Updated on 2024/02/02

写真a

 
YAMASHITA Yasushi
 
Organization
Faculty of Science and Engineering Professor
Other responsible organization
Mathematics Course of Graduate School of Science and Engineering, Master's Program
Mathematics Course of Graduate School of Science and Engineering, Doctoral Program
Contact information
The inquiry by e-mail is 《here
External link

Degree

  • 博士(理学) ( 東京工業大学 )

  • 理学修士 ( 東京工業大学 )

Education

  • 1991.6
     

    Tokyo Institute of Technology   doctor course   withdrawn before completion

  • 1991.3
     

    Tokyo Institute of Technology   master course   completed

  • 1989.3
     

    Tokyo Institute of Technology   graduated

Research History

  • 2022.4 - Now

    Nara Women's University

  • 2012.4 - Now

    Nara Women's University   Faculty Division of Natural Sciences   Professor

  • 2010.1 - 2012.3

    Nara Women's University   Faculty of Science   Professor

  • 2007.4 - 2009.12

    Nara Women's University   Faculty of Science   Associate Professor

  • 2005.8 - 2007.3

    Nara Women's University   Faculty of Science

  • 1996.1 - 2005.7

    Nara Women's University   Faculty of Science

  • 1991.7 - 1996.12

    Nara Women's University   Faculty of Science   Research Assistant

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Research Interests

  • knots

  • geometric structure

  • visualization

  • experimental mathematics

  • Kleinian groups

  • hyperbolic geometry

  • 3-manifolds

  • low-dimensional topology

  • Teichmuller theory

  • projective structure

  • configuration space

  • cone manifolds

  • hyperbolic Dehn surgery

  • geometric group theory

  • automatic groups

  • hyperbolic groups

  • growth function

Research Areas

  • Natural Science / Geometry

Papers

  • Random Kleinian Groups, II Two Parabolic Generators Reviewed

    Gaven Martin, Graeme O’Brien, Yasushi Yamashita

    Experimental Mathematics   29 ( 4 )   443 - 451   2020.12

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:Informa UK Limited  

    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.

    DOI: 10.1080/10586458.2018.1477079

    Web of Science

    arXiv

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  • The realization problem for Jørgensen numbers Reviewed

    Yasushi Yamashita, Ryosuke Yamazaki

    Conformal Geometry and Dynamics of the American Mathematical Society   23 ( 2 )   17 - 31   2019.2

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:American Mathematical Society (AMS)  

    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=<A,B>}.
    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.

    DOI: 10.1090/ecgd/331

    Web of Science

    arXiv

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  • The diagonal slice of Schottky space Reviewed

    Caroline Series, Ser Tan, Yasushi Yamashita

    Algebraic & Geometric Topology   17 ( 4 )   2239 - 2282   2017.8

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:Mathematical Sciences Publishers  

    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.

    DOI: 10.2140/agt.2017.17.2239

    Web of Science

    arXiv

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  • Cosmetic surgery and the link volume of hyperbolic 3–manifolds Reviewed

    Yo’av Rieck, Yasushi Yamashita

    Algebraic & Geometric Topology   16 ( 6 )   3445 - 3521   2016.12

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:Mathematical Sciences Publishers  

    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.)

    DOI: 10.2140/agt.2016.16.3445

    Web of Science

    Scopus

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  • Non-hyperbolic automatic groups and groups acting on CAT(0) cube complexes Reviewed

    Yoshiyuki Nakagawa, Makoto Tamura, Yasushi Yamashita

    International Journal of Algebra and Computation   24 ( 06 )   795 - 813   2014.9

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:World Scientific Pub Co Pte Lt  

    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.

    DOI: 10.1142/s0218196714500349

    Web of Science

    arXiv

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  • The link volume of 3–manifolds Reviewed

    Yo’av Rieck, Yasushi Yamashita

    Algebraic & Geometric Topology   13 ( 2 )   927 - 958   2013.4

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:Mathematical Sciences Publishers  

    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.

    DOI: 10.2140/agt.2013.13.927

    Web of Science

    arXiv

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  • CREATING SOFTWARE FOR VISUALIZING KLEINIAN GROUPS Reviewed

    Yasushi Yamashita

    Geometry, Topology and Dynamics of Character Varieties   159 - 190   2012.8

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    Language:English   Publishing type:Part of collection (book)   Publisher:WORLD SCIENTIFIC  

    DOI: 10.1142/9789814401364_0005

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  • Linear slices of the quasi-Fuchsian space of punctured tori Reviewed

    Yohei Komori, Yasushi Yamashita

    Conformal Geometry and Dynamics of the American Mathematical Society   16 ( 5 )   89 - 102   2012.4

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:American Mathematical Society (AMS)  

    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.

    DOI: 10.1090/s1088-4173-2012-00237-8

    Web of Science

    Scopus

    arXiv

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  • Finite planar emulators for K_4,5-4K_2 and K_1,2,2,2 and Fellows' conjecture Reviewed

    Yo’av Rieck, Yasushi Yamashita

    European Journal of Combinatorics   31 ( 3 )   903 - 907   2010.4

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:Elsevier BV  

    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.

    DOI: 10.1016/j.ejc.2009.06.003

    Web of Science

    arXiv

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    Other Link: http://arxiv.org/pdf/0812.3700v3

  • Punctured Torus Groups and 2-Bridge Knot Groups (I) Reviewed

    Lecture Notes in Mathematics   2007

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    Language:English   Publisher:Springer Berlin Heidelberg  

    DOI: 10.1007/978-3-540-71807-9

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    Other Link: http://link.springer.com/content/pdf/10.1007/978-3-540-71807-9

  • Computer experiments on the discreteness locus in projective structures Reviewed

    Yasushi Yamashita

    Lond. Math. Soc. Lec. Notes   329   375 - 390   2006

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    Language:English   Publishing type:Research paper (scientific journal)  

    DOI: 10.1017/CBO9781139106993.019

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  • Drawing Bers embeddings of the Teichmuller space of once-punctured tori Reviewed

    Y Komori, T Sugawa, M Wada, Y Yamashita

    EXPERIMENTAL MATHEMATICS   15 ( 1 )   51 - 60   2006

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:A K PETERS LTD  

    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.

    DOI: 10.1080/10586458.2006.10128951

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  • Jorgensen's picture of punctured torus groups and its refinement Reviewed

    H. Akiyoshi, M. Sakuma, M. Wada, Y. Yamashita

    Lond. Math. Soc. Lec. Notes   299   247 - 273   2003

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    Language:English   Publishing type:Research paper (scientific journal)  

    DOI: 10.1017/CBO9780511542817.012

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  • Configuration spaces of points on the circle and hyperbolic Dehn fillings, II Reviewed

    Yasushi Yamashita, Haruko Nishi, Sadayoshi Kojima

    GEOMETRIAE DEDICATA   89 ( 1 )   143 - 157   1999.7

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    Language:English   Publisher:KLUWER ACADEMIC PUBL  

    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.

    DOI: 10.1023/A:1014220417320

    Web of Science

    arXiv

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    Other Link: http://arxiv.org/pdf/math/9907163v1

  • Configuration spaces of points on the circle and hyperbolic Dehn fillings Reviewed

    Sadayoshi Kojima, Haruko Nishi, Yasushi Yamashita

    TOPOLOGY   38 ( 3 )   497 - 516   1998.9

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    Language:English   Publisher:PERGAMON-ELSEVIER SCIENCE LTD  

    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.

    Web of Science

    arXiv

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    Other Link: http://arxiv.org/pdf/math/9809147v1

  • An inequality for polyhedra and ideal triangulations of cusped hyperbolic 3-manifolds Reviewed

    M Wada, Y Yamashita, H Yoshida

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   124 ( 12 )   3905 - 3911   1996.12

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:AMER MATHEMATICAL SOC  

    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.

    DOI: 10.1090/S0002-9939-96-03563-0

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  • A System for Doing Knot Theory by Computer Reviewed

    Ochiai Mitsuyuki, Yamashita Yasushi, Yamada Syuji

    Transactions of the Japan Society for Industrial and Applied Mathematics   4 ( 4 )   337 - 348   1994.12

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    Language:Japanese   Publishing type:Research paper (scientific journal)   Publisher:The Japan Society for Industrial and Applied Mathematics  

    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.

    DOI: 10.11540/jsiamt.4.4_337

    CiNii Books

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  • SHAPES OF STARS Reviewed

    S KOJIMA, Y YAMASHITA

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   117 ( 3 )   845 - 851   1993.3

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:AMER MATHEMATICAL SOC  

    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.

    DOI: 10.1090/S0002-9939-1993-1111430-3

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Books

  • Punctured torus groups and 2-bridge knot groups (I)

    秋吉宏尚, 作間誠, 和田昌昭, 山下靖( Role: Joint author)

    Springer  2007  ( ISBN:9783540718062

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    Language:English  

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  • 3次元幾何学とトポロジー

    William P. Thurston, Silvio Levy, 小島定吉( Role: Joint translator)

    培風館  1999  ( ISBN:4563002720

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    Language:Japanese  

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MISC

  • Non-hyperbolic automatic groups and groups acting on CAT(0) cube complexes (Complex Analysis and Topology of Discrete Groups and Hyperbolic Spaces)

    Yamashita Yasushi

    RIMS Kokyuroku   1936   11 - 14   2015.4

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    Language:English   Publisher:Kyoto University  

    CiNii Books

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    Other Link: http://hdl.handle.net/2433/223709

  • The growth of torus link groups

    Yoshiyuki Nakagawa, Makoto Tamura, Yasushi Yamashita

    2014.1

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    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.

    arXiv

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    Other Link: http://arxiv.org/pdf/1401.3403v1

  • A VERY BRIEF INTRODUCTION TO VIRTUAL HAKEN CONJECTURE (Representation spaces, twisted topological invariants and geometric structures of 3-manifolds)

    YAMASHITA YASUSHI

    RIMS Kokyuroku   1836   192 - 199   2013.5

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    Language:English   Publisher:Kyoto University  

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  • 新入生のための数学書ガイド(分担) Invited

    山下靖

    数学セミナー   618   8 - 36   2013.4

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    Language:Japanese   Publishing type:Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)  

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  • The Link Volume of Hyperbolic 3-Manifolds

    Yo'av Rieck, Yasushi Yamashita

    2012.11

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    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).

    DOI: 10.2140/agt.2016.16.3445

    arXiv

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    Other Link: http://arxiv.org/pdf/1211.1839v1

  • A computer experiment on primitive stable representations (Integrated Research on Complex Dynamics)

    Yamashita Yasushi

    RIMS Kokyuroku   1807   87 - 93   2012.9

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    Language:English   Publisher:Kyoto University  

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  • 低次元トポロジーにおける分類 Invited

    山下靖

    数学セミナー   598 ( 7 )   18 - 22   2011.7

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    Language:Japanese   Publishing type:Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher:日本評論社  

    CiNii Books

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  • 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)) Invited

    山下 靖

    数理科学   47 ( 4 )   19 - 24   2009.4

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    Language:Japanese   Publisher:サイエンス社  

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  • OHT : A software for the dynamics of the modular group action on the character variety (Complex Dynamics and Related Topics)

    Yamashita Yasushi

    RIMS Kokyuroku   1586   18 - 25   2008.4

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    Language:English   Publisher:Kyoto University  

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  • On Negami's planar cover conjecture

    Yo'av Rieck, Yasushi Yamashita

    2006.12

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    Publishing type:Internal/External technical report, pre-print, etc.  

    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.

    arXiv

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    Other Link: http://arxiv.org/pdf/math/0612342v1

  • Searching for $\mathbb{Z+A}$ subgroups in non-hyperbolic automatic groups (Perspectives of Hyperbolic Spaces II)

    Nakagawa Yoshiyuki, Tamura Makoto, Yamashita Yasushi

    RIMS Kokyuroku   1387   110 - 117   2004.7

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    Language:Japanese   Publisher:Kyoto University  

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  • 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

    RIMS Kokyuroku   1163   9 - 17   2000.7

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    Language:English   Publisher:Kyoto University  

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  • 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

    RIMS Kokyuroku   1163   67 - 77   2000.7

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    Language:English   Publisher:Kyoto University  

    CiNii Books

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  • Configuration spaces of points on the circle and hyperbolic dehn fillings

    Sadayoshi Kojima, Haruko Nishi, Yasushi Yamashita

    Topology   38 ( 3 )   497 - 516   1999

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    Language:English   Publishing type:Internal/External technical report, pre-print, etc.   Publisher:Elsevier Ltd  

    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.

    DOI: 10.1016/S0040-9383(98)00022-6

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  • Punctured torus groups and two-parabolic groups (Analysis and Geometry of Hyperbolic Spaces)

    Akiyosi Hirotaka, Sakuma Makoto, Wada Masaaki, Yamashita Yasushi

    RIMS Kokyuroku   1065   61 - 73   1998.10

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    Language:English   Publisher:Kyoto University  

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  • ケーリーグラフの組み合せ的性質について

    山下 靖

    数理解析研究所講究録   1022   179 - 184   1997.12

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    Language:Japanese   Publisher:京都大学  

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  • THE UNIFORMATION THEOREM FOR CIRCLE PACKINGS

    YAMASHITA YASUSHI

    RIMS Kokyuroku   893   36 - 42   1995.1

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    Language:Japanese   Publisher:Kyoto University  

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  • Ideal triangulations of noncompact hyperbolic 3-manifolds(Complex Analysis on Hyperbolic 3-Manifolds)

    Yamashita Yasushi

    RIMS Kokyuroku   882   132 - 138   1994.8

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    Language:English   Publisher:Kyoto University  

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Presentations

  • Revisiting the moduli space of right-angled hyperbolic pentagon

    Yasushi Yamashita

    Geometry in low-dimensions 2022  2022.12 

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    Event date: 2022.12    

    Language:Japanese   Presentation type:Oral presentation (general)  

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  • Riley sliceと仲間たち Invited

    山下靖

    早稲田大学双曲幾何幾何学的群論セミナー  2022.6 

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    Event date: 2022.6    

    Language:Japanese   Presentation type:Oral presentation (invited, special)  

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  • Computer experiments on Mobius transformations and random Kleinian groups Invited

    Yasushi Yamashita

    MSJ Autumn Meeting  2021.9 

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    Event date: 2021.9    

    Language:Japanese   Presentation type:Oral presentation (invited, special)  

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  • The diagonal slice of Schottky space Invited

    Yasushi Yamashita

    Computational Problems in Low-dimensional Topology II  2019.4 

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    Event date: 2019.4    

    Language:English   Presentation type:Oral presentation (invited, special)  

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  • On discreteness problem of Mobius groups Invited

    Yasushi Yamashita

    2018.8 

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    Event date: 2018.8    

    Language:Japanese   Presentation type:Oral presentation (invited, special)  

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Research Projects

  • Invariants of pseudo-Anosov homeomorphisms

    Grant number:21K03259  2021.4 - 2024.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)  Waseda University

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    Grant amount: \4160000 ( Direct Cost: \3200000 、 Indirect Cost: \960000 )

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  • 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

    Grant number:20K03612  2020.4 - 2023.3

    日本学術振興会  科学研究費助成事業 基盤研究(C)  基盤研究(C)  奈良女子大学

    山下 靖

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    Grant amount: \1950000 ( Direct Cost: \1500000 、 Indirect Cost: \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。

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  • Geometry of discrete groups and its applications to 3-dimensional topology

    Grant number:17H02843  2017.4 - 2022.3

    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) 

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    Grant amount: \17290000 ( Direct Cost: \13300000 、 Indirect Cost: \3990000 )

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  • 結び目と3次元多様体の量子トポロジー

    Grant number:16H02145  2016.4 - 2021.3

    日本学術振興会  科学研究費助成事業 基盤研究(A)  基盤研究(A)  京都大学

    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

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    Grant amount: \33150000 ( Direct Cost: \25500000 、 Indirect Cost: \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。

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  • The geometry of character variety given by the dynamics of mapping class group action

    Grant number:17K05250  2017.4 - 2020.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)  Nara Women's University

    Yamashita Yasushi

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    Grant amount: \1950000 ( Direct Cost: \1500000 、 Indirect Cost: \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.

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  • Canonical fundamental domains and holonomy representations for cone hyperbolic manifolds

    Grant number:16K05153  2016.4 - 2019.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)  Osaka City University

    Akiyoshi Hirotaka, Sakuma Makoto, Yamashita Yasushi, Kanenobu Taizo

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    Grant amount: \4030000 ( Direct Cost: \3100000 、 Indirect Cost: \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.

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  • The geometric and dynamical decomposition of the character variety of surface groups

    Grant number:26400088  2014.4 - 2017.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)  Nara Women's University

    Yamashita Yasushi

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    Grant amount: \2340000 ( Direct Cost: \1800000 、 Indirect Cost: \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.

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  • Geometric structure and combinatorial structure of 3-dimensional manifolds

    Grant number:22340013  2010.4 - 2015.3

    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)  Hiroshima University

    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

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    Grant amount: \16250000 ( Direct Cost: \12500000 、 Indirect Cost: \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.

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  • The geometry of the mapping class group action on the character variety of surface groups

    Grant number:23540088  2011 - 2013

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)  Nara Women's University

    YAMASHITA Yasushi

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    Authorship:Principal investigator  Grant type:Competitive

    Grant amount: \2080000 ( Direct Cost: \1600000 、 Indirect Cost: \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.

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  • Indiscrete representations of discrete groups

    Grant number:21654011  2009 - 2011

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research  Grant-in-Aid for Challenging Exploratory Research  Hiroshima University

    MAKOTO Sakuma, SEIICHI Kamada, MASAKAZU Teragaito, KEN. ICHI Ohshika, TOSHIYUKI Sugawa, YASUSHI Yamashita, HIROTAKA Akiyoshi, HIROKI Sumi

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    Grant amount: \2740000 ( Direct Cost: \2500000 、 Indirect Cost: \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

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  • The mapping class group action on the space of representations of surface groups and complex dynamics

    Grant number:20540076  2008 - 2010

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)  Nara Women's University

    YAMASHITA Yasushi

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    Authorship:Principal investigator  Grant type:Competitive

    Grant amount: \1950000 ( Direct Cost: \1500000 、 Indirect Cost: \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.

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  • Heegaard structures and geometric structures of 3-manifolds

    Grant number:18340018  2006 - 2009

    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) 

    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

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    Grant amount: \9900000 ( Direct Cost: \8100000 、 Indirect Cost: \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.

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  • Research on 3-manifolds based on geometric techniques and its expanse

    Grant number:19540083  2007 - 2008

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)  Nara Women's University

    KOBAYASHI Tsuyoshi, YAMASHITA Yasushi, KATAGIRI Minnyou

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    Grant amount: \1950000 ( Direct Cost: \1500000 、 Indirect Cost: \450000 )

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  • The complex hyperbolic structures on the configuration spaces of points on the sphere and surface subgroup of mapping class groups

    Grant number:18540085  2006 - 2007

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)  Nara Women's University

    YAMASHITA Yasushi

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    Authorship:Principal investigator  Grant type:Competitive

    Grant amount: \1150000 ( Direct Cost: \1000000 、 Indirect Cost: \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

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  • 双曲構造と球面構造の双対性

    Grant number:17654016  2005 - 2007

    日本学術振興会  科学研究費助成事業 萌芽研究  萌芽研究 

    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

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    Grant amount: \2100000 ( Direct Cost: \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.

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  • Geometric structures of 3-manifolds and various related structures

    Grant number:17540077  2005 - 2006

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)  Nara Women's University

    KOBAYASHI Tsuyoshi, YAMASHITA Yasushi, KATAGIRI Minnyou

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    Grant amount: \2200000 ( Direct Cost: \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.

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  • 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

    Grant number:16740035  2004 - 2005

    日本学術振興会  科学研究費助成事業 若手研究(B)  若手研究(B)  奈良女子大学

    山下 靖

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    Authorship:Principal investigator  Grant type:Competitive

    Grant amount: \800000 ( Direct Cost: \800000 )

    (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。

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  • Hecguard Splittings and genetic structures of 3-manifolds

    Grant number:14340023  2002 - 2005

    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)  Osaka University

    SAKUMA Makoto, AKIYOSHI Hirotaka, WADA Masaki, YAMASHITA Yasushi, OHSHIKA Ken'ichi, KOBAYASHI Tsuyoshi

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    Grant amount: \10900000 ( Direct Cost: \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.

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  • Development of software to aid researches in Kleinnian groups and hyperbolic geometry

    Grant number:14540079  2002 - 2005

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)  Nara Women's University

    WADA Masaaki, YAMASHITA Yasushi

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    Grant amount: \2900000 ( Direct Cost: \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.

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  • Research on various geometric structures on 3-manifolds

    Grant number:15540073  2003 - 2004

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)  Nara Women's University

    KOBAYASHI Tsuyoshi, YAMASHITA Yasushi, KATAGIRI Minnyou, ICHIHARA Kazuhiro

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    Grant amount: \2100000 ( Direct Cost: \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_<i=1> K_i, Then we proved that there exists a subset I of {1,【triple bond】,n} such that #_<i∈1>/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.

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  • トポロジーにおける実験数学

    Grant number:15634003  2003    

    日本学術振興会  科学研究費助成事業 基盤研究(C)  基盤研究(C)  東京工業大学

    小島 定吉, 山下 靖, 阿原 一志, 和田 昌昭, 高沢 光彦

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    Grant amount: \1900000 ( Direct Cost: \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.

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  • 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

    Grant number:14740044  2002 - 2003

    日本学術振興会  科学研究費助成事業 若手研究(B)  若手研究(B)  奈良女子大学

    山下 靖

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    Authorship:Principal investigator  Grant type:Competitive

    Grant amount: \800000 ( Direct Cost: \800000 )

    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。

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  • Heegaad splittings of 3-dimensional manifolds and geometric structures

    Grant number:12440018  2000 - 2003

    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)  OSAKA UNIVERSITY

    NAMBA Makoto, WADA Masaaki, SAKUMA Makoto, KONNO Kazuhiro, KOMORI Yohei, YAMASHITA Yasushi

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    Grant amount: \14700000 ( Direct Cost: \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

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  • Representations of 3-manifolds and geometric informations derived from them

    Grant number:12640071  2000 - 2002

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)  Nara Women's University

    KOBAYASHI Tsuyoshi, KATAGIRI Minnyou, YAMASHITA Yasushi, OCHIAI Mitsuyuki

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    Grant amount: \2800000 ( Direct Cost: \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.

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  • 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

    Grant number:12740040  2000 - 2001

    日本学術振興会  科学研究費助成事業 奨励研究(A)  奨励研究(A)  奈良女子大学

    山下 靖

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    Authorship:Principal investigator  Grant type:Competitive

    Grant amount: \800000 ( Direct Cost: \800000 )

    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。

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  • 点配置空間の解析による双曲多様体の変形理論

    Grant number:10740030  1998 - 1999

    日本学術振興会  科学研究費助成事業 奨励研究(A)  奨励研究(A)  奈良女子大学

    山下 靖

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    Authorship:Principal investigator  Grant type:Competitive

    Grant amount: \1000000 ( Direct Cost: \1000000 )

    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.

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  • Heegaand spliffings and hyperbolic structures of 3-manifolds

    Grant number:09440033  1997 - 1999

    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)  Osaka University

    SAKUMA Makoto, MUSAKAMI Jun, EUOKI Tchiro, MABUCHI Toshiki, YAMASHITU Yasushi, WADA Masaaki

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    Grant amount: \11000000 ( Direct Cost: \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.

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  • Gemetric Structures on Manifolds and Global Analysis

    Grant number:09440034  1997 - 1999

    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) 

    KOBAYASHI Osamu, FUJIOKA Atsushi, KITAHARA Haruo, KODAMA Akio, KATO Shin, KATAGIRI Minyo

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    Grant amount: \9200000 ( Direct Cost: \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.

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  • Constructions of Matrix representations of Hecke algebras through W-graphs

    Grant number:08454019  1996    

    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)  NARA WOMEN'S UNIVERSITY

    OCHIAI Mitsuyuki, YAMASHITA Yasushi, KOBAYASHI Tsuyoshi, WADA Masaaki, KAKO Fujio

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    Grant amount: \6100000 ( Direct Cost: \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.

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  • 幾何構造の可視化

    Grant number:07304008  1995    

    日本学術振興会  科学研究費助成事業 総合研究(A)  総合研究(A)  東京工業大学

    小島 定吉, 山下 靖, 坪井 俊

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    Grant amount: \2900000 ( Direct Cost: \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.

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  • 共形構造及び射影構造に関する幾何学

    Grant number:06640141  1994    

    日本学術振興会  科学研究費助成事業 一般研究(C)  一般研究(C)  奈良女子大学

    小林 治, 山下 靖, 和田 昌昭, 静田 靖, 片桐 民陽, 落合 豊行

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    Grant amount: \1900000 ( Direct Cost: \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。

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  • 共形構造及び射影構造の幾何に関する研究

    Grant number:05640113  1993    

    日本学術振興会  科学研究費助成事業 一般研究(C)  一般研究(C)  奈良女子大学

    小林 治, 山下 靖, 藤田 収, 静田 靖, 加藤 信, 落合 豊行

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    Grant amount: \2000000 ( Direct Cost: \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。

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Allotted class

  • 2023   プログラミング言語2   Department

  • 2023   プログラミング言語3   Department

  • 2023   卒業研究Ⅰ   Department

  • 2023   卒業研究Ⅱ   Department

  • 2023   情報処理   Department

  • 2023   数値解析1   Department

  • 2023   数値計算法2   Department

  • 2023   計算の理論1   Department

  • 2023   数学特別演習第一   Graduate school

  • 2023   数学特別演習第二   Graduate school

  • 2023   数学特殊論文研修Ⅰ   Graduate school

  • 2023   数学特殊論文研修Ⅱ   Graduate school

  • 2023   数学特殊論文研修Ⅲ   Graduate school

  • 2023   数学特殊論文研修Ⅳ   Graduate school

  • 2023   数学特殊論文研修Ⅴ   Graduate school

  • 2023   数学特殊論文研修Ⅵ   Graduate school

  • 2023   数学特論   Graduate school

  • 2023   数学論文研修第一   Graduate school

  • 2023   数学論文研修第三   Graduate school

  • 2023   数学論文研修第二   Graduate school

  • 2023   数学論文研修第四   Graduate school

  • 2023   計算数学特論第三   Graduate school

  • 2023   計算数学特論第四   Graduate school

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Teaching Experience

  • Geometric Group Theory

    2019 - Now   Institution:Nara Women's University

  • Special lecture on hyperbolic geometry

    2018 - Now   Institution:Nara Women's University

  • Basic Science 2

    2016 - Now   Institution:Nara Women's University

  • The science you need to know before you enter the workforce

    2016 - Now   Institution:Nara Women's University

  • Basic Science 1

    2016 - Now   Institution:Nara Women's University

  • Graph Theory

    2015 - Now   Institution:Nara Women's University

  • Hyperbolic Geometry

    2015 - Now   Institution:Nara Women's University

  • Programming

    2015 - Now   Institution:Nara Women's University

  • SCORE

    2016 - 2019   Institution:Nara Women's University

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