Updated on 2024/02/02

写真a

 
TORII Takeshi
 
Organization
Faculty of Environmental, Life, Natural Science and Technology Professor
Position
Professor
External link

Degree

  • Ph.D. ( Johns Hopkins University )

  • Doctor (Science) ( Kyoto University )

Research Interests

  • 形式群

  • ボルディズム

  • Stable homotopy

  • 安定ホモトピー

  • Formal Group

  • Bordisum

Research Areas

  • Natural Science / Geometry

Research History

  • Okayama University

    2017

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  • Okayama University

    2007 - 2017

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  • Fukuoka University   Faculty of Science

    2001 - 2007

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Professional Memberships

 

Papers

  • Applications of Hochschild cohomology to the moduli of subalgebras of the full matrix ring Reviewed

    Kazunori Nakamoto, Takeshi Torii

    Journal of Pure and Applied Algebra   227 ( 11 )   107426 - 107426   2023.11

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

    DOI： 10.1016/j.jpaa.2023.107426

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  • A perfect pairing for monoidal adjunctions Reviewed

    Takeshi Torii

    Proceedings of the American Mathematical Society   2023.9

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

    <p>We give another proof of the fact that there is a dual equivalence between the -category of monoidal -categories with left adjoint oplax monoidal functors and that with right adjoint lax monoidal functors by constructing a perfect pairing between them.</p>

    DOI： 10.1090/proc/16460

    arXiv

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  • On graded $\mathbb{E}_{\infty}$-rings and projective schemes in spectral algebraic geometry Reviewed

    Mariko Ohara, Takeshi Torii

    Journal of Homotopy and Related Structures   17 ( 1 )   105 - 144   2022.1

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    We introduce graded $\mathbb{E}_{\infty}$-rings and graded modules over them,
    and study their properties. We construct projective schemes associated to
    connective $\mathbb{N}$-graded $\mathbb{E}_{\infty}$-rings in spectral
    algebraic geometry. Under some finiteness conditions, we show that the
    $\infty$-category of almost perfect quasi-coherent sheaves over a spectral
    projective scheme $\mathrm{Proj}\,(A)$ associated to a connective
    $\mathbb{N}$-graded $\mathbb{E}_{\infty}$-ring $A$ can be described in terms of
    $\mathbb{Z}$-graded $A$-modules.

    arXiv

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    Other Link： http://arxiv.org/pdf/1803.09389v4

  • On quasi-categories of comodules and Landweber exactness Reviewed

    Takeshi Torii

    Proceedings in Mathematics & Statistics   309   325 - 380   2020

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    In this paper we study quasi-categories of comodules over coalgebras in a
    stable homotopy theory. We show that the quasi-category of comodules over the
    coalgebra associated to a Landweber exact S-algebra depends only on the height
    of the associated formal group. We also show that the quasi-category of
    E(n)-local spectra is equivalent to the quasi-category of comodules over the
    coalgebra A\otimes A for any Landweber exact S_(p)-algebra A of height n at a
    prime p. Furthermore, we show that the category of module objects over a
    discrete model of the Morava E-theory spectrum in the K(n)-local discrete
    symmetric G_n-spectra is a model of the K(n)-local category, where G_n is the
    extended Morava stabilizer group.

    arXiv

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    Other Link： http://arxiv.org/pdf/1612.03265v1

  • Discrete G-spectra and embeddings of module spectra Reviewed

    Takeshi Torii

    JOURNAL OF HOMOTOPY AND RELATED STRUCTURES   12 ( 4 )   853 - 899   2017.12

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER HEIDELBERG  

    In this paper we study the category of discrete G-spectra for a profinite group G. We consider an embedding of module objects in spectra into a category of module objects in discrete G-spectra, and study the relationship between the embedding and the homotopy fixed points functor. We also consider an embedding of module objects in terms of quasi-categories, and show that the two formulations of embeddings are equivalent in some circumstances.

    DOI： 10.1007/s40062-016-0166-7

    Web of Science

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  • Comparison of power operations in Morava E-theories Reviewed

    Takeshi Torii

    Homology, Homotopy and Applications   19 ( 1 )   59 - 87   2017

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  • Virtual Hodge polynomials of the moduli spaces of representations of degree 2 for free monoids Reviewed

    Kazunori Nakamoto, Takeshi Torii

    Kodai Mathematical Journal   39 ( 1 )   80 - 109   2016

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    In this paper we study the topology of the moduli spaces of representations
    of degree $2$ for free monoids. We calculate the virtual Hodge polynomials of
    the character varieties for several types of $2$-dimensional representations.
    Furthermore, we count the number of isomorphism classes for each type of
    $2$-dimensional representations over any finite field ${\Bbb F}_q$, and show
    that the number coincides with the virtual Hodge polynomial evaluated at $q$.

    arXiv

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    Other Link： http://arxiv.org/pdf/1501.02933v2

  • Every K(n)-local spectrum is the homotopy fixed points of its Morava module Reviewed

    Daniel G. Davis, Takeshi Torii

    Proceedings of the American Mathematical Society   140 ( 3 )   1097 - 1103   2012

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    Let n \geq 1 and let p be any prime. Also, let E_n be the Lubin-Tate
    spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava
    K-theory spectrum. Then work of Devinatz and Hopkins and some results due to
    Behrens and the first author of this note, show that if X is a finite spectrum,
    then the localization L_{K(n)}(X) is equivalent to the homotopy fixed point
    spectrum (L_{K(n)}(E_n \wedge X))^{hG_n}, which is formed with respect to the
    continuous action of G_n on L_{K(n)}(E_n \wedge X). In this note, we show that
    this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for
    all such X, the strongly convergent Adams-type spectral sequence abutting to
    \pi_\ast(L_{K(n)}(X)) is isomorphic to the descent spectral sequence that abuts
    to \pi_\ast((L_{K(n)}(E_n \wedge X))^{hG_n}).

    arXiv

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    Other Link： http://arxiv.org/pdf/1101.5201v1

  • K(n)-localization of the K(n+1)-local En+1-Adams spectral sequences Reviewed

    T. Torii

    Pacific Journal of Mathematics   250 ( 2 )   439 - 471   2011

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  • HKR CHARACTERS, p-DIVISIBLE GROUPS AND THE GENERALIZED CHERN CHARACTER Reviewed

    Takeshi Torii

    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY   362 ( 11 )   6159 - 6181   2010.11

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

    In this paper we describe the generalized Chern character of classifying spaces of finite groups in terms of Hopkins-Kuhn-Ravenel generalized group characters. For this purpose we study the p-divisible group and its level structures associated with the K(n)-localization of the (n + 1)st Morava E-theory.

    Web of Science

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  • On E-infinity-structure of the generalized Chern character Reviewed

    Takeshi Torii

    BULLETIN OF THE LONDON MATHEMATICAL SOCIETY   42   680 - 690   2010.8

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：OXFORD UNIV PRESS  

    In this note we show that we can lift the generalized Chern character to a morphism of commutative S-algebras. Furthermore, we show that we can take a lifting that is equivariant with respect to the action of a Morava stabilizer group.

    DOI： 10.1112/blms/bdq026

    Web of Science

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  • Comparison of Morava E-theories Reviewed

    Takeshi Torii

    Mathematische Zeitschrift   266 ( 4 )   933 - 951   2010

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    In this note we show that the n-th Morava E-cohomology group of a finite
    spectrum with action of the n-th Morava stabilizer group can be recovered from
    the (n+1)-st Morava E-cohomology group with action of the (n+1)-st Morava
    stabilizer group.

    arXiv

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    Other Link： http://arxiv.org/pdf/0901.3396v1

  • Milnor operations and the generalized Chern character Reviewed

    Takeshi Torii

    Geometry & Topology Monographs   10   383 - 421   2007

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    We have shown that the n-th Morava K-theory K^*(X) for a CW-spectrum X with
    action of Morava stabilizer group G_n can be recovered from the system of some
    height-(n+1) cohomology groups E^*(Z) with G_{n+1}-action indexed by finite
    subspectra Z. In this note we reformulate and extend the above result. We
    construct a symmetric monoidal functor F from the category of
    E^{vee}_*(E)-precomodules to the category of K_{*}(K)-comodules. Then we show
    that K^*(X) is naturally isomorphic to the inverse limit of F(E^*(Z)) as a
    K_{*}(K)-comodule.

    DOI： 10.2140/gtm.2007.10.383

    arXiv

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    Other Link： http://arxiv.org/pdf/0903.4708v1

  • On relations between 1-lines of Adams-Novikov spectral sequences modulo invariant prime ideals Reviewed

    T. Torii

    Topology and its Applications   150 ( 1-3 )   33 - 57   2005

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  • Topology of the moduli of representations with Borel mold Reviewed

    K Nakamoto, T Torii

    PACIFIC JOURNAL OF MATHEMATICS   213 ( 2 )   365 - 387   2004.2

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：PACIFIC JOURNAL MATHEMATICS  

    We give descriptions of the moduli of representations with Borel mold for free monoids as fibre bundles over the configuration spaces. By using the associated Serre spectral sequences, we study the cohomology rings of the moduli. Also we calculate the virtual Hodge polynomials of them.

    Web of Science

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  • On degeneration of one-dimensional formal group laws and applications to stable homotopy theory Reviewed

    T Torii

    AMERICAN JOURNAL OF MATHEMATICS   125 ( 5 )   1037 - 1077   2003.10

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：JOHNS HOPKINS UNIV PRESS  

    In this note we study a certain formal group law over a complete discrete valuation ring F[u(n)-1] of characteristic p &gt; 0 which is of height n over the closed point and of height n-I over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on the generic point and the Honda group law Hn-1 of height n-1, we get a Galois extension of the quotient field of the discrete valuation ring with Galois group isomorphic to the automorphism group Sn-1 of Hn-1. We show that the automorphism group S-n of the formal group over the closed point acts on the quotient field, lifting to an action on the Galois extension which commutes with the action of Galois group. We use this to construct a ring homomorphism from the cohomology of Sn-1 to the cohomology Of S-n with coefficients in the quotient field. Applications of these results in stable homotopy theory and relation to the chromatic splitting conjecture are discussed.

    Web of Science

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  • The geometric fixed point spectrum of (Z/p)^k Borel cohomology for E_n and its completion Reviewed

    T. Torii

    Contemporary Mathematics   293, 343-369   2002

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  • Rational homotopy type of the moduli of representations with Borel mold Reviewed

    K. Nakamoto, T. Torii

    Forum Mathematicum   24 ( 3 )   507 - 538   2012

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  • Topology of the representation varieties with Borel mold for unstable cases Reviewed

    K. Nakamoto, T. Torii

    Journal of the Australian Mathematical Society   91 ( 1 )   55 - 87   2011

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  • Algebraic vector bundles on SL(3,C) Reviewed

    K. Nakamoto, T. Torii

    The Rocky Mountain Journal of Mathematics   37 ( 2 )   587 - 596   2007

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  • Topological realization of level structures of the formal group law over E(n) Reviewed

    T Torii

    JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY   39 ( 3 )   577 - 587   1999.10

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：KINOKUNIYA CO LTD  

    Web of Science

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  • Level structure over E n and stable splitting by Steinberg idempotent Reviewed

    T Torii

    JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY   39 ( 3 )   589 - 596   1999.10

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：KINOKUNIYA CO LTD  

    Web of Science

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  • Topological realization of the integer ring of local field Reviewed

    T Torii

    JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY   38 ( 4 )   781 - 788   1998.12

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：KINOKUNIYA CO LTD  

    Web of Science

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MISC

  • Uniqueness of monoidal adjunctions

    Takeshi Torii

    2023.2

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    There are two dual equivalences between the $\infty$-category of
    $\mathcal{O}$-monoidal $\infty$-categories with right adjoint lax
    $\mathcal{O}$-monoidal functors and that with left adjoint oplax
    $\mathcal{O}$-monoidal functors, where $\mathcal{O}$ is an $\infty$-operad. We
    study the space of equivalences between these two $\infty$-categories, and show
    that the two equivalences equipped with compatible $\mathcal{O}$-monoidal
    presheaf functors are canonically equivalent.

    arXiv

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    Other Link： http://arxiv.org/pdf/2302.02035v2

  • Hecke operators in Morava $E$-theories of different heights

    Takeshi Torii

    2022.10

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    There is a natural action of a kind of Hecke algebra $\mathcal{H}_n$ on the
    $n$th Morava $E$-theory of spaces. We construct Hecke operators in an
    amalgamated cohomology theory of the $n$th and the $(n+1)$st Morava
    $E$-theories. These operations are natural extensions of the Hecke operators in
    the $(n+1)$st Morava $E$-theory, and they induce an action of the Hecke algebra
    $\mathcal{H}_{n+1}$ on the $n$th Morava $E$-theory of spaces. We study a
    relationship between the actions of the Hecke algebras $\mathcal{H}_n$ and
    $\mathcal{H}_{n+1}$ on the $n$th Morava $E$-theory, and show that the
    $\mathcal{H}_{n+1}$-module structure is obtained from the
    $\mathcal{H}_n$-module structure by the restriction along an algebra
    homomorphism from $\mathcal{H}_{n+1}$ to $\mathcal{H}_n$.

    arXiv

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    Other Link： http://arxiv.org/pdf/2210.06625v1

  • Duoidal $\infty$-categories of operadic modules

    Takeshi Torii

    2022.4

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    In this paper we study duoidal structures on $\infty$-categories of operadic
    modules. Let $\mathcal{O}^{\otimes}$ be a small coherent $\infty$-operad and
    let $\mathcal{P}^{\otimes}$ be an $\infty$-operad. If a
    $\mathcal{P}\otimes\mathcal{O}$-monoidal $\infty$-category
    $\mathcal{C}^{\otimes}$ has a sufficient supply of colimits, then we show that
    the $\infty$-category ${\rm Mod}_A^{\mathcal{O } }(\mathcal{C})$ of
    $\mathcal{O}$-$A$-modules in $\mathcal{C}^{\otimes}$ has a structure of
    $(\mathcal{P},\mathcal{O})$-duoidal $\infty$-category for any
    $\mathcal{P}\otimes\mathcal{O}$-algebra object $A$.

    arXiv

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    Other Link： http://arxiv.org/pdf/2204.11152v1

  • On higher monoidal $\infty$-categories

    Takeshi Torii

    2021.10

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    In this paper we introduce a notion of $\mathbf{O}$-monoidal
    $\infty$-categories for a finite sequence $\mathbf{O}^{\otimes}$ of
    $\infty$-operads, which is a generalization of the notion of higher monoidal
    categories in the setting of $\infty$-categories. We show that the
    $\infty$-category of coCartesian $\mathbf{O}$-monoidal $\infty$-categories and
    right adjoint lax $\mathbf{O}$-monoidal functors is equivalent to the opposite
    of the $\infty$-category of Cartesian $\mathbf{O}_{\rm rev}$-monoidal
    $\infty$-categories and left adjoint oplax $\mathbf{O}_{\rm rev}$-monoidal
    functors, where $\mathbf{O}^{\otimes}_{\rm rev}$ is a sequence obtained by
    reversing the order of $\mathbf{O}^{\otimes}$.

    arXiv

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    Other Link： http://arxiv.org/pdf/2111.00158v1

  • On duoidal $\infty$-categories

    Takeshi Torii

    2021.6

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    A duoidal category is a category equipped with two monoidal structures in
    which one is (op)lax monoidal with respect to the other. In this paper we
    introduce duoidal $\infty$-categories which are counterparts of duoidal
    categories in the setting of $\infty$-categories. There are three kinds of
    functors between duoidal $\infty$-categories, which are called bilax, double
    lax, and double oplax monoidal functors. We make three formulations of
    $\infty$-categories of duoidal $\infty$-categories according to which functors
    we take. Furthermore, corresponding to the three kinds of functors, we define
    bimonoids, double monoids, and double comonoids in duoidal $\infty$-categories.

    arXiv

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    Other Link： http://arxiv.org/pdf/2106.14121v1

  • Hochschild cohomology of Nm

    T. Itagaki, K. Nakamoto, T. Torii

    Proceedings of the 53rd Symposium on Ring Theory and Representation Theory   116 - 123   2022

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  • An application of Hochschild cohomology to the moduli of subalgebras of the full matrix ring II

    K. Nakamoto, T. Torii

    Ring theory 2019   176 - 187   2021

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  • An application of Hochschild cohomology to the moduli of subalgebras of the full matrix ring

    K, Nakamoto, T. Torii

    Proceedings of the 51st Symposium on Ring Theory and Representation Theory   110 - 118   2019

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  • The moduli of subalgebras of the full matrix ring of degree 3

    K. Nakamoto, T. Torii

    Proceedings of the 50th Symposium on Ring Theory and Representation Theory   137 - 149   2018

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  • Equivariance of generalized Chern characters

    Takeshi Torii

    2009.4

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    In this note some generalization of the Chern character is discussed from the
    chromatic point of view. We construct a multiplicative G_{n+1}-equivariant
    natural transformation \Theta from some height (n+1) cohomology theory E^*(-)
    to the height n cohomology theory K^*(-)\hat{\otimes}_F L, where K^*(-) is
    essentially the n-th Morava K-theory. As a corollary, it is shown that the
    G_n-module K^*(X) can be recovered from the G_{n+1}-module E^*(X). We also
    construct a lift of \Theta to a natural transformation between characteristic
    zero cohomology theories.

    arXiv

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    Other Link： http://arxiv.org/pdf/0904.1647v1

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Research Projects

  • トランス・クロマティック・ホモトピー論の研究

    Grant number：23K03113  2023.04 - 2028.03

    日本学術振興会  科学研究費助成事業  基盤研究(C)

    鳥居 猛

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    Grant amount：\4680000 （ Direct expense: \3600000 、 Indirect expense：\1080000 ）

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  • Equivariant Schubert calculus for p-compact groups

    Grant number：23K03092  2023.04 - 2026.03

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

    中川 征樹, 西本 哲, 鳥居 猛, 奥山 真吾

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    Grant amount：\3510000 （ Direct expense: \2700000 、 Indirect expense：\810000 ）

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  • Moduli of representations and related topics (4)

    Grant number：20K03509  2020.04 - 2024.03

    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)

    中本 和典, 鳥居 猛, 面田 康裕, 奥山 真吾

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    Grant amount：\4420000 （ Direct expense: \3400000 、 Indirect expense：\1020000 ）

    Quadratic monomial algebraであるN_m (m≧3)と名付けたKoszul algebraについて、板垣智洋氏（高崎経済大）と鳥居猛氏（岡山大）と共同で、そのHochschild cohomology ringの代数構造を決定し、無限生成代数であることを示した。その成果を「第53回環論および表現論シンポジウム」および日本数学会2022年度年会（埼玉大学）にて発表した。さらにGerstenhaber bracketを計算し、Gerstenhaber bracketと整合性のあるBatalin-Vilkovisky構造が入らないことを確認した。これらの結果について論文を執筆中である。
    面田康裕氏（明石高専）との共著“The classification of thick representations of simple Lie groups”がKodai Mathematical Journalに掲載受理された。連結複素単純Lie群の有限次元複素thick表現を分類した論文である。
    また、城野悠志氏（山梨大）との共著“Decision tree-based estimation of the overlap of two probability distributions"について、学術雑誌に投稿中である。その内容について、「非可換代数幾何学の大域的問題とその周辺」高知小研究集会で発表した。

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  • 一般の安定ホモトピー論における余加群の研究

    Grant number：17K05253  2017.04 - 2024.03

    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)

    鳥居 猛

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    Grant amount：\4680000 （ Direct expense: \3600000 、 Indirect expense：\1080000 ）

    一般の安定ホモトピー論におけるガロア群と導来淡中双対性および導来群スキームの表現のモジュライについて研究するために、デュオイダル圏およびデュオイダル圏におけるホップ亜代数とその余加群の無限大圏への一般化について研究を行った。
    2つのモノイダル構造をもち、一方のモノイダル構造が他方のモノイダル構造に関して、ラックスモノイダルになっている、あるいは同値であるが、一方のモノイダル構造が他方のモノイダル構造に関して、コラックスモノイダルになっているような圏をデュオイダル圏と呼ぶ。デュオイダル圏は双代数を定義できる最小の構造のみを備えた圏と考えることができる。今年度はデュオイダル圏の無限大圏への一般化について研究を行った。二つの無限大オペラッド上のモノイダル圏の構造をもち、一方のモノイダル構造が他方のモノイダル構造に関して、ラックスモノイダルになっている無限大圏を定式化した。また、無限大オペラッド上のモノイダル無限大バイカテゴリーに対して、ループ構成により、デュオイダル無限大圏が得られることを示した。また、デュオイダル無限大圏における双亜代数がホップ亜代数になるための条件をその余加群の無限大圏の性質により特徴づける研究を行った。
    また、研究集会「ホモトピー沖縄」、研究集会「空間の代数的・幾何的モデルとその周辺」、「非可換代数幾何学の大域的問題とその周辺」高知小研究集会、高知ホモトピー論談話会、福岡ホモトピー論セミナーなどに参加し、様々な研究者と研究課題について議論を行った。さらに、研究集会「ホモトピー沖縄」および高知ホモトピー論談話会ではデュオイダル無限大圏とホップ亜代数について講演を行った。

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  • Research on the stable homotopy category using quasi-categories

    Grant number：25400092  2013.04 - 2017.03

    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)

    Torii Takeshi

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    Grant amount：\4940000 （ Direct expense: \3800000 、 Indirect expense：\1140000 ）

    I have studied the stable homotopy category and its localizations by means of quasi-categories. Through spectral sequences, the stable homotopy category and its Bousfield localizations are considered to be related to the categories of representations for some groups and their derived categories. I gave a formulation of this relationship through the theories of model categories and quasi-categories. I have also constructed a functor between algebraic models of Bousfield localizations of the stable homotopy category via Morava K-theories. Furthermore, based on these, I have also studied more general moduli spaces of representations.

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  • Chromatic redshift and homotopical algebraic geometry

    Grant number：22540087  2010 - 2012

    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)

    TORII Takeshi

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    Grant amount：\3120000 （ Direct expense: \2400000 、 Indirect expense：\720000 ）

    I studied the relationship among the layers of the chromatic filtration in the stable homotopy category bymeans of homotopical algebraic geometry. I obtained the relationship between the Hecke operators on the Morava E-theories with different heights. I also showed that any spectrum which is local with respect to the Morava K-theory can be obtained as the homotopy fixed point spectrum for its Morava module in collaboration with D.G.Davis. Furthermore, I obtained results on the embeddings of the module categories for the Galois extensions of commutative S-algebras.

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  • 安定ホモトピー圏の大域的構造の研究

    Grant number：18740040  2006 - 2007

    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)

    鳥居 猛

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    Grant amount：\1700000 （ Direct expense: \1700000 ）

    安定ホモトピー圏の大域的構造の理解を目標とし、素ボルディズム関手および形式群を用いて安定ホモトピー圏の代数的モデルや数論的構造について研究を行った。特にクロマチックレベルが一つずれているMoravaK理論で局所化された安定ホモトピー圏の間の関係について調べた。
    Devinatz,Hopkins は Morava安定化群G_nの任意の閉部分群によるMoravaE理論E』のホモトピー固定点スペクトラムを構成し、特にMorava 安定化群全体によるホモトピー固定点スペクトラムが球面スペクトラムのMoravaK理論K(n)による局所化と一致することを示した。しかしながらDevinatz,Hopkins によるホモトピー固定点スペクトラムは、ホモトピー固定点スペクトラムが持つべき性質を満たすように技巧的に構成されている。Davis はこれを本来の固定点スペクトラムの観点から見直し、副有限群作用をもつ離散スペクトラムのモデル圏の構造を用いて連続スペクトラムのホモトピー固定点スペクトラムを定義した。さらに Davis はMorava E理論の Morava 安定化群によるホモトピー固定点スペクトラムの場合には、Devinatz,Hopkins のホモトピー固定点スペクトラムと一致することを示した。
    今年度の研究では Morava E理論 E_{n+1} のMorava K理論K(n)による局所化L_{K(n)}E_{n+1} のDavis の意味でのホモトピー固牢点スペクトラムについて考察し次のことを得た。
    (1) L_{K(n)}E_{n+1} は Morava 安定化群 G_{n+1} の作用に関して連続スペクトラムである。(2) Morava安定化群の任意の閉部分群に関する Davisの意味でのホモトピー固定点スペクトラムはDevinatz,Hopkinsの意味のホモトピー固定点スペクトラムのK(n)局所化と一致する。(3) L_{K(n)}E_{n+1} のホモトピー固定点スペクトル系列は K(n+1)局所 E_{n+1}-Adamsスペクトル系列のK(n)局所化に一致する。

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  • 安定ホモトピー圏と形式群のモジュライ

    Grant number：16740041  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)

    鳥居 猛

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    Grant amount：\1800000 （ Direct expense: \1800000 ）

    前年度に引き続き、複素ボルディズム函手および形式群を用いた安定ホモトピー圏の代数化についての研究を行った。安定ホモトピー圏は複素ボルディズム函手を通して形式群のモジュライ空間上の層の圏と密接な関係にある。この関係を通して形式群のモジュライ空間の構造が安定ホモトピー圏の構造に関して非常に強い代数的制限を課していることが知られている。また、安定ホモトピー圏をMorava K-理論で局所化した圏は形式群を通して整数論の局所理論と深い関係にあることが知られている。Honda group lawと呼ばれる基本的な形式群の自己同型群はMorava stabilizer groupと呼ばれ、Morava E-理論の乗法的な一次作用素のなす群として現れる。この群のコホモロジーはMorava K-理論により局所化された安定ホモトピー圏の最も基本的な不変量である。今年度は前年度に得られた結果の応用として、異なる形式群の高さに対応するMorava E-理論の間についての比較定理を得た。これは高さ(n+1)のMorava E-cohomologyの係数環上の加群の構造と一次作用素の作用の様子から高さnのMorava E-理論の係数環上の加群の構造と一次作用素の作用の様子が得られることを主張している。また、Morava K-理論K(n)に関するBousfield局所化L_<K(n)>Xのホモトピー群に収束するスペクトル系列とL_<K(n)>L_<K(n+1)>Xのホモトピー群に収束するスペクトル系列の間に射を構成し、E_<2^->項における射をMorava stabilizer groupのコホモロジーの間のある種のinflation mapとして記述した。また、Rognesにより定式化されたstrictly commutative ring spectrumの間のガロア理論を用いて、これまでに構成した一般化されたChern指標の性質について研究し、そのE_<∞^->構造や新しい記述についての結果を得た。

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  • Higher dimensional category and its applications

    Grant number：19540075  2007 - 2009

    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)

    NISHIDA Goro, KONO Akira, FUKAYA Kinji, NAKAJIMA Hiraku, MORIWAKI Atsushi, YOSHIDA Hiroyuki, WAKANO Isao, MINAMI Norihiko, TORII Takeshi

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    Grant amount：\4420000 （ Direct expense: \3400000 、 Indirect expense：\1020000 ）

    It is shown that a based mapping space from a finite Postnikov space to a finite complex is contractible. This implies that either homology groups or homotopy groups is not bounded up to certain dimension. In particular the Serre conjecture, homotopy groups of a finite complex with torsion homology is not bounded, is proved.

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  • グロタンディークデッサンと悲合同的タイヒミュラー被覆の数論

    Grant number：19654005  2007 - 2009

    日本学術振興会  科学研究費助成事業 挑戦的萌芽研究  挑戦的萌芽研究

    中村 博昭, 鳥居 猛, 鈴木 武史, 吉野 雄二, 山田 裕史, 松崎 克彦, 廣川 真男, 石川 佳弘

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    Grant amount：\3200000 （ Direct expense: \3200000 ）

    昨年度に基礎を確立した複素および1進の反復積分の関数等式の導出法(Wojtkowiak氏との共同研究)を延長して,具体的な実例計算をさらに検証した.とりわけ古典的な高次対数関数について知られている分布関係式(distribution relation)の1進版を導出することに成功した.分布関係式は,様々な特殊値を代入することで,高次対数関数の特殊値の間に成立する様々な関係式を組織的に生み出す重要なものであり,1進の場合にも並行してガロア群上の関数族(1-コチェイン)を統御する要となることが期待されるが,前年度までに得られた関数等式との整合性についても検証を行った.8月にケンブリッジのニュートン数理科学研究所で行われた研究集会"Anabelian Geometry"において口頭発表を行った.このときの講演に参加していたH.Gangl氏,P.Deligne氏から今後の研究指針を考える上で有用になると思われるコメントを頂戴することが出来た.また分布関係式の低次項の解消問題に関連して,有理的な道に沿った解析接続の概念にっいて考察を進める必要が生じた.こうしたテーマに関連して研究分担者の鳥居氏には,有理ホモトピー論に関する情報収集を担当していただき,また研究分担者の鈴木氏には,量子代数やKZ方程式との関連で組みひも群の数理についての情報収集を担当していただいた.以上の研究成果の一部は,共同研究者のWojtkowiak氏と協力して,"On distribution formula of complex and 1-adic polylogarithms"という仮題の草稿におおよその骨子をまとめたが,まだ完成に至っていない.周辺にやり残した問題(楕円ポリログ版など)もあり,これらについて一定の目処をつけてから公表までの工程を相談する予定になっている.

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  • A study of stable Hopf invariants and Hopf constructions

    Grant number：19540106  2007 - 2009

    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)

    ODA Nobuyuki, ISHIGURO kenshi, IWASE Norio, TORII Takeshi, HIRASHIMA Yasumasa

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    Grant amount：\4420000 （ Direct expense: \3400000 、 Indirect expense：\1020000 ）

    Formulas are obtained among box brackets and matrix Toda brackets. Generalizations of classical formulas by Toda are obtained including relations with the Hopf invariants. A class of spaces is defined which is closely connected with the LS category. Relations between the class and covering spaces are clarified which are useful to study examples. Topologies of function spaces which are defined by the exponential topology are studied. Exponential homeomorphisms are proved which hold for any topological spaces and applications of the theorem are obtained.

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  • Reconstruction of generalized cohomology theories by means of the notion of continuous functors

    Grant number：19540087  2007 - 2009

    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)

    SHIMAKAWA Kazuhisa, MIMURA Mamoru, OKUYAMA Shingo, TORII Takeshi

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    Grant amount：\4420000 （ Direct expense: \3400000 、 Indirect expense：\1020000 ）

    We constructed a category of topological spaces which satisfies the properties suitable for developing homotopy theory, and used it to establish a method for defining generalized cohomology theories by means of the notion of bivariant functors. In comparison with the conventional construction given by spectra, our new construction is more systematic and is easier to extend to categories other than that of topological spaces ; hence we may expect it to be applied to a variety of problems.

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  • Classifying spaces of compact Lie groups topology of p-compact groups

    Grant number：16540088  2004 - 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)

    ISHIGURO Kenshi, ODA Nobuyuki, KUROSE Takashi, TORII Takeshi

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    Grant amount：\3500000 （ Direct expense: \3500000 ）

    The research on the classifying spaces of compact Lie groups is one of the major area in Homotopy Theory. Our results obtained during 2004 through 2005 are basically concerned with maps between classifying spaces and their applications, as well as the invariant theory. Dwyer--Wilkerson introduced the notion of p-compact group and studied its properties. The purely homotopy theoretic object appears to be a good generalization of a compact Lie group. A p-compact group has rich structure, such as a maximal torus, a Weyl group, etc. A note written by Moeller in the AMS Bulletin summarizes their work. Further progress on the homotopy theory of p-compact groups are being made.
    We state here our main results. First, we considered a further generalization of a result of Dror Farjoun and Zabrodsky on the relationship between fixed point sets and homotopy fixed point sets, which is related to the generalized Sullivan Conjecture : Suppose a-space X with an action of a finite p-group is p-complete and the cohomological dimension is finite. Then the fixed point set is an empty set if and only if the homotopy fixed point set is empty. Next, we consider a problem on the conditions of a compact Lie group G that its loop space of the p-completed classifying space be a p-compact group for a set of primes. In particular, we discuss the classifying spaces BG that are p-compact for all primes when the groups are certain subgroups of simple Lie groups. Finally, we discuss the invariant theory and the cohomology of classifying spaces. The cohomology can be expressed as an invariant ring under the action of the Weyl group. All Weyl groups are reflection groups. We obtained some results on certain reflection groups.

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  • Homotopy Methods in a Generalization of Lie Group Theory

    Grant number：14540096  2002 - 2003

    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)

    ISHIGURO Kenshi, TORII Takeshi, KUROSE Takashi, ODA Nobuyuki

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    Grant amount：\2800000 （ Direct expense: \2800000 ）

    The research on the classifying spaces of compact Lie groups is one of the major area in Homotopy Theory. Our results obtained during 2002 through 2003 are basically concerned with maps between classifying spaces and their applications Dwyer-Wilkerson defined a p-compact group and studied its properties. The purely homotopy theoretic object appears to be a good generalization of a compact Lie group. A p-compact group has rich structure, such as a maximal torus, a Weyl group, etc. A note written by Moeller in the AMS Bulletin summarizes their work. Further progress on the homotopy theory of p-compact groups are being made. We state here our main results. First, we generalize a result of Dror Farjoun and Zabrodsky on the relationship between fixed point sets and homotopy fixed point sets, which is related to the generalized Sullivan Conjecture : For a finite p-group Π, suppose a Π -space X is F_p-complete and cd_p(X) is finite. Then X^πis an empty set if and only if the homotopy fixed point set X^<hπ> is empty. As an application, we discuss extension problems considering actions on homogeneous spaces of p-compact groups. Next, we consider a problem on the conditions of a compact Lie group G that its loop space of the p-completed classifying space be a p-compact group for a set of primes. In particular, we discuss the classifying spaces BG that are p-compact for all primes when the groups are certain subgroups of simple Lie groups. A necessary and sufficient condition that BG be p-compact toral for all primes has been obtained. We ask if BH is p-compact for a set of primes when H is a subgroup of a simple Lie group G and obtaine certain results.

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