Researchers Database

YAMASHITA Yasushi

FacultyFaculty Division of Natural Sciences Research Group of Mathematics
PositionProfessor
Last Updated :2022/10/06

researchmap

Profile and Settings

  • Name (Japanese)

    Yamashita
  • Name (Kana)

    Yasushi

Degree

  • Doctor of Science, Tokyo Institute of Technology, Mar. 1996

Research Interests

  • growth function
  • experimental mathematics
  • visualization
  • geometric structure
  • cone manifolds
  • knots
  • hyperbolic groups
  • low-dimensional topology
  • projective structure
  • Teichmuller theory
  • automatic groups
  • configuration space
  • hyperbolic Dehn surgery
  • Kleinian groups
  • 3-manifolds
  • hyperbolic geometry
  • geometric group theory

Research Areas

  • Natural sciences, Geometry

Research Experience

  • Apr. 2022, 9999, Nara Women's University, STEAM・融合教育開発機構, 機構長, Japan
  • Apr. 2012, 9999, Nara Women's University, Faculty Division of Natural Sciences, 教授
  • Jan. 2010, Mar. 2012, Nara Women's University, Faculty of Science, 教授, Japan
  • Apr. 2007, Dec. 2009, Nara Women's University, Faculty of Science, 准教授
  • Aug. 2005, Mar. 2007, Nara Women's University, Faculty of Science, 助教授
  • Jan. 1996, Jul. 2005, Nara Women's University, Faculty of Science, 講師
  • Jul. 1991, Dec. 1996, Nara Women's University, Faculty of Science, 助手

Teaching Experience

  • Geometric Group Theory, Nara Women's University, 99 2019
  • Special lecture on hyperbolic geometry, Nara Women's University, 99 2018
  • Basic Science 2, Nara Women's University, 99 2016
  • The science you need to know before you enter the workforce, Nara Women's University, 99 2016
  • Basic Science 1, Nara Women's University, 99 2016
  • Graph Theory, Nara Women's University, 99 2015
  • Hyperbolic Geometry, Nara Women's University, 99 2015
  • Programming, Nara Women's University, 99 2015
  • SCORE, Nara Women's University, 19 2016

Ⅱ.研究活動実績

Published Papers

  • Refereed, Algebraic & Geometric Topology, Mathematical Sciences Publishers, The diagonal slice of Schottky space, Caroline Series; Ser Tan; Yasushi Yamashita, An irreducible representation of the free group on two generators X,Y into SL(2,C) is determined up to conjugation by the traces of X,Y and XY. We study the diagonal slice of representations for which X,Y and XY have equal trace. Using the three-fold symmetry and Keen-Series pleating rays we locate those groups which are free and discrete, in which case the resulting hyperbolic manifold is a genus-2 handlebody. We also compute the Bowditch set, consisting of those representations for which no primitive elements in the group generated by X,Y are parabolic or elliptic, and at most finitely many have trace with absolute value at most 2. In contrast to the quasifuchsian punctured torus groups originally studied by Bowditch, computer graphics show that this set is significantly different from the discreteness locus., 03 Aug. 2017, 17, 4, 2239, 2282, Scientific journal
  • Refereed, Algebraic & Geometric Topology, Mathematical Sciences Publishers, Cosmetic surgery and the link volume of hyperbolic 3–manifolds, Yo’av Rieck; Yasushi Yamashita, We prove that for any V > 0 there exists a hyperbolic manifold M-V such that Vol(M-V) < 2.03 and LinkVol (M-V) > V. This was conjectured by the authors in [Algebr. Geom. Topol. 13 (2013) 927-958, Conjecture 1.3].The proof requires study of cosmetic surgery on links (equivalently, fillings of manifolds with boundary tori). There is no bound on the number of components of the link (or boundary components). For statements, see the second part of the introduction. Here are two examples of the results we obtain:(1) Let K be a component of a link L in S-3. Then "most" slopes on K cannot be completed to a cosmetic surgery on L, unless K becomes a component of a Hopf link.(2) Let X be a manifold and epsilon > 0. Then all but finitely many hyperbolic manifolds obtained by filling X admit a geodesic shorter than epsilon. (Note that it is not true that there are only finitely many fillings fulfilling this condition.), 15 Dec. 2016, 16, 6, 3445, 3521, Scientific journal
  • Refereed, International Journal of Algebra and Computation, World Scientific Pub Co Pte Lt, Non-hyperbolic automatic groups and groups acting on CAT(0) cube complexes, Yoshiyuki Nakagawa; Makoto Tamura; Yasushi Yamashita, We discuss a problem posed by Gersten: Is every automatic group which does not contain ℤ × ℤ subgroup, hyperbolic? To study this question, we define the notion of "n-track of length n", which is a structure like ℤ × ℤ, and prove its existence in the non-hyperbolic automatic groups with mild conditions. As an application, we show that if a group acts freely, cellularly, properly discontinuously and cocompactly on a CAT(0) cube complex and its quotient is "weakly special", then the above question is answered affirmatively., Sep. 2014, 24, 06, 795, 813, Scientific journal
  • Refereed, Algebraic & Geometric Topology, Mathematical Sciences Publishers, The link volume of 3–manifolds, Yo’av Rieck; Yasushi Yamashita, We view closed orientable 3-manifolds as covers of S^3 branched over hyperbolic links. For a p-fold cover M \to S^3, branched over a hyperbolic link L, we assign the complexity p Vol(S^3 minus L) (where Vol is the hyperbolic volume). We define an invariant of 3-manifolds, called the link volume and denoted LV, that assigns to a 3-manifold M the infimum of the complexities of all possible covers M \to S^3, where the only constraint is that the branch set is a hyperbolic link. Thus the link volume measures how efficiently M can be represented as a cover of S^3. We study the basic properties of the link volume and related invariants, in particular observing that for any hyperbolic manifold M, Vol(M) < LV(M). We prove a structure theorem that is similar to (and relies on) the celebrated theorem of Jorgensen and Thurston. This leads us to conjecture that, generically, the link volume of a hyperbolic 3-manifold is much bigger than its volume. Finally we prove that the link volumes of the manifolds obtained by Dehn filling a manifold with boundary tori are linearly bounded above in terms of the length of the continued fraction expansion of the filling curves., 05 Apr. 2013, 13, 2, 927, 958, Scientific journal
  • Refereed, Conformal Geometry and Dynamics of the American Mathematical Society, American Mathematical Society (AMS), Linear slices of the quasi-Fuchsian space of punctured tori, Yohei Komori; Yasushi Yamashita, After fixing a marking (V, W) of a quasifuchsian punctured torus group G, the complex length l_V and the complex twist tau_V,W parameters define a holomorphic embedding of the quasifuchsian space QF of punctured tori into C^2. It is called the complex Fenchel-Nielsen coordinates of QF. For a complex number c, let Q_gamma,c be the affine subspace of C^2 defined by the linear equation l_V=c. Then we can consider the linear slice L of QF by QF \cap Q_gamma,c which is a holomorphic slice of QF. For any positive real value c, L always contains the so called Bers-Maskit slice BM_gamma,c. In this paper we show that if c is sufficiently small, then L coincides with BM_gamma,c whereas L has other components besides BM_gamma,c when c is sufficiently large. We also observe the scaling property of L., 04 Apr. 2012, 16, 5, 89, 102, Scientific journal
  • Refereed, Geometry, Topology and Dynamics of Character Varieties, WORLD SCIENTIFIC, CREATING SOFTWARE FOR VISUALIZING KLEINIAN GROUPS, Yasushi Yamashita, Aug. 2012, 159, 190, In book
  • Refereed, European Journal of Combinatorics, Elsevier BV, Finite planar emulators for K_4,5-4K_2 and K_1,2,2,2 and Fellows' conjecture, Yo’av Rieck; Yasushi Yamashita, In 1988 Fellows conjectured that if a finite, connected graph admits a finite planar emulator, then it admits a finite planar cover. We construct a finite planar emulator for K_{4,5} - 4K_2. Archdeacon showed that K_{4,5} - 4K_2 does not admit a finite planar cover; thus K_{4,5} - 4K_2 provides a counterexample to Fellows' Conjecture. It is known that Negami's Planar Cover Conjecture is true if and only if K_{1,2,2,2} admits no finite planar cover. We construct a finite planar emulator for K_{1,2,2,2}. The existence of a finite planar cover for K_{1,2,2,2} is still open., Apr. 2010, 31, 3, 903, 907, Scientific journal
  • Refereed, Lecture Notes in Mathematics, Springer Berlin Heidelberg, Punctured Torus Groups and 2-Bridge Knot Groups (I), 2007
  • Refereed, Lond. Math. Soc. Lec. Notes, Computer experiments on the discreteness locus in projective structures, Yasushi Yamashita, 2006, 329, 375, 390, Scientific journal
  • Refereed, EXPERIMENTAL MATHEMATICS, A K PETERS LTD, Drawing Bers embeddings of the Teichmuller space of once-punctured tori, Y Komori; T Sugawa; M Wada; Y Yamashita, We present a computer-oriented method of producing pictures of Bers embeddings of the Teichmuller space of once-punctured tori. The coordinate plane is chosen in such a way that the accessory parameter is hidden in the relative position of the origin. Our algorithm consists of two steps. For each point in the coordinate plane, we first compute the corresponding monodromy representation by numerical integration along certain loops. Then we decide whether the representation is discrete by applying Jorgensen's theory on the quasi-Fuchsian space of once-punctured tori., 2006, 15, 1, 51, 60, Scientific journal
  • Refereed, Lond. Math. Soc. Lec. Notes, Jorgensen's picture of punctured torus groups and its refinement, H. Akiyoshi; M. Sakuma; M. Wada; Y. Yamashita, 2003, 299, 247, 273, Scientific journal
  • Refereed, GEOMETRIAE DEDICATA, KLUWER ACADEMIC PUBL, Configuration spaces of points on the circle and hyperbolic Dehn fillings, II, Yasushi Yamashita; Haruko Nishi; Sadayoshi Kojima, In our previous paper, we discussed the hyperbolization of the configuration space of n(> 4) marked points with weights in the projective line up to projective transformations. A variation of the weights induces a deformation. It was shown that this correspondence of the set of the weights to the Teichm\"uller space when n = 5 and to the Dehn filling space when n= 6 is locally one-to-one near the equal weight. In this paper, we establish its global injectivity., 24 Jul. 1999, 89, 1, 143, 157
  • Refereed, TOPOLOGY, PERGAMON-ELSEVIER SCIENCE LTD, Configuration spaces of points on the circle and hyperbolic Dehn fillings, Sadayoshi Kojima; Haruko Nishi; Yasushi Yamashita, A purely combinatorial compactification of the configuration space of n (>4) distinct points with equal weights in the real projective line was introduced by M. Yoshida. We geometrize it so that it will be a real hyperbolic cone-manifold of finite volume with dimension n-3. Then, we vary weights for points. The geometrization still makes sense and yields a deformation. The effectivity of deformations arisen in this manner will be locally described in the existing deformation theory of hyperbolic structures when n-3 = 2, 3., 25 Sep. 1998, 38, 3, 497, 516
  • Refereed, PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, AMER MATHEMATICAL SOC, An inequality for polyhedra and ideal triangulations of cusped hyperbolic 3-manifolds, M Wada; Y Yamashita; H Yoshida, It is not known whether every noncompact hyperbolic 3-manifold of finite volume admits a decomposition into ideal tetrahedra. We give a partial solution to this problem: Let M be a hyperbolic 3-manifold obtained by identifying the faces of n convex ideal polyhedra P-1, ..., P-n. If the faces of P-1, ..., P-n-1 are glued to P-n, then M can be decomposed into ideal tetrahedra by subdividing the P-i's., Dec. 1996, 124, 12, 3905, 3911, Scientific journal
  • Refereed, Transactions of the Japan Society for Industrial and Applied Mathematics, The Japan Society for Industrial and Applied Mathematics, A System for Doing Knot Theory by Computer, Ochiai Mitsuyuki; Yamashita Yasushi; Yamada Syuji, We made a new method and a data structure to draw knots and links rapidly. We also developed a computer software which realizes our ideas in order to assist reserchers in knot theory. As an example of using our software in knot theory, we explain computational results of polynomial invariants which can recognize mutant knots of 3-4 braids, Kinoshita-Terasaka knot and Conway knot., 15 Dec. 1994, 4, 4, 337, 348, Scientific journal
  • Refereed, PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, AMER MATHEMATICAL SOC, SHAPES OF STARS, S KOJIMA; Y YAMASHITA, In this paper we construct a natural geometric structure for the space of shapes of a star-shaped polygon. Roughly speaking we find: The set of similarity classes of marked stars forms a hyperbolic right angle pentagon bundle over the space of external angle sets of inscribed pentagons. The assignment of the shape of its fiber to each angle set forms a hyperbolic plane bundle over the Teichmuller space of hyperbolic right angle pentagons. Any automorphism induced by renumbering is compatible with these geometric structures., Mar. 1993, 117, 3, 845, 851, Scientific journal
  • Refereed, Experimental Mathematics, Informa UK Limited, Random Kleinian Groups, II Two Parabolic Generators, Gaven Martin; Graeme O’Brien; Yasushi Yamashita, In earlier work we introduced geometrically natural probability measures on the group of all M\"obius transformations in order to study "random" groups of M\"obius transformations, random surfaces, and in particular random two-generator groups, that is groups where the generators are selected randomly, with a view to estimating the likely-hood that such groups are discrete and then to make calculations of the expectation of their associated parameters, geometry and topology. In this paper we continue that study and identify the precise probability that a Fuchsian group generated by two parabolic M\"obius transformations is discrete, and give estimates for the case of Kleinian groups generated by a pair of random parabolic elements which we support with a computational investigation into of the Riley slice as identified by Bowditch's condition, and establish rigorous bounds., 01 Dec. 2020, 29, 4, 443, 451, Scientific journal
  • Refereed, Conformal Geometry and Dynamics of the American Mathematical Society, American Mathematical Society (AMS), The realization problem for Jørgensen numbers, Yasushi Yamashita; Ryosuke Yamazaki, Let G be a two generator subgroup of PSL(2,C). The Jorgensen number J(G) of G is defined by J(G)=inf{ |tr^2 A-4|+|tr[A,B]-2| ; G=}. If G is a non-elementary Kleinian group, then J(G) >= 1. This inequality is called Jorgensen's inequality. In this paper, we show that, for any r >= 1, there exists a non-elementary Kleinian group whose Jorgensen number is equal to r. This answers a question posed by Oichi and Sato. We also present our computer generated picture which estimates Jorgensen numbers from above in the diagonal slice of Schottky space., 25 Feb. 2019, 23, 2, 17, 31, Scientific journal

MISC

  • Not Refereed, 数学セミナー, 新入生のための数学書ガイド(分担), 山下靖, Apr. 2013, 618, 8, 36, Introduction commerce magazine
  • Not Refereed, RIMS Kokyuroku, Kyoto University, A computer experiment on primitive stable representations (Integrated Research on Complex Dynamics), Yamashita Yasushi, Sep. 2012, 1807, 87, 93
  • Not Refereed, 数学セミナー, 日本評論社, 低次元トポロジーにおける分類, 山下靖, Jul. 2011, 598, 7, 18, 22, Introduction commerce magazine
  • Not Refereed, 数理科学, サイエンス社, 対称性と結晶 (特集 現代数学はいかに使われているか(幾何編)), 山下 靖, Apr. 2009, 47, 4, 19, 24
  • Not Refereed, RIMS Kokyuroku, Kyoto University, OHT : A software for the dynamics of the modular group action on the character variety (Complex Dynamics and Related Topics), Yamashita Yasushi, Apr. 2008, 1586, 18, 25
  • Not Refereed, On Negami's planar cover conjecture, Yo'av Rieck; Yasushi Yamashita, Given a finite cover f:tilde{G} \to G and an embedding of tilde{G} in the plane, Negami conjectures that G embeds in P^2. Negami proved this conjecture for regular covers. In this paper we define two properties (Propserties V and E), depending on the cover tilde{G} and its embedding into S^2, and generalize Negami's result by showing: (1) If Properties V and E are fulfilled then G embeds in P^2. (2) Regular covers always fulfill Properties V and E. We give an example of an irregular cover fulfilling Properties V and E. Covers not fulfilling Properties V and E are discussed as well., 13 Dec. 2006, Technical report
  • Not Refereed, RIMS Kokyuroku, Kyoto University, Searching for $\mathbb{Z+A}$ subgroups in non-hyperbolic automatic groups (Perspectives of Hyperbolic Spaces II), Nakagawa Yoshiyuki; Tamura Makoto; Yamashita Yasushi, Jul. 2004, 1387, 110, 117
  • Not Refereed, RIMS Kokyuroku, Kyoto University, DRAWING BERS EMBEDDINGS OF THE TEICHMULLER SPACE OF ONCE PUNCTURED TORI (Hyperbolic Spaces and Related Topics II), Komori Yohei; Sugawa Toshiyuki; Wada Masaaki; Yamashita Yasushi, Jul. 2000, 1163, 9, 17
  • Not Refereed, RIMS Kokyuroku, Kyoto University, FORD DOMAINS OF PUNCTURED TORUS GROUPS AND TWO-BRIDGE KNOT GROUPS (Hyperbolic Spaces and Related Topics II), Akiyoshi Hirotaka; Sakuma Makoto; Wada Masaaki; Yamashita Yasushi, Jul. 2000, 1163, 67, 77
  • Not Refereed, Topology, Elsevier Ltd, Configuration spaces of points on the circle and hyperbolic dehn fillings, Sadayoshi Kojima; Haruko Nishi; Yasushi Yamashita, A purely combinatorial compactification of the configuration space of n( ≥ 5) distinct points with equal weights in the real projective line was introduced by M. Yoshida. We geometrize it so that it will be a real hyperbolic cone-manifold of finite volume with dimension n - 3. Then, we vary weights for points. The geometrization still makes sense and yields a deformation. The effectivity of deformations arisen in this manner will be locally described in the existing deformation theory of hyperbolic structures when n - 3 = 2, 3. © 1999 Elsevier Science Ltd. All rights reserved., 1999, 38, 3, 497, 516, Technical report
  • Not Refereed, RIMS Kokyuroku, Kyoto University, Punctured torus groups and two-parabolic groups (Analysis and Geometry of Hyperbolic Spaces), Akiyosi Hirotaka; Sakuma Makoto; Wada Masaaki; Yamashita Yasushi, Oct. 1998, 1065, 61, 73
  • Not Refereed, 数理解析研究所講究録, 京都大学, ケーリーグラフの組み合せ的性質について, 山下 靖, Dec. 1997, 1022, 179, 184
  • Not Refereed, RIMS Kokyuroku, Kyoto University, THE UNIFORMATION THEOREM FOR CIRCLE PACKINGS, YAMASHITA YASUSHI, Jan. 1995, 893, 36, 42
  • Not Refereed, RIMS Kokyuroku, Kyoto University, Ideal triangulations of noncompact hyperbolic 3-manifolds(Complex Analysis on Hyperbolic 3-Manifolds), Yamashita Yasushi, Aug. 1994, 882, 132, 138
  • The growth of torus link groups, Yoshiyuki Nakagawa; Makoto Tamura; Yasushi Yamashita, Let $G$ be a finitely generated group with a finite generating set $S$. For $g\in G$, let $l_S(g)$ be the length of the shortest word over $S$ representing $g$. The growth series of $G$ with respect to $S$ is the series $A(t) = \sum_{n=0}^\infty a_n t^n$, where $a_n$ is the number of elements of $G$ with $l_S(g)=n$. If $A(t)$ can be expressed as a rational function of $t$, then $G$ is said to have a rational growth function. We calculate explicitly the rational growth functions of $(p,q)$-torus link groups for any $p, q > 1.$ As an application, we show that their growth rates are Perron numbers., 15 Jan. 2014
  • The Link Volume of Hyperbolic 3-Manifolds, Yo'av Rieck; Yasushi Yamashita, We prove that for any V>0, there exist a hyperbolic manifold M_V, so that Vol(M_V) < 2.03 and LinVol(M_V) > V. The proof requires study of cosmetic surgery on links (equivalently, fillings of manifolds with boundary tori). There is no bound on the number of components of the link (or boundary components). For statements, see the second part of the introduction. Here are two examples of the results we obtain: 1) Let K be a component of a link L in S^3. Then "most" slopes on K cannot be completed to a cosmetic surgery on L, unless K becomes a component of a Hopf link. 2) Let X be a manifold and \epsilon>0. Then all but finitely many hyperbolic manifolds obtained by filling X admit a geodesic shorter than \epsilon\ (note that this finite set may correspond to an infinitely many fillings)., 08 Nov. 2012
  • RIMS Kokyuroku, Kyoto University, Non-hyperbolic automatic groups and groups acting on CAT(0) cube complexes (Complex Analysis and Topology of Discrete Groups and Hyperbolic Spaces), Yamashita Yasushi, Apr. 2015, 1936, 11, 14
  • RIMS Kokyuroku, Kyoto University, A VERY BRIEF INTRODUCTION TO VIRTUAL HAKEN CONJECTURE (Representation spaces, twisted topological invariants and geometric structures of 3-manifolds), YAMASHITA YASUSHI, May 2013, 1836, 192, 199

Books etc

  • Punctured torus groups and 2-bridge knot groups (I), Springer, 秋吉宏尚; 作間誠; 和田昌昭; 山下靖, 2007, Not Refereed, 9783540718062, cinii_books
  • 3次元幾何学とトポロジー, 培風館, William P. Thurston; Silvio Levy; 小島定吉, 1999, Not Refereed, 4563002720, cinii_books

Presentations

Research Projects

  • Grant-in-Aid for Scientific Research (C), 2011, 2013, 23540088, Principal investigator, The geometry of the mapping class group action on the character variety of surface groups, YAMASHITA Yasushi, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), Nara Women's University, 2080000, 1600000, 480000, We studied the SL(2,C)-character variety of once punctured torus. In particular, we investigate the relation between the Q-condition due to Bowditch, the discreteness of the corresponding representation, the complexity of the dynamics of the mapping class group action on the character variety. We published a joint paper with Yohei Komori on the global structure of the discreteness loci of the linear slices of the character variety. We carried our a computer experiments on primitive stableness, which was introduced recently by Minsky and measures the complexity of the dynamics of the mapping class group action, and compared our results with Q-condition., Competitive research funding, url
  • Grant-in-Aid for Scientific Research (C), 2008, 2010, 20540076, Principal investigator, The mapping class group action on the space of representations of surface groups and complex dynamics, YAMASHITA Yasushi, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), Nara Women's University, 1950000, 1500000, 450000, I was succeeded in constructing a counter example to the conjecture on the end invariant of the character variety of the free group of rank two. This was a joint word with Prof. S.P.Tan and Prof.M.Sakuma. I have found a set of rays similar to the pleating rays in the diagonal slice of the character variety with Prof.C.Series and Prof.S.P.Tan. With Prof.Y.Rieck, I showed that the complexity of the word problem for the automorphism groups of right-angled Artin groups is bounded from above by a polynomial., Competitive research funding, url
  • Grant-in-Aid for Scientific Research (C), 2006, 2007, 18540085, Principal investigator, The complex hyperbolic structures on the configuration spaces of points on the sphere and surface subgroup of mapping class groups, YAMASHITA Yasushi, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), Nara Women's University, 1150000, 1000000, 150000, (1) Structures of non-hyperbolic automatic groups (Joint work with Y. Nakagawa, M. Tamura) Let G be a finitely presented group. If G contains a Z + Z subgroup, then it is well known that G cannot be word hyperbolic. A natural question is that "is Z + Z the only obstruction for a finitely presented group to be word hyperbolic?" In other words, "if G does not contain any Z + Z subgroups, is it word hyperbolic?" Baumslag-Solitar groups are counter examples to this question. Thus it would be better to restrict our attention to some good class of groups. Here we focus on automatic groups. Note that Baumslag-Solitar groups are not automatic. Our problem is indicated in the list of open problems and attributed to Gersten. We call this problem "Gersten's problem". Recall that the class of all automatic groups contains the class of all hyperbolic groups, all virtually abelian groups and all geometrically finite hyperbolic groups. A geometrically finite hyperbolic group is, in some sense, similar to hyperbolic groups, but it might contain a Z + Z subgroup. Thus the class of automatic groups is a nice target to consider the question mentioned before. We define the notion of "n-tracks of length n", which suggests a clue of the existence of Z + Z subgroup and shows its existence in every non-hyperbolic automatic groups with mild conditions. (2) The character variety of one-holed torus (Joint work with S.P. Tan) The quasifuchsian space of punctured torus groups is deeply studied by many people and some of the major conjectures on them are solved in the last decade. But, for general "one-holed" cases, not much is known. In this study, we produced computer software to investigate the character variety of one holed torus and were able to find many interasting phenomena, Competitive research funding, url
  • 若手研究(B), 2004, 2005, 16740035, Principal investigator, 球面上の重みつき点配置空間の上の複素双曲構造の変形理論の構築, 山下 靖, 日本学術振興会, 科学研究費助成事業 若手研究(B), 奈良女子大学, 800000, 800000, 0, (1)オートマティック群に関するGerstenの問題の研究 (研究協力者:田村誠氏、中川義行氏) 閉3次元多様体の基本群に対する弱双曲化予想(Perelmanの仕事を認めれば定理)とは基本群は(1)有限群(2)Z+Z(階数2の自由アーベル群)を部分群として含む(3)語双曲群のいずれかになる、というものである。(技術的には有限群は語双曲群だが、ここでは分けて考えた。)この「閉3次元多様体の基本群」を「オートマティック群」に置き換えて同じ現象が起こるかどうかを問うのがGerstenの問題である。この研究では、この問題について考察を行った。群がZ+Zを部分群として含む場合は、Z+Zの格子が群の中にあることになる。本研究では、「n-track」というZ+Zの格子に似ている構造を導入し、オートマティック群が語双曲的でない場合はほとんどいつも、n-trackが群の中に見つかることを示した。さらにオートマティック構造が比較的単純な場合として「prime-starred」というオートマティック構造のクラスを導入し、この場合は、上記Gerstenの問題が(技術的な条件付で)肯定的に解けることを示した。 (2)4色問題と球面の分岐被覆に関する研究 (研究協力者:Yo'av Rieck氏) 平面グラフに関する4色問題は1970年代に計算機を用いた方法で証明されているが、実際に与えられた平面グラフを4彩色するための効果的なアルゴリズムはよく知られていない。本研究では、球面の分岐被覆から定まるデータと遺伝的アルゴリズムを組み合わせた方法を考察し、その効果について検討を行った。, Competitive research funding, url
  • 若手研究(B), 2002, 2003, 14740044, Principal investigator, 点配置空間の上に定義される複素双曲構造の空間の記述, 山下 靖, 日本学術振興会, 科学研究費助成事業 若手研究(B), 奈良女子大学, 800000, 800000, 0, 本研究代表者は、東京工業大学の小島定吉氏、九州大学の西晴子氏らとの共同研究で、円周上の点の配置空間に自然に定まる双曲構造を定義した。ただし双曲構造を与えるためには元の配置空間に適当な構造(重み)を与えておく必要がある。そしてこの重みを動かすことにより、配置空間の双曲構造も変形され、この変形の様子を記述する研究も行った。より詳しく述べると、点の数が5個および6個の場合は得られる配置空間の次元が2次元および3次元になるため、上記の変形と双曲多様体の変形空間であるタイヒミュラー空間や、クライン群の変形の空間との局所的および大域的な関係について調べてきた。 後者のクライン群の変形については、多様体内の結び目による錘特異点を許す場合の変形がサーストンの双曲デーン手術理論によって与えられていた。それとは別に考えている多様体の無限大の場合の変形で、中でも特殊なクライン群の場合、すなわち1点穴あきトーラスの擬正則変形空間が、ヨルゲンセンが具体的な記述を与えている。これら研究に刺激されて、双曲幾何学において研究が活発に進められてきた。本研究代表者は、大阪大学の作間誠氏、奈良女子大学の和田昌昭氏らとこの理論の精密化の研究を行った。 具体的には、ヨルゲンセンによる1点穴あきトーラスの基本領域の記述の精密化と、これに基づき、ベンディングラミネーションと呼ばれる方法による多様体の記述に関するある予想を提出し、部分的な回答と、計算機実験による検証を行った。, Competitive research funding, url
  • 奨励研究(A), 2000, 2001, 12740040, Principal investigator, 球面上の点配置空間から生じる複素双曲多様体の変形理論, 山下 靖, 日本学術振興会, 科学研究費助成事業 奨励研究(A), 奈良女子大学, 800000, 800000, 0, 円周上の点の配置空間に自然に定まる双曲構造について研究を行った.元の配置空間に適当な構造を付加することによって、その付加した構造を動かすことにより配置空間の双曲構造も変形しすることが以前の研究で分かっていた.特に現れる多様体(配置空間)の次元が2または3のときは、付加される構造の空間からTeichmuller空間および、character varietyへの写像が自然に定義されるが、この写像が大域的に単射であることを、本研究課題の研究代表者による研究および東京工業大学・小島定吉氏・九州大学・西晴子氏との共同研究で議論した。しかし上記の議論には、途中の部分に若干の誤りがあることが分かり、これらの修正を行った。これにより、あらためて、写像が大域的に単射であることを、研究協力者と共に示した。この研究成果は現在出版予定となっている。 また、大阪大学の作間誠氏、秋吉宏尚氏、奈良女子大学の和田昌昭氏らと、1点穴あきトーラス群から定まる多様体の双曲構造に関して、特にその標準的な多面体分割に関する議論をおこない、多面体分割を得るための方法として知られていた代表的な2種類の方法の比較を行うとともに、関連した話題について計算機による実験も行った。, Competitive research funding, url
  • 奨励研究(A), 1998, 1999, 10740030, Principal investigator, 点配置空間の解析による双曲多様体の変形理論, 山下 靖, 日本学術振興会, 科学研究費助成事業 奨励研究(A), 奈良女子大学, 1000000, 1000000, 0, 円周上の点の配置空間に自然に定まる双曲構造について研究を行った.元の配置空間に適当な構造を付加することによって、その付加した構造を動かすことにより配置空間の双曲構造も変形しすることが以前の研究で分かっていた.特に現れる多様体(配置空間)の次元が2または3のときは、付加される構造の空間からTeichmuller空間および、character varietyへの写像が自然に定義されるが、この写像が大域的に単射であることを、本研究課題の研究代表者による研究および東京工業大学・小島定吉氏・九州大学・西晴子氏との共同研究より示した.より具体的には、写像の定義域にあたる空間の座標系を適当に定めることと、写像の値域にあたる空間からもとの定義域の空間への逆写像を、上記座標系を利用して幾何学的に構成することにより議論をすすめた. 設備備品費は当初予定していた幾何学関係図書・位相幾何学関係図書等の購入を中心に行った.消耗品は計算データ結果の保存用に、計算機記憶媒体の購入が主であった.国内旅費は、共同研究者との研究連絡を学会等で行うための利用と共に、第45回トポロジーシンポジウムや研究集会「リーマン面・不連続群」での発表など、研究発表にも使用した.なお、上記共同研究成果は現在投稿中である., Competitive research funding, url
  • 基盤研究(C), 01 Apr. 2020, 31 Mar. 2023, 20K03612, 指標多様体上の幾何と写像類群作用を用いた算術的クライン群の分類, 山下 靖, 日本学術振興会, 科学研究費助成事業 基盤研究(C), 奈良女子大学, 1950000, 1500000, 450000, 曲面Σの基本群πからリー群Gへの表現全体の空間Hom(π,G)には群Gが共役により作用する。この作用による幾何学的不変式論の意味での商空間Xを指標多様体という。この指標多様体を、表現の像が離散群になる部分とそうでない部分に分割すると、前者はΣのG構造の変形の空間とみなすことができる。特にGがSL(2,C)の場合は双曲幾何構造の変形空間であり、重要な研究対象である。特に離散部分群はクライン群とよばれ、重要な研究対象である。また、指標多様体には曲面Σの写像類群が自然に作用し、この作用の複雑さによっても指標多様体は2つに分割される。これら2つの関係は未解明な部分が多い。 今年度は、SL(2,C)の部分群で楕円型の元2つによって生成されるクライン群で、算術的とよばれる条件をみたすものの分類のための研究を行った。楕円型の元はその位数で特徴づけることができるが、特に位数が6以下の場合において、どのような算術的クライン群が存在しうるかについて、計算機を用いた実験を行った。クライン群は指標多様体のパラメータを用いて記述され、それが算術的になるためにはそのパラメータが代数的整数であって四元数代数等に関する一定の条件をみたす必要があることが知られている。さらに、指標多様体上の写像類群作用に関して、BowditchのQ条件というものをみたさなければならないことが予想されている。そのため、これらに関する計算機実験を進めることで、算術的クライン群の完全分類に向けた候補を与えるための研究を進展させた。, rm:presentations
  • Grant-in-Aid for Scientific Research (B), 01 Apr. 2017, 31 Mar. 2022, 17H02843, Geometry of discrete groups and its applications to 3-dimensional topology, 大鹿 健一; 作間 誠; 宮地 秀樹; 山下 靖; 森藤 孝之, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B), 17290000, 13300000, 3990000, 繰り越しをした年度も含めた期間で以下のような研究を遂行した. 大鹿はPapadopoulosと写像類群の作用の剛性についての研究を進め,曲面上のgeodesic laminations全体が作る空間に非対称Hausdorff距離を入れたものについての剛性を示した.さらにPapdopoulos,Yi HuangとともにThurstonの非対称計量の下での,Teichmuller空間のRoydenの剛性と類似の無限小剛性を証明した.また大鹿,作間は秋吉らと共同で2元生成Klein群のうち自由群とならないものを分類するというAgolのプログラムを完成させた.宮地は擬等角写像の無限小空間を擬等角写像の空間における力学系による極限集合と理解してその基本的性質を得た.さらに,宮地はNewYork市立大学のDragomir Saric氏と共同研究によりTeichmuller円板の境界挙動を調べた.森藤は双曲絡み目のねじれアレキサンダー多項式が持つ性質と比較するために,3次元球面内のトーラス絡み目のねじれアレキサンダー多項式を詳しく考察して明示公式を与えるとともに,それがSL(2,C)-指標代数多様体上で局所定数になることを示した.山下はSL(2,C)の部分群で、2つの楕円的な元で生成されるものについて,算術的クライン群になるものの分類(数え上げ)のための研究を行った.さらにある種の条件をみたす整数係数モニック多項式の数え上げを行うための計算機を用いた研究を進めた. これらの研究をロシア,韓国などで開催された国際研究集会で発表するとともに,金沢大学で国際研究集会を主催することにより,今後の研究の発展につなげる努力をした.
  • 基盤研究(A), 01 Apr. 2016, 31 Mar. 2021, 16H02145, 結び目と3次元多様体の量子トポロジー, 大槻 知忠; 金信 泰造; 伊藤 哲也; 谷山 公規; 藤原 耕二; 逆井 卓也; 大山 淑之; 山下 靖; 茂手木 公彦; 森藤 孝之; 玉木 大; 志摩 亜希子, 日本学術振興会, 科学研究費助成事業 基盤研究(A), 京都大学, 33150000, 25500000, 7650000, 結び目のKashaev不変量と双曲体積を関連づける体積予想は、量子トポロジーと双曲幾何を結びつける懸案の予想であり、最近15年間世界的にこの分野の中心的な話題となってきた。本研究の目標は、体積予想を多くの結び目について解決し、Kashaev不変量の漸近展開として得られるべき級数を新しい結び目不変量として研究することである。これにより、量子トポロジーと双曲幾何を融合する新しい研究テーマが創出されることが期待される。また、3次元多様体の量子不変量の漸近展開に双曲体積が現れることを主張する「3次元多様体の体積予想」も近年定式化され、これについての研究もすすめた。とくに、漸近展開の準古典極限の項にはReidemeister torionが現れることが観察され、いくつかの例に対してそれを証明した。 また、国際会議「East Asian Conference on Geometric Topology」と、研究集会「Intelligence of Low-dimensional Topology」「結び目の数理」「トポロジーシンポジウム」「トポロジー新人セミナー」「Topology and Geometry of Low-dimensional Manifolds」「トポロジーとコンピュータ」「東北結び目セミナー」を開催した。これらの国際会議と研究集会では、国内外の研究者による活発な研究交流が行われ、十分な成果を挙げた。
  • Grant-in-Aid for Scientific Research (C), 01 Apr. 2017, 31 Mar. 2020, 17K05250, The geometry of character variety given by the dynamics of mapping class group action, Yamashita Yasushi, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), Nara Women's University, 1950000, 1500000, 450000, Hyperbolic geometry is important in studying two and three-dimensional manifolds. To understand this geometric structure, we studied the character variety of the fundamental group of two-dimensional manifolds. In particular, we studied the realization problem of Jorgensen numbers of the Kleinian groups generated by two elements. Also, we performed a computer experiment on the problem of when randomly generated two parabolic elements give a Kleinian group., url;rm:presentations;rm:presentations
  • Grant-in-Aid for Scientific Research (C), 01 Apr. 2016, 31 Mar. 2019, 16K05153, Canonical fundamental domains and holonomy representations for cone hyperbolic manifolds, Akiyoshi Hirotaka; Sakuma Makoto; Yamashita Yasushi; Kanenobu Taizo, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), Osaka City University, 4030000, 3100000, 930000, The aim of this project is to generalize Jorgensen's theory on punctured torus groups to cone hyperbolic structures, by carefully preparing basic theory on the deformation of cone hyperbolic structures. We established the concepts of Ford domains and compact closed convex cores for cone hyperbolic manifolds, and showed a kind of stability that Ford and Dirichlet domains have. As for coned torus manifolds, we obtained a deep understanding for Fuchsian and thin representations. We also obtained a numerical result which strongly suggests the existence of a way from coned tori to 2-bridge cone manifolds., url
  • Grant-in-Aid for Scientific Research (C), 01 Apr. 2014, 31 Mar. 2017, 26400088, The geometric and dynamical decomposition of the character variety of surface groups, Yamashita Yasushi, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), Nara Women's University, 2340000, 1800000, 540000, The hyperbolic geometry is important in studying the geometry of two and three dimensional manifolds. To understand this geometry, we studied the character variety of the fundamental group of a surface. In particular, we defined a new kind of volume for closed three dimensional manifolds using hyperbolic geometry, and studied the basic structure of this invariant. Moreover, using CAT(0) cube complexes, we found conditions for infinite discrete groups, such as fundamental groups of manifolds, to became hyperbolic in the sense of Gromov., url
  • Grant-in-Aid for Scientific Research (B), 01 Apr. 2010, 31 Mar. 2015, 22340013, Geometric structure and combinatorial structure of 3-dimensional manifolds, SAKUMA MAKOTO; SHIMADA Ichiro; DOI Hideo; YASUI Koichi; HIRANOUCHI Toshiro; KAMADA Seiichi; KONO Masaharu; NIKKUNI Ryo; AKIYOSHI Hirotaka; HIRASAWA Mikami; OHSHIKA Ken'ICHI; WADA Masaaki; MIYACHI Hideki; KIN Eiko; KOBAYASHI Tsuyoshi; YAMASHITA Yasushi; MORIMOTO Kanji; NAKANISHI Toshihiro; KOMORI Yohei; SUGAWA Toshiyuki; SHACKLETON Kenneth, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B), Hiroshima University, 16250000, 12500000, 3750000, (1) Joint work with Donghi Lee: We established a variation of McShane’s identity for 2-bridge links. Moreover, we introduced the Heckoid orbifolds and proved that they are hyperbolic, and gave a systematic construction of epimorphisms from 2-bridge link groups onto Heckoid groups. Furthermore, we proved that these are the only upper-meridian pair preserving epimorphisms onto even Heckoid groups. (2) Joint work with Ken’ichi Ohshika: We proved that for a Heegarrd surface S of a 3-manifold M with high Hempel distance, a certain natural mapping class group associated with S has a natural free decomposition. We also proved that if S is of bounded combinatorics then there is a nonempty open set U of the projective measured lamination space of S, such that any simple loop in U is not null-homntopic in M and that any two distinct simple loops in U are not homotopic in M., url
  • Grant-in-Aid for Challenging Exploratory Research, 2009, 2011, 21654011, Indiscrete representations of discrete groups, MAKOTO Sakuma; SEIICHI Kamada; MASAKAZU Teragaito; KEN. ICHI Ohshika; TOSHIYUKI Sugawa; YASUSHI Yamashita; HIROTAKA Akiyoshi; HIROKI Sumi, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research, Hiroshima University, 2740000, 2500000, 240000, We completely determined those simple loops on the 2-bridge spheres of 2-bridge links to be null-homotopic or peripheral in the link complements. We also completely determined when two simple loops on the 2-bridge spheres of 2-bridge links to be homotopic in the link complements. As an application of these results, we established a variation of McShane' s identity for 2-bridge links, which gives a formula to express the modulus of the cusp of a 2-bridge link in terms of the complex translation lengths of closed geodesics, url
  • Grant-in-Aid for Scientific Research (B), 2006, 2009, 18340018, Heegaard structures and geometric structures of 3-manifolds, SAKUMA Makoto; KAMADA Seiichi; NAGAI Toshitaka; MATSUMOTO Takao; UMEHARA Masaaki; OHSHIKA Ken'ichi; KONNO Kazuhiro; MABUCHI Toshiki; WADA Masaaki; MIYACHI Hideki; KOBAYASHI Tsuyoshi; YAMASHITA Yasushi; MORIMOTO Kanji; NAKANISHI Toshihiro; KOMORI Yohei; AKIYOSHI Hirotaka, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B), 9900000, 8100000, 1800000, We have concentrated on the study of the once-punctured torus, the simplest hyperbolic surface, believing that it would bring us to deep understanding of general hyperbolic surfaces, and obtained the following results. (1) We gave a complete description and proof to Jorgensen's theory on quasifuchsian punctured torus groups. (2) We found an intimate relation between the following two tessellations associated with a punctured torus bundles over the circle ; the Cannon-Thurston-Dicks fractal tessellation and the cusp triangulation induced by the canionical decomposition. We also proposed a conjecture concerning the canonical decompositions of punctured surface bundles over the circle. (3) We gave a complete characterization of those essential simple loops on the bridge sphere of a 2-bridge knot which are null-homotopic in the knot complement., url;url
  • 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 Grant-in-Aid for Scientific Research (C), 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., url
  • 萌芽研究, 2005, 2007, 17654016, 双曲構造と球面構造の双対性, 作間 誠; 秋吉 宏尚; 和田 昌昭; 山下 靖; 大鹿 健一; 難波 誠; 吉田 正章, 日本学術振興会, 科学研究費助成事業 萌芽研究, 2100000, 2100000, (1)Cannon-Thurston Mapの研究 Warren Dicksとのe-mail文通により,穴あきトーラス束から生じるCannon-Thurston Mapに付随する平面のフラクタル分割と,標準的分割が導くカスプ三角形分割との間に自然な関係があることを証明した.現在共著論文を執筆中である. (2)強可逆結び目の同変種数の研究 任意の周期的結び目は,周期写像で不変な最小種数ザイフェルト曲面を持つことがA.Edmondsにより証明されているが,強可逆結び目に対しては,対応する結果が成立しないことがわかる.しかし強可逆結び目に対して,対合で不変なザイフェルト曲面は存在するので,同変種数が定義出来る.この同変種数と通常の種数の差はいくらでも大きくなり得ることを証明した.この研究はToulouseで開催された国際研究集会で発表した. (3)垣水複体の研究 垣水複体の単連結性を証明したJ.Schultensの議論を拡張することにより,K.Shackletonとの共同研究により,垣水複体の2次元ホモトピー群が消えていることを証明した. (4)あるMontesinos結び目補空間のvirtual fiber性の証明 最近,Boyer-Zhangによりオイラー数が0であるMontesinos結び目補空間のvirtual fiber性が証明された.A'Campo-石川にdivide理論を応用することにより,オイラー数が0でないあるMontesinos結び目補空間のvirtual fiber性を証明した.
  • 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 Grant-in-Aid for Scientific Research (C), 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.
  • Grant-in-Aid for Scientific Research (B), 2002, 2005, 14340023, Hecguard Splittings and genetic structures of 3-manifolds, SAKUMA Makoto; AKIYOSHI Hirotaka; WADA Masaki; YAMASHITA Yasushi; OHSHIKA Ken'ichi; KOBAYASHI Tsuyoshi, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B), Osaka University, 10900000, 10900000, The main results obtained by this project are as follows. 1.Akiyoshi, Sakuma, Wada and Yamashita have completed a preprint (256 pages) which gives a full exposition of Jorgensen's theory for the Ford domains of quasifuchsian punctured torus groups, including a full proof. We plan to write a sequel of the paper to explain our extension of his theory to the outside of the quasifuchsian punctured torus space and to explain the relationship between the bridge structure of a 2-bridge knot and the complete hyperbolic structure of its complement. 2.Epstein-Penner has introduced the Euclidean decompositions of finite-volume cusped hyperbolic manifolds through a convex hull construction in the Minkowski space. Akiyoshi-Sakuma has generalized the construction to (possibly) infinite-volume cusped hyperbolic manifolds and introduced EPH-decompositions of these manifolds. Moreover, relation between the EPH-decompositions and the bending laminations of cusped hyper-bolic manifolds were studied by Akiyoshi-Sakuma-Wada Yamashita. 3.Akiyoshi-Miyachi-Sakuma have generalized Bowditch's variation of McShane's identity for hyperbolic punctured torus bundles to general hyperbolic punctured surface bundles.
  • Grant-in-Aid for Scientific Research (C), 2002, 2005, 14540079, Development of software to aid researches in Kleinnian groups and hyperbolic geometry, WADA Masaaki; YAMASHITA Yasushi, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), Nara Women's University, 2900000, 2900000, The head investigator is developing and distributing publicly the program OPTi. The program has been used by many researchers in Kleinnian groups and hyperbolic geometry, including such famous names as Troels Jorgensen (Columbia Univ.), Albert Marden (Univ.Minnesota), and William Thurston (UC Davis). The goal of this research project was to further refine OPTi to make it a more useful tool for researchers in Kleinnian groups and hyperbolic geometry. At the same time, we also wanted to make OPTi an effective educational device by improving its user interface and making OPTi easy to use for both researchers and students. After four years of continued improvements and diligent popularization of the program, we think the goal has been achieved. OPTi's web site (http://vivaldi.ics.nara-wu.ac.jp/~wada/OPTi/) is accessed from all over the world, as can be seen from the fact that the web site is ranked highly in the Topology Software category of Google's directory. Nowadays, OPTi is one of the most important research tools and educational aids for Topologists of the world.
  • 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 Grant-in-Aid for Scientific Research (C), 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.
  • 基盤研究(C), 2003, 2003, 15634003, トポロジーにおける実験数学, 小島 定吉; 山下 靖; 阿原 一志; 和田 昌昭; 高沢 光彦, 日本学術振興会, 科学研究費助成事業 基盤研究(C), 東京工業大学, 1900000, 1900000, 本研究は,トポロジーにおける実験数学の研究形態のプロトタイプを提案することを主眼として,この1年間企画調査を行った. 当初の予定通り,夏にイギリスを訪問し,Experimental Mathematics誌の初代編集長であったD.Epstein教授,および実験数学を代表する書物Indra's Pearlsの著者のであるC.Series教授,D.Wright教授とトポロジーにおける実験数学の現状について意見交換し,米国および英国の情報を収集した.その結果,実験数学の裾野が拡がる過程では,実装するアルゴリズムに話を絞るのが数学上の問題と計算上の問題を同じ土俵で議論するのに有効であり,さらに協調的な実験数学の研究につながる例が多かったことを知った. そこで12月に予定していた研究集会「トポロジーとコンピュータ」は,このことを念頭においてプログラムを組み東工大で開催した.とくに,多項式解法プログラムの作成者と基本群の表現の研究者の共同研究の発表では,当初は違う問題を解く目的で設計されたアルゴリズムがこの場合に妥当であるかどうかを,実験成果だけからでなくより実証的に示せないかなどの,数学と計算の双方で新たな課題が出るという討論の展開があった. 確かにアルゴリズムは,論証を重んじる数学と技術を重視する計算を結びつけるスポットであり,それを主役に置くことにが実験数学の研究およびその発表形態のプロトタイプになり得ることが確認できた.今後はこの企画調査の成果を,サマースクール形式でのプログラミング技術講習会,およびアルゴリズム指向の新しい研究集会の企画につなげ,平成17年度に実行に移す予定である.
  • Grant-in-Aid for Scientific Research (B), 2000, 2003, 12440018, Heegaad splittings of 3-dimensional manifolds and geometric structures, NAMBA Makoto; WADA Masaaki; SAKUMA Makoto; KONNO Kazuhiro; KOMORI Yohei; YAMASHITA Yasushi, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B), OSAKA UNIVERSITY, 14700000, 14700000, (1)Fundamental groups and branched coverings. M.Namba gave with H.Tsuchihashi a method for concrete computations of fundamental groups of the compliments of curves in the complex projective plane and finite branched Galois coverings branching along the curves, and gave a new example of Zariski pair using the method. (2)Generaliztion of Epsein-Penner decomposition. H.Akiyoshi and M.Sakuma gave a generalization of the Epstein-Penner decompositions of cusped hyperbol manifolds of finite volume to those of infinite volume, and studied relation with the convex cores. They collaborated with M.Wada and Y.Yamashita and gave partial answer and experimental evidences to their conjecture that the pleating loci would determine the generalized Epstein-Pener decompositions for punctured torus groups. (3)H.Akiyoshi, M.Miyachi and M.Sakuma have established a variation of McShane's identity for punctued surface bundles over a circle, which expresses the modulus of cusptori in terms of the complex translation lengths of essential simple loops of the fiber surfaces. (4)Drawing the 3D slices of the quasifuchsian punctured torus space. M.Wada and Y.Yamashita developed a software to draw (real) 3-dimensional slices of the quasifuchsian punctured torus space
  • 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 Grant-in-Aid for Scientific Research (C), 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.
  • Grant-in-Aid for Scientific Research (B), 1997, 1999, 09440033, Heegaand spliffings and hyperbolic structures of 3-manifolds, SAKUMA Makoto; MUSAKAMI Jun; EUOKI Tchiro; MABUCHI Toshiki; YAMASHITU Yasushi; WADA Masaaki, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B), Osaka University, 11000000, 11000000, The study of Heegaard splittings of 3-manifolds has been one of the most important themes in 3-manifold theory, and we already have deep understanding of the Heegaard splittings of "non-hyperbolike" 3-manifolds. However, our understanding of those of hyperbolic manifolds is far from satisfaction. In particular, as far as we know, no relationship between the hyperbolic structures and the Heegaard splittigs had been known. In this project we have proved that the hyperbolic structure of a 2-bridge knot complement is intimately related with its bridge structure, which is a kind of Heegaard splitting. In fact, we have given a concrete construction of the hyperbolic structure of a 2-bridge knot complement by using the 2-bridge structure. To be more precise, we have constructed a continuous family of hyperbolic cone-manifold structures on a 2-bridge knot complement which have singularities along the upper and lower tunnels, where the cone angle varies from 0 to 2π. The cone-manifold structure with cone angle 0 corresponds to a rational boundary group of the quasi-Fuchsian once-punctured torus space and that with cone angle 2π gives the hyperbolic structure of the 2-bridge knot complement. To establish this result, we have given an explicit formulation and a full proof to (a part of) the theory announced by Jorgensen on the quasi-Fuchsian once-punctured torus groups, and generalized the theory to that for the groups outside the quasi-Fuchsian once-punctured space. The computer program "OPTI" developed by Masaaki Wada for this project visualizes Jorgensen's theory and its generalization, and it has been an indispensable tool not only for this project but also for the study of Teichmuller spaces. We hope the result we have obtained in this project is the beginning of the study of the relationship between the hyperbolic structures and the Heegaard splittings of 3-manifolds.
  • 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 Grant-in-Aid for Scientific Research (B), 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.
  • Grant-in-Aid for Scientific Research (B), 1996, 1996, 08454019, Constructions of Matrix representations of Hecke algebras through W-graphs, OCHIAI Mitsuyuki; YAMASHITA Yasushi; KOBAYASHI Tsuyoshi; WADA Masaaki; KAKO Fujio, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B), NARA WOMEN'S UNIVERSITY, 6100000, 6100000, The purpose of this research is to construct a software to assist researches about Knot Theory. It assists to compute all polynomial invariants and in particular, parallel polynomial invariants related with knots and links. We had constructed a computer software, Knot Theory by Computer which works on Windows 95 and Windows NT.The software has the following features : (1)to construct knots and links by mouse tracking, (2)to deform knots by mouse operations, (3)to visualize knots and links by 3-dimensional computer graphics, (4)to compute all polynomial invariants which have already known, (5)to compute 3-parallel polynomial invariants associated with 3,4, and 5 braids (6)to recognize to whether a knot to be trivial or not (but not complete), This software will be distributed worldwide through leternet (ftp.ics.nara-wu.ac.jp) by the end of April, 1998. In future research, we will construct new features to compute 4-parallel polynomial invariants associated with 4-braids and establish a method to construct any irreducible representations associated with Hecke algebras. The well known Lascoux-Schutzenberger's method is correct by up to 13 of braid index but false greater than 13.
  • 総合研究(A), 1995, 1995, 07304008, 幾何構造の可視化, 小島 定吉; 山下 靖; 坪井 俊, 日本学術振興会, 科学研究費助成事業 総合研究(A), 東京工業大学, 2900000, 2900000, 「幾何構造の可視化」の研究に関連して企画された以下の6つの研究集会を後援した:阿部孝順(信州大)主催の「コンピュータを援用するトポロジーの研究」,津久井康之(専修大)主催の「Graphと3次元多様体の研究」,北野晃朗(東工大)主催の「基本群の表現空間の幾何」,根上生也(横国大)主催の「位相幾何学的グラフ理論研究集会」,小山晃(大阪教育大)主催の「高次元位相多様体論」,合田洋(東京理大)主催の「結び目の位置と3次元多様体の構造」.いずれも少人数の集会で,幾何学におけるコンピュータの役割に関する突っ込んだ討論がなされた.とくに日本ではこの分野の研究支援の研究が遅れているという指摘が複数の集会であった.また,深谷賢治(京都大)代表の科研費総合A(研究課題:幾何学と種々の数学の関わり)と共同で研究集会「Surveys in Geometry無限群と幾何学」を開催した.ここでは,計算機科学が幾何学的群論へ理論的に貢献するという,従来とは趣のことなる数学と計算機科学の相互作用が一つのテーマとなり,将来の発展に期待が集まった. これらの活動と連携しながら,研究組織では幾何構造の可視化のため具体的な実験試行を繰り返し,計算機支援の環境整備を目指した.とくに,境界付き双曲多様体のDehn手術変形の境界への影響をディスプレイ上に可視化するアルゴリズムを作った.ただしコンピュータ上実装については計算量の問題を残している.
  • 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 一般研究(C), 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。
  • 一般研究(C), 1993, 1993, 05640113, 共形構造及び射影構造の幾何に関する研究, 小林 治; 山下 靖; 藤田 収; 静田 靖; 加藤 信; 落合 豊行, 日本学術振興会, 科学研究費助成事業 一般研究(C), 奈良女子大学, 2000000, 2000000, 1)球面曲線のMobius幾何として,閉曲線の頂点の研究。この課題に対しては今年度としては十分な成果を得たといえる(研究代表者による論文Geometry of Scrolls(共著)は,現在投稿中であるため研究発表欄への記載はない)具体的な成果として単純ループの複合体での頂点数の最良評価,2つの無頂点曲線の交差の決定.応用としていわゆる4頂点定理の最終版に到達したことなどがある。 2)リーマン計量の共形変形とスカラー曲率については研究分担者加藤による研究が注目に値する。中でも与えられた計量が山辺計量であるための新しい十分条件を見いだしたことは,これが比較的単純な観察結果であるにもかかわらず,この方面の今後の研究の中でその価値が認識されるであろうことが期待される。 3)アフィン構造,射影構造の内在的微分幾何に関しては上記1)の研究に力点をうつしたため今年度の具体的成果はない。来年度に継続する課題としたい。 4)共形幾何,射影幾何に関連する双曲幾何について,研究分担者山下による双曲多様体の多面体分割についての研究が,トポロジーからの視点であるが,なされた。これは分割の仕方の組み合わせ論的制限と双曲多様体の位相について論じたものである。
  • Grant-in-Aid for Scientific Research (C), 01 Apr. 2021, 31 Mar. 2024, 21K03259, Invariants of pseudo-Anosov homeomorphisms, 小島 定吉; 山下 靖, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), Waseda University, 4160000, 3200000, 960000


Copyright © MEDIA FUSION Co.,Ltd. All rights reserved.