Akutagawa, IZEKI Hiroyasu, Mathematical Society of Japan, 2013.12

    Scope: Chapter 9

  • 文科系のための自然科学総合実験

    IZEKI Hiroyasu, 2008.03

    Scope: 163-173

  • 数学事典 第4版

    IZEKI Hiroyasu, NAYATANI Shin, 2007.03

    Scope: 252-255

  • 21世紀の数学 --- 幾何学の未踏峰

    IZEKI Hiroyasu, 2004.07

    Scope: 168-182

Papers 【 Display / hide

  • A fixed point property of random groups

    IZEKI Hiroyasu

    京都大学数理解析研究所講究録  2018.04

    Research paper (conference, symposium, etc.), Single Work

  • Poisson boundary and rigidity of discrete groups

    IZEKI Hiroyasu

    Geometry and Analysis, Fukuoka 1   117 - 122 2015.12

    Research paper (conference, symposium, etc.), Single Work

  • Fixed-point property of random quotients by plain words

    IZEKI Hiroyasu

    Groups, Geometry, and Dynamics (EUROPEAN MATHEMATICAL SOC)  8 ( 4 ) 1101 - 1140 2014.12

    Research paper (scientific journal), Single Work, Accepted,  ISSN  1661-7207

     View Summary

    We show a fixed-point property of certain random groups for a wide class of CAT(0) spaces. The model of random groups under consideration is given as the set of presentations (S, R), where S is a generating set and the set of relations R is a subset of the set of all plain words of the same length with suitably fixed density. Our main theorem says that, with high probability, groups obtained by such presentations have the fixed-point property for all CAT(0) spaces having bounded singularities.

  • N-step energy of maps and fixed-point property of random groups

    IZEKI Hiroyasu, KONDO Takefumi, and NAYATANI Shin

    Geometry, Groups, and Dynamics 6 ( 4 ) 701-736 2012

    Research paper (scientific journal), Joint Work, Accepted

  • N-step energy of maps and the fixed-point property of random groups

    Hiroyasu Izeki, Takefumi Kondo, Shin Nayatani

    GROUPS GEOMETRY AND DYNAMICS (EUROPEAN MATHEMATICAL SOC)  6 ( 4 ) 701 - 736 2012

    Accepted,  ISSN  1661-7207

     View Summary

    We prove that a random group of the graph model associated with a sequence of expanders has the fixed-point property for a certain class of CAT (0) spaces. We use Gromov's criterion for the fixed-point property in terms of the growth of n-step energy of equivariant maps from a finitely generated group into a CAT.(0) space, for which we give a detailed proof. We estimate a relevant geometric invariant of the tangent cones of the Euclidean buildings associated with the groups PGL (m, Q(r)), and deduce from the general result above that the same random group has the fixed-point property for all of these Euclidean buildings with m bounded from above.

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Papers, etc., Registered in KOARA 【 Display / hide

Reviews, Commentaries, etc. 【 Display / hide

  • Isometric group actions with vanishing rate of escape on CAT(0) spaces

    Hiroyasu Izeki

     2022.04

     View Summary

    Let $\Gamma$ be a finitely generated group equipped with a symmetric and
    nondegenerate probability measure $\mu$ with finite second moment, and $Y$ a
    CAT(0) space which is either proper or of finite telescopic dimension. We show
    that if an isometric action of $\Gamma$ on $Y$ has vanishing rate of escape
    with respect to $\mu$ and does not fix a point in the boundary at infinity of
    $Y$, then there exists a flat subspace in $Y$ which is left invariant under the
    action of $\Gamma$. In the proof of this result, an equivariant $\mu$-harmonic
    map from $\Gamma$ into $Y$ plays an important role.

Presentations 【 Display / hide

  • 有限生成群からCAT(0)空間への調和写像と境界写像

    井関 裕靖

    大阪大学幾何セミナー, 

    2022.06

    Oral presentation (invited, special)

     View Summary

    有限生成群がCAT(0)空間に無限遠境界に固定点を持たないような作用をしているとき、不変平坦部分空間が存在するか、あるいは群のポアソン境界からCAT(0)空間の境界への標準的同変写像が存在するかのいずれかが成り立つことを解説した。

  • Boundary maps and harmonic maps from finitely generated groups into CAT(0) spaces

    Hiroyasu Izeki

    Workshop on Surface Theory -- UY60 --, 

    2022.03

    Oral presentation (invited, special)

     View Summary

    有限生成群がCAT(0)空間に無限遠境界に固定点を持たないような作用をしているとき、不変平坦部分空間が存在するか、あるいは群のポアソン境界からCAT(0)空間の境界への標準的同変写像が存在するかのいずれかが成り立つことを解説した。

  • Boundary maps from finitely generated groups to CAT(0) spaces

    Hiroyasu Izeki

    Boston-Keio-Tsinghua Workshop, 

    2021.06
    -
    2021.07

    Oral presentation (invited, special)

     View Summary

    有限生成群がCAT(0)空間に無限遠境界に固定点を持たないような作用をしているとき、不変平坦部分空間が存在するか、あるいは群のポアソン境界からCAT(0)空間の境界への標準的同変写像が存在するかのいずれかが成り立つことを解説した。

  • 離散群からの調和写像・境界写像と剛性

    井関 裕靖

    早稲田大学双曲幾何幾何学的群論セミナー, 

    2018.12

    Public lecture, seminar, tutorial, course, or other speech

  • 離散群の境界写像と剛性

    IZEKI Hiroyasu

    幾何学阿蘇研究集会, 

    2018.09

    Oral presentation (keynote)

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Research Projects of Competitive Funds, etc. 【 Display / hide

  • 幾何学的剛性理論の深化

    2020.04
    -
    2025.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, 井関 裕靖, Grant-in-Aid for Scientific Research (B), Principal investigator

  • Rigidity of non-isometric actions of discrete groups and non-linear spectral gap

    2017.04
    -
    2022.03

    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B), Grant-in-Aid for Scientific Research (B), No Setting

     View Summary

    昨年度に引き続き, 球面S^n内の高種数の極小閉曲面を構成する問題について研究を行った. 研究の動機として, ラプラシアンの第1固有値を最大化する閉曲面上の計量が球面への極小はめこみによる誘導計量として与えられることが知られており, したがって, 球面内の高種数の極小閉曲面を構成することで, そのような最大化計量の候補が得られるということがある. 種数3以上の場合に最大化計量が未知であるので, とくに種数が低い例を与えることを目指している.
    LawsonによるS^3内の高種数極小閉曲面の構成を高次元化し, S^nの3角形分割として正2^{n+1}胞体を用いることで高種数の極小閉曲面を構成した. ただ, この構成で得られる極小閉曲面は自己交叉点を多く持ち, しかも種数が高くなる傾向がある(S^5, S^6で種数5, S^7で種数49等). 最大化計量を与える極小閉曲面は, 高い対称性を持つとともに, 自己交差が少ないことが期待されるので, 今後, 別の構成法を考案して, 自己交叉が少なく種数も低い例を見出したい.
    また, 昨年度に引き続き, 有限グラフのユークリッド空間への埋め込みに関する最適化問題と線形スペクトルギャップの最大化問題について研究を行なった. この研究は, 有限グラフの非線形スペクトルギャップの研究に動機付けられたものである. 類似の埋め込み最適化問題としてGoering-Wappler-Helmbergによるものが知られていたが, 両者の埋め込み最適化問題の関係を明らかにすることにより, 我々の埋め込み最適化問題の最適値と第線形スペクトルギャップの最大値の間にもある等式が成立することが示せた. また, アルキメデス多面体を始めとして対称性の高い様々な多面体について, Goering達および我々の埋め込み最適化問題の解を求めることができた.

  • An approach to the superrigidity of infinite discrete groups via random groups

    2013.04
    -
    2018.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, IZEKI Hiroyasu, NAYATANI Shin, KONDO Takefumi, Grant-in-Aid for Scientific Research (B), Principal investigator

     View Summary

    The group is an algebraic object which also gives a description of symmetries of spaces. Some important and interesting groups often admits a property called "superrigidity", which we tried to understand as an extremal property among that involving infinite discrete groups. We could show that a fixed-point property, which should be considered to be an important aspect of superrigidity, is shared by finitely presented groups with overwhelming probability.

  • Geometric structures related to neutral metrics

    2012.04
    -
    2016.03

    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), Kamada Hiroyuki, AIHARA YOSHIHIRO, NAYATANI SHIN, NAKAGAWA YASUHIRO, IZEKI HIROYASU, NAKATA FUMINORI, Grant-in-Aid for Scientific Research (C), No Setting

     View Summary

    A pseudo-Riemannian metric on a manifold is called a neutral metric if it has neutral signature, and a family of local neutral metrics that conicide, except for multiplication by -1, on the overlaps, is called a neutral structure. Davidov et al. obtained examples of compact complex surfaces with a quaternion-like structure (parahypercomplex structure) that admit compatible neutral structures, but never admit any compatible neutral metric. Then we show that their example of a hyperelliptic surface can be deformed to a compatible neutral structure, which is not locally conformal parahyperkahler. Also, we introduce the notion of strong integrability for a quaternionic CR manifold (of dimension greater than 7), and show that, under ultra pseudoconvexity and strong integrability, a partially integrable almost CR structure (called the twistor almost CR structure) is defined naturally on its twistor space.

  • New approach to discrete geometry --- capturing the shape of finite groups

    2012.04
    -
    2015.03

    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research, IZEKI Hiroyasu, Grant-in-Aid for Challenging Exploratory Research, No Setting

     View Summary

    The aim of this research was capturing the higher dimensional structure of discrete metric spaces and getting some rigidity results as applications of this structure. We could not obtain a satisfactory result describing the higher dimensional structure of discrete sets, however, using an observation and results obtained in the course of this research, we proved that certain random groups have strong fixed-point property for a wide class of nonpositively curved metric spaces. Also we gave some good estimates of nonlinear spectral gap for some embeddings of discrete metric spaces into nonpositively curved metric spaces.

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Awards 【 Display / hide

  • 日本数学会建部賢弘賞

    IZEKI Hiroyasu, 1998.09, 日本数学会

    Type of Award: Award from Japanese society, conference, symposium, etc.

  • 日本数学会建部賢弘賞

    IZEKI Hiroyasu, 1998.09, 日本数学会

    Type of Award: Award from Japanese society, conference, symposium, etc.

 

Courses Taught 【 Display / hide

  • TOPICS IN LIFE INSURANCE MATHEMATICS

    2022

  • MATHEMATICS 2B

    2022

  • MATHEMATICS 2A

    2022

  • INTRODUCTION TO LIFE INSURANCE

    2022

  • INTRODUCTION TO GEOMETRY

    2022

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Courses Previously Taught 【 Display / hide

  • MATHEMATICS 2B

    Keio University

    2022.04
    -
    2023.03

  • TOPICS IN GEOMETRY C

    Keio University

    2021.04
    -
    2022.03

  • INTRODUCTION TO LIFE INSURANCE

    Keio University

    2020.04
    -
    2021.03

  • TOPICS IN LIFE INSURANCE MATHEMATICS

    Keio University

    2020.04
    -
    2021.03

  • FOUNDATION FOR MATHEMATICAL SCIENCE 1 AND EXERCISE

    Keio University

    2020.04
    -
    2021.03

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Memberships in Academic Societies 【 Display / hide

  • Mathematical Society of Japan, 

    1990.04
    -
    Present
  • Mathematical Society of Japan, 

    1990.04
    -
    Present

Committee Experiences 【 Display / hide

  • 2012.03
    -
    2013.02

    全国区代議員(評議員), 日本数学会

  • 2012.03
    -
    2013.02

    全国区代議員(評議員), 日本数学会

  • 1990.04
    -
    Present

    Member, Mathematical Society of Japan

  • 1990.04
    -
    Present

    Member, Mathematical Society of Japan