Researchers Database

KATAGIRI Minyo

FacultyFaculty Division of Natural Sciences Research Group of Mathematics
PositionAssociate Professor
Last Updated :2024/04/03

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Profile and Settings

  • Name (Japanese)

    Katagiri
  • Name (Kana)

    Minyo

Research Interests

  • graph polinomial, categorification, cohomology group

Research Areas

  • Natural sciences, Geometry

Research Experience

  • 1996, 1999, 奈良女子大学理学部講師
  • 1996, 1999, Nara Women's University, Assistant Professor
  • 1999, 奈良女子大学理学部助教授
  • 1999, Nara Women's University, Assosiate Professor
  • 1994, 1996, 奈良女子大学理学部助手
  • 1994, 1996, Nara Women's University, Assistant

Education

  • 1994, Keio University, Graduate School, Division of Science and Engineering, 数理科学
  • 1990, Keio University, Faculty of Science and Engineering, 数理科学/数学

Association Memberships

  • Nihon Origami Association
  • Society for Science on Form
  • Mathematical Society of Japan

Ⅱ.研究活動実績

Published Papers

  • Refereed, PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES, JAPAN ACAD, On conformally flat critical Riemannian metrics for a curvature functional, M Katagiri, The normalized L-2-norm of the traceless part of the Ricci curvature defines a Riemannian functional on the space of Riemannian metrics. In this paper, we will consider the critical Riemannian metrics with a flat conformal structure for this functional., Feb. 2005, 81, 2, 27, 29, Scientific journal
  • Refereed, PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES, JAPAN ACAD, On critical Riemannian metrics for a curvature functional on 3-manifolds, M Katagiri, The normalized L-2-norm of the traceless part of the Ricci curvature defines a Riemannian functional on the space of metrics. In this paper, we will consider this functional on 3-manifolds., Apr. 2002, 78, 4, 43, 45, Scientific journal
  • Refereed, PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES, JAPAN ACAD, On the topology of the moduli space of negative constant scalar curvature metrics on a Haken manifold, M Katagiri, Sep. 1999, 75, 7, 126, 128, Scientific journal
  • Refereed, PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES, JAPAN ACAD, On compact conformally flat Einstein-Weyl manifolds, M Katagiri, Jun. 1998, 74, 6, 104, 105, Scientific journal
  • Refereed, Tokyo Journal of Mathematics, On the uniqueness of a Weyl Structure with Prescribed Ricci Curvature, Minyo Katagiri, 1998, 21, 2, 453, 455, Scientific journal, 10.3836/tjm/1270041825
  • Refereed, Tokyo Journal of Mathematics, On deformations of Einstein-Weyl structures, Minyo Katagiri, 1998, 21, 2, 457, 461, Scientific journal, 10.3836/tjm/1270041826
  • Refereed, Functional Analysis and Global Analysis, Estimate of singularities of the Yang-Mills gradient flow, Minyo Katagiri, 1997, 162
  • Refereed, JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, MATH SOC JAPAN, ON THE EXISTENCE OF YANG-MILLS CONNECTIONS BY CONFORMAL CHANGES IN HIGHER DIMENSIONS, M KATAGIRI, Jan. 1994, 46, 1, 139, 147, Scientific journal
  • Refereed, Annual report of Graduate School of Humanities and Sciences Nara Women's University, A categorification for the dichromatic polynomial of graphs, Minyo Katagiri, Mar. 2022, 37, 37, 111, 118, Research institution
  • Refereed, Annual Reports of Graduate of Humanities and Sciences, On the group of the dichromatic cohomology of graphs, Minyo Katagiri, Mar. 2024, 39, 1, 15, 23, Research institution
  • Refereed, Annual Reports of Graduate School of humanities and Sciences, A decomposition of the space of Riemannian metrics and Riemannian functionals, Minyo Katagiri, Mar. 2020, 35, 113, 120
  • Refereed, Annual Report of Graduate School of Humanities and Sciences Nara Women's University, A categorification for the flow polynomial of graphs, Minyo Katagiri, Mar. 2018, 33, 113-121
  • Refereed, Annual Report of Graduate School of Humanities and Sciences Nara Women's University, 奈良女子大学大学院人間文化研究科, Upper bounds for the Roman bondage number of graphs on closed surfaces, KATAGIRI Minyo, Let G be a simple graph, and its vertex sets is denoted by V (G). A set D V (G)is the dominating set if every vertex not in D is adjacent to at least one vertex in D.The minimum cadinality of a dominatin set of G is the dominationg number (G).Clearly, for any spanning subgraph H of G, (H) (G). The bondage numberof G, denoted by b(G), is the minimum cardinality of a set of edges B E(G)such that (G − B) > (G), where G − B is the graph with V (G − B) = V (G)and E(G − B) = E(G) \ B.A function f : V (G) {0, 1, 2} is a Roman dominating function if every vertexv for which f(v) = 0 is adjacent to at least one vertex u for which f(u) = 2. Theweight of a Roman dominating function is the value v V (G) f(v). The Romandomination number of a graph G, denoted by R(G), is the minimum weight of aRoman dominating function of G. The Roman bondage number bR(G) of a graph Gis the cardinality of a smallest set of edges B E(G) for which R(G−B) > R(G),where V (G − B) = V (G) and E(G − B) = E(G) \ B.In this paper, for a graph G on a closed surface M, we get an upper bound forthe Roman bondage number bR(G) of G by Euler characteristic (M) of M., Mar. 2017, 32, 32, 119,124, 124

MISC

  • Not Refereed, 教育システム研究, 数学的活動を通じた数列学習の実践検討―高等学校数学科教育における高大連携授業研究の試み, KATAGIRI Minyo, Mar. 2016, 12, 123-129

Books etc

  • 科学の言語としての数学(LADy SCIENCE BOOKLET 8), 奈良女子大学理系女性教育開発共同機構, KATAGIRI Minyo, Mar. 2016, 97,109, Not Refereed

Research Projects

  • 2010, グラフ多項式のカテゴリー化に関する研究, 0, 0, 0, Competitive research funding
  • Grant-in-Aid for Scientific Research (C), 2007, 2009, 19540082, Developments of nilpotent geometry and nilpotent analysis, MORIMOTO Tohru; KOISO Miyuki; ARAKAWA Tomoyuki; ISHIKAWA Goo; FURUTANI Kenro; MACHIDA Yoshinori; KIYOHARA Kazuyoshi; AGAOKA Yoshio; KISO Kazuhiro; NAKANISHI Nobutada; KATAGIRI Minyo, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 3250000, 2500000, 750000, Based on nilpotent geometry and analysis, a trinity of representation of Lie algebras, integrable system of linear differential equations and extrinsic geometry in flag manifolds is established. Moreover, we obtain a general method to calculate the invariants of an integrable system of linear differential equations (or those of a submanifold in a flag manifold) associated with a representation of a Lie algebra, if the Lie algebra is simple., kaken
  • Grant-in-Aid for Scientific Research (C), 2007, 2008, 19540083, Research on 3-manifolds based on geometric techniques and its expanse, KOBAYASHI Tsuyoshi; YAMASHITA Yasushi; KATAGIRI Minnyou, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Nara Women's University, 1950000, 1500000, 450000, 片桐ははリーマン計量全体の中の臨界リーマン計量に関して研究を行った. 山下は2元生成メビウス変換群と3次元双曲幾何学との関連について研究を行った。小林は三次元多様体のHeegaard分解, 写像類群を利用した流体の混合に関する研究を行った. これらに関して例えば, 高いHempel距離を持つHeegaard分解を許容する三次元多様体を境界で貼りあわせて得られる三次元多様体の既約なHeegaard分解は必ずこれらのHeegaard分解の融合(amalgamation)になっていることが分かった, 等の結果が得られた., kaken
  • Grant-in-Aid for Scientific Research (C), 2005, 2006, 17540077, Geometric structures of 3-manifolds and various related structures, KOBAYASHI Tsuyoshi; YAMASHITA Yasushi; KATAGIRI Minnyou, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Nara Women's University, 2200000, 2200000, In this research project, we obtained the following results. 1. We defined a numerical invariant, called growth rate of tunnel numbers, of knots in 3-manifolds. For m-small knots, we obtained the following. Suppose K is a m-small knot in. a 3-manifold M. Let g = g(X)-g(M), and b_i (i =1,..., g) be the bridge index of K with respect to genus g(X) - i Heegaard surface of M. Then the growth rate of K is given by min_i=_<1,..., n>{1-i/(b_i)}. 2. Heegaard splittings of exteriors of knots. ・ Let K_1, K_2 be knots in 3-manifolds, and T_1,T_2 tunnel systems of K_1, K_2 respectively. We gave a necessary and sufficient condition for the tunnel system t_1 ∪ T_2 of K_1#K_2 giving a stabilized Heegaard splitting. ・ For each natural number n, there exists a knot K such that the equality g(nK) = gt(K) holds, where nK denotes the connected sum of n copies of K. This implies the existence of counterexample to Morimoto's Conjecture concerning super additive phenomina of tunnel number of knots. 3. We showed that for any link L in the 3-sphere, there is a Seifert surface S for L such that S is obtained from a disk by successively plumbing flat annuli, where all of the attaching regions are contained in the disk. 4. We made research on Gersten's Problem : each automatic group is either (1) a finite group, (2) contains a free abelian group of rank 2. or (3) a word hyperbolic group. We showed that for the n-starred automatic groups this assertion holds. 5. Growth function of 2-bridge link groups We made computar experiments on the growth functions of 2-bridge link groups, and posed conjectures on the structure of the growth functions., kaken
  • Grant-in-Aid for Scientific Research (C), 2003, 2004, 15540073, Research on various geometric structures on 3-manifolds, KOBAYASHI Tsuyoshi; YAMASHITA Yasushi; KATAGIRI Minnyou; ICHIHARA Kazuhiro, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Nara Women's University, 2100000, 2100000, 1.Morimoto's Conjecture on the tunnel numbers of composite knots in 3-manifolds Let t(K) be the tunnel number of a knot K in a 3-manifold. Suppose for m-small knots K_1,…,K_n, the super additivity of tunnel number does not hold for #^n_ K_i, Then we proved that there exists a subset I of {1,【triple bond】,n} such that #_/K_i admits a primitive meridian. 2.The growth rate of tunnel number of knots For a knot K in a 3-manifold M, we defined a numerical invariant called the growth rate of the tunnel numbers of K, and proved the following. Suppose that the Heegaard genus of K is greater than the Heegaard genus of M. Then the growth rate of the tunnel numbers of K is less than 1. 3.Gersten's Problem for automatic group Gersten posed the following problem "Each automatic group is eithr (1)a finite group, (2)contains free abelian group of rank 2,or (3)a word hyperbolic group." We showed that for a class of automatic group (called n-starred groups) this problem is solved affirmatively. 4.Heegaard gradients Seifert fibered spaces We completely determined for which Seifert fibered space, the Heegaard gradient vanish., kaken
  • Grant-in-Aid for Scientific Research (C), 2000, 2002, 12640071, Representations of 3-manifolds and geometric informations derived from them, KOBAYASHI Tsuyoshi; KATAGIRI Minnyou; YAMASHITA Yasushi; OCHIAI Mitsuyuki, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Nara Women's University, 2800000, 2800000, 1. Graphic of 3-manifolds Kobayashi make use of the graphic defined by Rubinstein-Scharlemann to give a complete classification of Heegaard splittings of the exteriors of the 2-bridge knots. 2. Local detection of strong irreducibility of Heegaard splittings by using knot exteriors Kobayashi together with, Yo'av Rieck analyzed how strongly irreducible Heegaard splittings can intersect the exteriors of non-trivial knots in the 3-sphere and showed that such Heegaard surface intersect the knot exteriors in meridional annuli. 3. Research on Morimoto's Conjecture Kobayashi together with Yo'av Rieckstudied about Morimoto's Conjecture concerned with the connectedsums of knots in 3-manifolds and the tunnel numbers. 4. Algorithm for decompositions of attaching homeomorphisms of Heegaard splittings into Dehn twists Ochiai gave an algorithm for giving a decomposition of given attaching homeomorphisms of genus two Heegaard splittings into standard Dehn twists. 5. Moduli space of metrics of Riemannian manifolds Katagiri studied about Riemannian functional via Ricci curvature and showed that Einstein metric is a critical point of this functional, however there exist critical points that are not Einstein metric. He also gave a sufficient condition for critical points to be Einstein metrics., kaken
  • 萌芽的研究, 1999, 2001, 11874011, Schwarz微分の幾何, 小林 治; 片桐 民陽, 日本学術振興会, 科学研究費助成事業, 1200000, 1200000, 今年度でようやく研究の方向づけができた。投稿中の論文が3編あるが、いまだ投稿中なので裏面にかけないことが残念である。今年度の研究の結果は、今後とりくむべき問題が明確になってきたことにある。 (1)Schwarz微分を用いて多様体のMobius幾何を展開すること、例えばHopf Rinuw型の定理「任意の3点を通す測地円が存在する」か?など基本的なところで未解決問題が多い。 (2)(1)と関係しているがMobius円の変分問題的特徴が不備である。 (3)本Schwarz微分はこれまでのものと異なり、共形的でない写像にも適用可能である。この点に注目すると異なる共形類の違いを定量的に表現する機能がある。これを用いて共形類のモデュライ、特に正スカラー曲率計量を含むもののモデュライの研究に役に立つ可能性がある。 (4)我々は2種類の新しいSchwarz微分を創出したが、本当に機能するものはこれらを統合的に一般化された第3のSchwarz微分であろう。 研究当初から未解決問題は多くあったが、それらの問題を解決するには至ってないがより深い理解ができたことが基本的成果である。, kaken
  • Grant-in-Aid for Scientific Research (C), 1998, 1999, 10640076, Combinatorial structures of low dimensional manifolds, KOBAYASHI Tsuyoshi; KATAGIRI Minnyou; WADA Masaaki; OCHIAI Mitsuyuki; NIIDE Naoyuki, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Nara Women's University, 3800000, 3800000, Recently "low dimensional topology theory" is far going out of the framework of "geometry" and finding out intimate relations between group theory, complex analysis, dynamical system, and even some fields out of mathematics like theoretical physics, and computer science. Within the relations, there are many (very huge, in general) combinatorial structures, for example, train tracks which give coordinates on Teichmuller spaces, canonical decomposition of hyperbolic 3-manifolds by ideal cells (Epstein-Penner), constructions of representations of Hecke algebra by Young diagram (Jones), automatic group theory (Thurston), and normal surface theory by Haken. In connection with these phenomena, it seems that recent development of low dimensional topology and of computer enables us to treat these objects directly and concretely. In view of these situations, in this research, we intended to study 2 and 3 dimensional manifolds from geometrical can combinatorial viewpoint. Concretely speaking, we studied about the following topics. ・Analyzing 3-manifolds and knots via Heegaard splitting (particularly, with using "graphic" introduced by Rubinstein- Scharlemann), and obtaining useful informations on unknotting tunnels of knots, ・Studying hyperbolic structures on 3-manifolds via triangulations, particularly on hyperbolic structures on 2-bridge knot complements starting from a very simple hyperbolic structure, ・Studying about the relations between moduli spaces of certain kind of Riemannian metrics of 3-manifolds and geometric structures, ・Studying about algorithms (that the computer can handle) to decompose the attaching homeomorphisms of the given Heegaard splittings into canonical Dehn twists., kaken
  • Grant-in-Aid for Scientific Research (B), 1997, 1999, 09440034, Gemetric Structures on Manifolds and Global Analysis, KOBAYASHI Osamu; FUJIOKA Atsushi; KITAHARA Haruo; KODAMA Akio; KATO Shin; KATAGIRI Minyo, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 9200000, 9200000, Among many geometric structures of a manifold we are mainly interested in those structures which are closely related to the conformal geometry. Here are some of main results of this research project : 1. The scalar curvature equation. This equation describes the scalar curvature under a conformal change of a Riemannian metric. A systematic analysis has been done on non-compact manifolds, and the space of complete confomal metrics with prescribed scalar curvature is made clearer. 2. The Weyl structure. This is a torsion free affine connection that is compatible with a given conformal class. It is shown that the Ricci curvature is a complete invariant of a Weyl structure. Also conformally flat Einstein-Weyl structures on compact manifolds are classified. 3. Moebius geometry. The minimum number of vertices of a regular closed curve on the sphere with given topological type is completely determined in the case when the curve has at most five self-inter-sections. Also we introduce a Schwarzian derivative of a regular curve. This leads to new proofs of injectivity results of Nehari type. A gist is that a confomal strucutre of a manifold induces an integrable projective structure of a regular curve on the manifold. It is shown that injectivity of the projective development map of the curve implies the injectivity of the immersion to Moebius spaces., kaken
  • 奨励研究(A), 1996, 1996, 08740055, 共形構造と接続の幾何学に関する研究, 片桐 民陽, 日本学術振興会, 科学研究費助成事業, 奈良女子大学, 1000000, 1000000, 20世紀初頭,ワイルは統一場理論を定式化する上で,リーマン多様体の一般化として,今日,ワイル多様体と呼ばれる概念を導入した.物理現象の記述の可能性は別として,ワイルの理論は数学的には興味ある対象であると考えられる.ところが,現代的立場に立ったとき,今世紀における,大域解析学を用いたリーマン幾何学の発展と比較すると,ワイル多様体の幾何学は置き去りされてきた感がある.本研究の問題意識は,共形構造と接続の幾何学として,ワイル多様体の,近年の大域解析学の技術を用いた展開を試みることにあった.今年度に行った研究により得られた研究成果は以下のとおりである. 1.アインシュタイン計量におけるアインシュタイン・ワイル構造の変形についてのある種の剛性に関する研究. 2.ワイル構造に対する,リッチ曲率の完全不変量としての特徴付け. 3.共形的に平坦なアインシュタイン・ワイル多様体の分類. これらの結果は,今後,この研究を進めて行く上で,基礎的な部分を占めるものであると考えられる.更に,今年度,本研究を行うことにより,アインシュタイン・ワイル構造に対する,ある不変量と作用素が,これらの研究を進めて行く上で非常に重要な対象である,ということを認識するに至った.基礎的な問題であるにも関らず,今年度の本研究で明らかにされなっかた問題についても,上に述べた不変量や作用素をより詳しく調べることにより,新たな視点を与えるものであると考えられる., kaken
  • 一般研究(C), 1995, 1995, 07640114, 低次元多様体の組合せ的構造の研究, 小林 毅; 篠田 正人; 片桐 民陽; 和田 昌昭; 小林 治; 落合 豊行, 日本学術振興会, 科学研究費助成事業, 奈良女子大学, 2400000, 2400000, 今年度の研究により次のような結果が得られた。 1.低次元(2,3,4次元)の多様体の構造の幾何的観点からの研究. 曲面上の(単純でない)閉曲線のガウス語から閉曲線の正則ホモトピー類の不変量を構成した(小林(治),なお,この結果に関しては1996年度日本数学会年会で報告の予定).結び目の(通常の)数種と,canonicalな種数,freeな種数は本質的に異なるものであることを明らかにした(小林(毅)).三次元球面内の結び目に対して“局所的にthin"と呼ばれる概念を定義し全ての結び目はその様な位置にもって行けることをアルゴリズム的に証明した(小林(毅),なお,この結果に関しては加太で行われた集会“結び目の諸問題と最近の成果"で報告が行われた).3次元多様体のHeegarrd分解に関する結果を拡張して2橋結び目の種数の1の1橋表現は標準的なものしかないことを明らかにした(小林(毅)). 2.低次元(2,3,4次元)の多様体の構造の組み合わせ的観点からの研究. pre-Sierpinskiガスケット上のパーコレーションに関する研究(篠田,この結果に関しては関西確率論セミナーで報告が行われた).W-graphを利用してヘッケ環H(q,n)の表現をn=15まで書き下す方法を与えた(落合.これに関しては賢島で行われた研究集会“Art of low dimensional topology"で報告が行われた).双曲的三次元多様体の理想的単体による分割が理想的な単体による分割に細分される為の十分条件を求めた(和田,山下)., kaken
  • 一般研究(C), 1994, 1994, 06640141, 共形構造及び射影構造に関する幾何学, 小林 治; 山下 靖; 和田 昌昭; 静田 靖; 片桐 民陽; 落合 豊行, 日本学術振興会, 科学研究費助成事業, 奈良女子大学, 1900000, 1900000, 球面上の閉曲線のトポロジーと幾何については,すべての自己交点において2つの単純ループに分解可能な閉曲線の最小頂点数を決定した。この結果により自己交点数が5以下のすべての閉曲線の位相型について最小自己交点数が明らかになった。この研究と関連してトーラス上の閉曲線の回転指数についての新たな公式を得た。これは正則ホモトピーについての結果であり,今後の高次元化へ進む足がかりとなりうるものである。 共形変換で不変な変分問題に関する研究として,研究分担者の片桐はYang-Mills接続の存在定理を5次元以上のRiemann多様体において示した。これは5次元以上ではこの変分問題が共形変換での不変性を失うことに着眼点をおき得られたものである。 射影構造に関する内在的な幾何の研究については研究は継続中である。射影反転の多様体での定式化がこの報告書を書いている時点での課題である。 双曲幾何に関しては,関連する3次元多様体論,結び目理論から分担者の落合,山下,和田による成果があった。落合,山下は結び目理論研究支援システムの設計を行い,また和田は新たな結び目不変量を定義し,それによって樹下・寺阪結び目とConway結び目が区別できるという成果を得た。, kaken


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