Updated on 2022/03/17

写真a

 
TANAKA Katsumi
 
Organization
Institute for Promotion of Education and Campus Life Professor
Position
Professor
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Degree

  • Ph. D ( 1988.3   Kobe University )

  • Master of Science ( 1983.3   Shizuoka University )

Research Interests

  • mathematical logic

  • model theory

  • 大学入試

Professional Memberships

 

Papers

  • Ladder Index of Groups Reviewed

    Kazuhiro Ishikawa, Hiroshi Tanaka, Katsumi Tanaka

    Mathematical Journal of Okayama University   44   37 - 41   2002

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    Authorship:Last author   Language:English   Publishing type:Research paper (scientific journal)  

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  • Bounded word length of linear groups Reviewed

    Katsumi Tanaka

    Proceedings of Model Theory at St. Cahterine College Kobe Institute, Tokai University   1999.2

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    Authorship:Lead author   Language:English   Publishing type:Research paper (international conference proceedings)  

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  • 2重可移群のモデル理論

    田中克己

    岡山大学医療技術短期大学部紀要   8   1 - 6   1997.9

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    Authorship:Lead author   Language:Japanese   Publishing type:Research paper (bulletin of university, research institution)  

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  • 安定群上の位相について

    田中克己

    岡山大学医療技術短期大学部紀要   8   27 - 29   1993.1

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    Authorship:Lead author   Language:Japanese   Publishing type:Research paper (bulletin of university, research institution)  

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  • A two decomposition of a bounded metric space Reviewed

    Kazuaki Kitahara, Katsumi Tanaka

    Applied Mathematics Letters   4 ( 1 )   21 - 23   1991.11

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    Authorship:Last author   Language:English   Publishing type:Research paper (scientific journal)  

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  • On locally finite stable groups

    Katsumi Tanaka

    Bulletin of Osaka Prefectural College of Technology   24   137 - 144   1990.10

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  • Non-abelian groups of Morley rank 2 Reviewed

    Katsumi Tanaka

    Mathematica Japonica   33 ( 4 )   627 - 635   1988.12

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  • Some local properties for omega-stable groups Reviewed

    Katsumi Tanaka

    Archive for Mathematical Logic   21 ( 1 )   45 - 47   1988.11

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    Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)  

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  • On the theory of ordered groups Reviewed

    Katsumi Tanaka

    Kobe Journal of Mathematics   5 ( 1 )   117 - 122   1988.6

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    Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)  

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  • omega-stable groups and CZ-groups

    644   146 - 156   1988.2

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    Authorship:Lead author   Language:Japanese   Publishing type:Research paper (conference, symposium, etc.)  

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  • Some structure theorems for omega-stable groups Reviewed

    Katsumi Tanaka

    Proceeding of The Japan Academy   63 ( 7 )   254 - 257   1987.9

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  • Algebraic groups and omega-stable groups

    Katsumi Tanaka

    608   152 - 166   1987.2

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Presentations

  • 岡山大学の国際バカロレア入試 Invited

    田中克己

    APシンポジウム(テーマIII)  2019.10.26 

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    Event date: 2019.10.26

    Language:Japanese   Presentation type:Oral presentation (general)  

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Research Projects

  • Combinatorics and Representation Theory of Nonlinear Differential Equations

    Grant number:17540026  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)

    YAMADA Hiro-fumi, YOSHINO Yuji, NAKAMURA Hiroaki, HIRANO Yasuyuki, TANAKA Katsumi, IKEDA Takeshi

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

    I focused on the applications of the representation theory of the symmetric groups to certain nonlinear systems of differential equations. More precisely I investigated the Cartan matrices of the symmetric groups which play an important role in modular representation theory. It has been known that the coefficients of Q-functions appearing in the expansion of 2-reduced Schur functions are non-negative integers. These are called the Stembridge coefficients. I noticed that the matrices of Stembridge coefficients are "similar" to the decomposition matrices for the 2-modular representations of the symmetric groups. I proved that they are transformed to each other by simple column operations, and that the elementary divisors of the Cartan matrices and those of the so-called "Gartan matrices" coincide. Next I introduced the "compound basis" for the space of the symmetric functions and expanded (non-reduced) Schur functions in terms of our new basis. I found that the appearing coefficients are all integers. This compound basis arose naturally, at least for me, from representation theory of certain affine Lie algebras, which I have been studying for many years. At the present moment our basis is obtained only for the case of characteristic 2, but it is plausible that this exists for any characteristic p. A natural problem occurs: What is the transition matrix between the two bases, i.e., Schur function basis and our compound basis ? In a joint work with Mizukawa and Aokage, it is proved that the determinant of this transition matrix is a power of 2. This is a non-trivial fact.

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  • Research on Operator Analysis for Nonrelativistic Quantum Dynamics in High Energy Region

    Grant number:16540155  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)

    HIROKAWA Masao, HIROSHIMA Fumio, TAMURA Hideo, SATO Ryotaro, TANAKA Katsumi

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    Grant amount:\3700000 ( Direct expense: \3700000 )

    There are some mathematical problems in nonrelativistic quantum electrodynamics. Especially, we have been concerned with ultraviolet catastrophe in high energy region and moreover infrared catastrophe after removing ultraviolet cutoff.
    Hirokawa established some problems on ultraviolet catastrophe and/or infrared castastrophe in mathematics. Hiroshima and Spohn investigated the renormalized mass of electron. Hirokawa, Hiroshima, and Spohn have succeeded in removing both, ultraviolet and infrared cutoffs for the so-called Nelson model, which describes the electron coupled with photons in such as hydrogen-like atom. Also, Hirokawa clarified that the result by the regular perturbation theory is still available in the above model after removing both cutoffs in collaboration with Hainzl and Spohn. They employed a nonperturbative method to prove it.
    Hirokawa studied scattering theory with Tamura, problem on singularity of ultraviolet catastrophe with Tanaka, and uniqueness of the ground state with Sato.

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  • モジュライ空間の算術幾何に対する種種のグラフ複合体の関与の研究

    Grant number:15654007  2004 - 2005

    日本学術振興会  科学研究費助成事業 萌芽研究  萌芽研究

    中村 博昭, 山田 裕史, 吉野 雄二, 田中 克己, 勝田 篤, 廣川 真男

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

    昨年度に赤澤尋樹氏が行ったグラスマン代数のある商代数にあらわれる斜交表現の不変式のなす環の三叉グラフの生成関係式をつかった表示の研究に基づいて,研究代表者が過去に米国のガロウファリディス教授との共同研究のなかで,より立ち入った議論を必要とする部分について簡単なまとめを行った.
    長年の懸案であるモーデル型の楕円曲線の1パラメータ族についてGrothendieck-Teichmueller理論で有用なものの探索を引き続いて精力的に行ったが,残念ながらまだ十分に議論が展開できていない.一方でグロタンディーク・デッサンに付随する代数曲線の不変量の計算法について,主に三角群に付随して注目に値する実例について保型関数との関連で議論を進め,いくつか注目に値する知見を深めることが出来た.
    6月には,カリフォルニア工芸州立大学の加藤五郎氏の岡山来訪を実現し,物質,空間,時間の前層化に基づく氏による非常に前衛的な理論に関する講演"Extension Type Yoneda Lemma for Relativistic t-Topos"を通して数理物理におけるカテゴリー理論の可能性について知見を深めた.
    研究分担者には、それぞれの専門の立場から研究課題に関連する内容の発展について協力を頂いていた.特に田中克己助教授には,国内出張を通して,情報収集に協力していただいた.
    また本研究課題と関連する必要な図書備品・消耗品の購入を通して,各分担者の研究活動を支援した.

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  • Combinatorial Representation Theory which Center of Schur Functions

    Grant number:15540030  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)

    YAMADA Hirofumi, YOSHINO Yuji, NAKAMURA Hiroaki, HIRANO Yasuyuki, TANAKA Katsumi, IKEDA Takeshi

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    Grant amount:\2900000 ( Direct expense: \2900000 )

    This is an effort for understanding the role of Schur functions and Schur's Q functions, the projective analogue of Schur functions, in representation theory. To be more precise, we proved the following theorem. Schur functions associated with the rectangular Young diagrams occur as weight vectors of the basic representation of the affine Lie algebra of type D^{(2)}_2. And also, they turn out to be the homogeneous tau functions of the nonlinear Schroedinger hierarchy. The key idea for proving the above is to write down the representation spaces and operators in terms of fermions, and derive polynomials via the boson-fermion correspondence. We have succeeded in verifying the similar phenomena for the case of the affine Lie algebra of type A^{(2)}_2. In 2004 we considered the following problem. Clarify the nature of the coefficients in the 2 reduced Schur functions when expanded in terms of Schur's Q functions. Through some experimental computations in small rank cases, I had been convinced that these coefficients are of great interests, both from representation theoretical and combinatorial points of view. Finally we realized that these coefficients are nothing but the so-called Stembridge numbers. As a result we could relate these numbers with the representation theory of affine Lie algebras. Looking carefully at the table of these numbers, we found a simple formula for the elementary divisors of the Cartan matrices of the symmetric groups. More than 10 years ago, Olsson in Copenhagen gave a formula for those, which is expressed in terms of a generating function and is rather complicated. Our version is more direct and combinatorial.

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  • Galois groups and fundamental groups in anabelian geometry

    Grant number:14340017  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)

    NAKAMURA Hiroaki, YAMADA Hiro-fumi, YOSHINO Yuji, TANAKA Katsumi, KATSUDA Atsushi, HIROKAWA Masao

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    Grant amount:\10000000 ( Direct expense: \10000000 )

    We studied a measure function that describes the meta-abelian quotient of the monodromy representation associated with the universal family of elliptic curves and its relation with generalized Dedekind sums.
    In particular, we showed a congruence formula that describes moment integrals of the measure function along variation of weights. Equations in the Grothendieck-Teichmueller group satisfied by the Galois image were investigated.
    Using genus zero non-Galois covers, we found a new type equation. Utilizing the Magnus-Gassner type representation, another new type equation was found to hold in the topological matrix ring in two variables.
    In a collaboration with H.Tsunogai, using a characterization of the lemniscate elliptic curve as a Grothendieck dessin, we studied the behavior of Galois parameters of the Grothendieck-Teichmueller group, and described the decomposition of the standard parameter into a product of mutually transposed harmonic parameters in terms of adelic beta functions. In a collaboration with P.Lochak and L.Schneps, we replaced a toplogical path from the standard tangential basepoint to the five cyclic point by a composition of algebraic paths that are transformed by the Galois group with Grothendieck-Teichmueller parameters. Then, we succeeded in interpreting the five cyclic decomposition of the standard parameter in the fundamental group of the moduli spaces of the 5-pointed projective lines. Comparing the method of Ihara-Matsumoto with a paper by Gerritzen-Herrlich-Put about stable compactification of moduli spaces of n-pointed projective lines, we obtained a natural interpretation of tangential base points on those moduli spaces. Through discussions with Wojtkowiak at Nice University, a new direction of investigation and perspectives about 1-adic itereated integrals was obtained.

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  • OPERATOR-ANALYTICAL STUDY OF SINGULARITIES OF HAMILTONIANS IN QUANTUM PHYSICS

    Grant number:13640215  2001 - 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)

    HIROKAWA Masao, TAMURA Hideo, SATO Ryotaro, TANAKA Katsumi, HIROSHIMA Fumio, YAMADA Hirofumi

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    Grant amount:\2900000 ( Direct expense: \2900000 )

    We studied Nelson's model derived from the Pauli-Fierz model through several physical approximations. The Pauli-Fierz model describes an electron coupled with the quantized radiation field in nonrelativistic quantum electrodynamics, when we regard the electron as a nonrelativistic particle. We proved that the Nelson model has infrared catastrophe when its Hamiltonian has the Coulomb potential appearing in the structure of atoms. Developing the proof and using the Carleman operator, we clarified and characterized a mathematical mechanism which causes infrared safe or infrared divergence. The Carleman operator is derived from the so-called pull-through formula, and we gave the exact operator-theoretical proof for the formula, which is the fast to succeed in it. By this proof, we can investigate mathematical properties of the domain of the Carleman operator and pull-through formula, which resulted in our results. Because Nelson's model has infrared catastrophe by our results, we find another representation in which the model has a ground state. This representation describes the actual physical phenomenon. So, we removed both, infrared and ultraviolet cutoffs, and proved the Nelson model without both cutoffs has a ground state in the representation.
    We studied the norm resolvent convergence for the Hamiltonian describing relativistic particle coupled with the Aharonov-Bohm field in 2-dim. space. We investigated which self-adjoint extension has the most suitable representation to the actual physics among several self-adjoint extensions corresponding to the boundary conditions around singularities.

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  • Study of control theory of the fixed point sets on spheres

    Grant number:09640110  1997 - 1999

    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)

    MORIMOTO Masaharu, NAKAJIMA Atsushi, NODA Ryuzaburo, SHIMOKAWA Kazuhisa, TANAKA Katsumi, IKEHATA Shuichi

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

    The purpose of this research was to study the following three : (1) (P(G), L(G))-controlled equivariant surgery, cobordism and representation theories and a theory to control isotropy subgroups appearing on manifolds ; (2) Dress'induction of the equivariant cobordism theory of equivariant framed normal maps ; (3) the injection maps IndィイD3G(/)HィエD3 among various finite groups H ⊂ G ; and determine the G-fixed point manifolds of smooth G-actions on spheres for Oliver groups G. We obtained the following results in the research. (1) We proved a deleting-inserting theorem of fixed point components on disks and spheres for Oliver Groups. In a joint work K. Pawalowski, we proved an extension theory of (P(G), L(G))-vector bundles on finite G-CW complexes. Using the equivariant thickening theory with this extension theory, we developed a theory to control isotropy subgroups on disks. (2) We proved that Bak-Morimoto's surgery obstruction group is a Mackey functor on which a Green functor acts, and algebraic Dress'induction works for the obstruction group. In addition, we proved that the cobordism invariance of the surgery obstruction and show that geometric Dress'induction works. (3) In joint works with T. Sumi and M. Yanagihara, we studied the induction maps IndィイD3G(/)HィエD3 for various finite groups H ⊂ G, we constructed (P(G), L(G))-matched pairs and (P(G), L(G))-gap modules for many G. Putting all this together, we determined the G-fixed point manifolds of smooth G-actions on spheres for various Oliver groups G.

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  • Spectral and Scattering Theory and its Application

    Grant number:09640151  1997 - 1998

    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)

    TAMURA Hideo, KAWASHITA Mishio, ONISHI Kazuei, IWATSUKA Akira, TANAKA Katumi, KATSUDA Atsushi

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

    The present project has been devoted to the study on the following three subjects related to the spectral and scattering theory for Schr_dinger operators.
    (1) For exponential product formula(Lie-Trotter-Kato product formula), the convergence in operator norm has been proved and the error estimate has been also established. The 9btained results have been applied to Schr_dinger semi-groups or propagators with singular or time-dependent potentials.
    (2) The unperturbed Pauli operator without electric potentials has zero eigenvalue with infinite multiplicities as its bottom of essential spectrum. When the operators are perturbed by potentials falling off at infinity, the asymptotic distribution of discrete eigenvalues near the origin has been studied. The special emphasis is placed on the case that Pauli operators do not necessarily have constant magnetic fields.
    (3) The asymptotic behavior at low energy of scattering amplitudes has been analysed for scattering by two dimensional magnetic fields and the relation to scattering by magnetic fields with small support has been also discussed.

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  • 一般コホモロジー論

    Grant number:07640117  1995

    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)

    藤井 道一, 田中 克己, 田坂 隆士, 吉岡 巌, 島川 和久, 三村 護

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    Grant amount:\1300000 ( Direct expense: \1300000 )

    1.Gがコンパクト可換リー群の場合に,複素G同変同境理論を局所化して得られるG同変コホモロジー理論では,複素G同変ベクトル東に対して分解原理が成り立つ.このことを用いて形式群を構成し,その構造について研究した.これは複素G同境理論の係数環を調べるために必要である.種々の結果を得たが,まだ発表できる段階に到らない。
    2.(1)リー群,特にコンパクト例外群および射影古典群のMorava K-理論の決定問題は,これまでN.Yagita,J.R.Hunton等により行われてきたが,未決定なところが少なからずある.これについて,Atiyah-Hirzebruch-Serreスペクトル系列を用いて決定した.これはJ.R.Hunton-T.Nishimoto-B.Schusterとの共同研究として発表される予定である.
    (2)R.Body-D.Sullivanにより約20年前に予想された,p-普遍空間に関する結果(特徴付け)に完全な証明を与えた.これはR.Body-H.Shiga-D.Sullivanとの共同研究として発表される予定である.
    3.任意のコンパクトリー群Gに対して,G同変束の分類空間のモデルをカテゴリカルな構成法を用いて与えた.このモデルは構造群および作用する群の双方に関して完全に凾手的である,という利点を持っている.
    4.minimal usco写像に関連する研究は1982年以来J.P.R.Christensen等により盛んに行われている.minimal usco写像に関する,ある条件を満たす位相空間の族についての特徴付けを行った.
    5.ガロアコホモロジーについて研究し,いくつかの結果を得た.

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  • ω安定群のモデル理論的研究

    Grant number:06740156  1994

    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)

    田中 克己

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    Grant amount:\900000 ( Direct expense: \900000 )

    1.モーレーランク有限のω(オメガ)安定群は極小群を持てば定義可能であることが知られているが.また極大部分群を持つとも限らないし,たとえ持っていてもそれが定義可能になるかどうかはわからない.しかし,有限のモーレーランクを持つω安定群は,定義可能な部分群のうちで極大なものがあればそれが極大部分群であることを示した.
    この応用として次のことが明らかになった.モーレーランクが3でランク2の部分群を持たない群とくにそのような単純群,いわゆるBad group,はその存在も分かっていない.Bad groupのランク1の定義可能な部分群は定義可能な部分群のなかで極大であるので,先の結果から,これらはアブストラクトな群として極大部分群となることが分かった.
    2.任意のべき零群にたいしあるリー環を構成する方法はすでに知られており,ゼルマノフはこのことを使って,べき指数が素数のベキである群について『制限されたバーンサイド問題』を解決したが,このリー環の構成は定義可能であることを示した.したがって,もとのべき零群とそれから構成されるリー環の安定性のクラスは同じであることが分かった.
    この応用として,「すべてのべき指数有限のω安定群はべき零群である」という予想について,このリー環を使って解決されるのではないかと期待される.

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Class subject in charge

  • Model Theory (2021academic year) Prophase  - その他

  • Seminar in Algebra (2021academic year) Year-round  - その他

  • Mathematical Logic (2021academic year) Late  - 火1,火2

  • Model Theory (2020academic year) Prophase  - その他

  • Seminar in Algebra (2020academic year) Year-round  - その他

  • Mathematical Logic (2020academic year) Late  - 火1,火2

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