Updated on 2024/05/19

写真a

 
ORITA Ryuma
 
Organization
Academic Assembly Institute of Science and Technology Fundamental Sciences Assistant Professor
Faculty of Science Department of Science Assistant Professor
Title
Assistant Professor
Contact information
メールアドレス
External link

Degree

  • Ph.D. in Mathematical Sciences ( 2017.3   The University of Tokyo )

Research Interests

  • Floer Theory

  • Morse Theory

  • symplectic manifolds

  • Hamiltonian Dynamics

  • Topological complexity

  • Persistence modules

Research Areas

  • Natural Science / Geometry  / Topology

  • Natural Science / Geometry  / Symplectic Geometry

Research History (researchmap)

  • Niigata University   Faculty of Science   Assistant Professor

    2020.3

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    Country:Japan

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  • Aoyama Gakuin University   College of Science and Engineering   Teaching Associate (adjunct)

    2019.4 - 2020.2

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    Country:Japan

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  • Tokyo Metropolitan University   Graduate School of Science   JSPS Research Fellow (PD)

    2018.4 - 2020.2

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    Country:Japan

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  • National Center for Theoretical Sciences (NCTS)   Mathematics Division   Postdoctoral Fellow

    2017.8 - 2018.3

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    Country:Taiwan, Province of China

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  • Kyushu University   Institute of Mathematics for Industry   Postdoctoral Researcher

    2017.4 - 2017.7

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    Country:Japan

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  • The University of Tokyo   Graduate School of Mathematical Sciences   JSPS Research Fellow (DC1)

    2014.4 - 2017.3

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    Country:Japan

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Research History

  • Niigata University   Faculty of Science Department of Science   Assistant Professor

    2020.3

Education

  • The University of Tokyo   Graduate School of Mathematical Sciences   Doctoral Course

    2014.4 - 2017.3

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    Country: Japan

    Notes: 数物フロンティア・リーディング大学院コース生

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  • The University of Tokyo   Graduate School of Mathematical Sciences   Master Course

    2012.4 - 2014.3

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    Country: Japan

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  • Kyushu University   School of Science   Department of Mathematics

    2008.4 - 2012.3

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    Country: Japan

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Professional Memberships

  • The Mathematical Society of Japan

    2020.4

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Papers

  • Rigid fibers of integrable systems on cotangent bundles Reviewed International journal

    Morimichi Kawasaki, Ryuma Orita

    Journal of the Mathematical Society of Japan   74 ( 3 )   829 - 847   2022.7

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:Mathematical Society of Japan (Project Euclid)  

    DOI: 10.2969/jmsj/84278427

    arXiv

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  • Existence of pseudoheavy fibers of moment maps Reviewed International journal

    Morimichi Kawasaki, Ryuma Orita

    Communications in Contemporary Mathematics   23 ( 05 )   2050047 - 2050047   2020.7

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

    In this paper, we introduce the notion of pseudoheaviness of closed subsets of closed symplectic manifolds and prove the existence of pseudoheavy fibers of moment maps. In particular, we generalize Entov and Polterovich’s theorem, which ensures the existence of non-displaceable fibers. As its application, we provide a partial answer to a problem posed by them, which asks the existence of heavy fibers. Moreover, we obtain a family of singular Lagrangian submanifolds in [Formula: see text] with various rigidities.

    DOI: 10.1142/s0219199720500479

    arXiv

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  • On the existence of infinitely many non-contractible periodic orbits of Hamiltonian diffeomorphisms of closed symplectic manifolds Reviewed International journal

    Ryuma Orita

    Journal of Symplectic Geometry   17 ( 6 )   1893 - 1927   2020.1

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

    We show that the presence of a non-contractible one-periodic orbit of a
    Hamiltonian diffeomorphism of a connected closed symplectic manifold
    $(M,\omega)$ implies the existence of infinitely many non-contractible simple
    periodic orbits, provided that the symplectic form $\omega$ is aspherical and
    the fundamental group $\pi_1(M)$ is either a virtually abelian group or an
    $\mathrm{R}$-group. We also show that a similar statement holds for Hamiltonian
    diffeomorphisms of closed monotone or negative monotone symplectic manifolds
    under the same conditions on their fundamental groups. These results generalize
    some works by Ginzburg and G\"urel. The proof uses the filtered Floer--Novikov
    homology for non-contractible periodic orbits.

    DOI: 10.4310/JSG.2019.v17.n6.a9

    arXiv

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  • Disjoint superheavy subsets and fragmentation norms Reviewed International journal

    Morimichi Kawasaki, Ryuma Orita

    Journal of Topology and Analysis   13 ( 02 )   443 - 468   2019.6

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

    We present a lower bound for a fragmentation norm and construct a bi-Lipschitz embedding [Formula: see text] with respect to the fragmentation norm on the group [Formula: see text] of Hamiltonian diffeomorphisms of a symplectic manifold [Formula: see text]. As an application, we provide an answer to Brandenbursky’s question on fragmentation norms on [Formula: see text], where [Formula: see text] is a closed Riemannian surface of genus [Formula: see text].

    DOI: 10.1142/S179352532050017X

    arXiv

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  • Application of fragmentation norms to transported points by Hamiltonian isotopies

    Morimichi Kawasaki, Ryuma Orita

    RIMS Kôkyûroku   2098   2018.6

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

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  • Morse-Bott inequalities for manifolds with boundary Reviewed International journal

    Ryuma Orita

    Tokyo Journal of Mathematics   41 ( 1 )   113 - 130   2017.12

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

    In the present paper, we define Morse-Bott functions on manifolds with
    boundary which are generalizations of Morse functions and show Morse-Bott
    inequalities for these manifolds.

    DOI: 10.3836/tjm/1502179256

    arXiv

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  • Non-contractible periodic orbits in Hamiltonian dynamics on tori Reviewed International journal

    Ryuma Orita

    Bulletin of the London Mathematical Society   49 ( 4 )   571 - 580   2017.4

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

    We show that the presence of one non-degenerate, non-contractible periodic orbit of a Hamiltonian on the standard symplectic torus implies the existence of infinitely many simple non-contractible periodic orbits.

    DOI: 10.1112/blms.12054

    arXiv

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  • Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories Reviewed International journal

    Morimichi Kawasaki, Ryuma Orita

    Journal of Modern Dynamics   11   313 - 339   2017.4

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

    The first author introduced a relative symplectic capacity $C$ for a
    symplectic manifold $(N,\omega_N)$ and its subset $X$ which measures the
    existence of non-contractible periodic trajectories of Hamiltonian isotopies on
    the product of $N$ with the annulus $A_R=(R,R)\times\mathbb{R}/\mathbb{Z}$. In
    the present paper, we give an exact computation of the capacity $C$ of the
    $2n$-torus $\mathbb{T}^{2n}$ relative to a Lagrangian submanifold
    $\mathbb{T}^n$ which implies the existence of non-contractible Hamiltonian
    periodic trajectories on $A_R\times\mathbb{T}^{2n}$. Moreover, we give a lower
    bound on the number of such trajectories.

    DOI: 10.3934/jmd.2017013

    arXiv

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Books

  • Linear Mathematics, Third Edition

    Tamaki Tanaka, Hideo Kojima, Akinari Hoshi, Ryuma Orita, Nobuhiro Innami, Hisao Yoshihara( Role: Joint author)

    Baifukan  2022.4  ( ISBN:9784563012410

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    Language:Japanese

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MISC

  • Floer-type bipersistence modules and rectangle barcodes

    Kanta Koeda, Ryuma Orita, Kanon Yashiro

    2023.12

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    Language:English   Publisher:arXiv  

    In this paper, we show that the pointwise finite-dimensional two-parameter
    persistence module $\mathbb{HF}_*^{(\bullet,\bullet]}$, defined in terms of
    interlevel filtered Floer homology, is rectangle-decomposable. This allows for
    the definition of a barcode $\mathcal{B}_*^{(\bullet,\bullet]}$ consisting only
    of rectangles in $\mathbb{R}^2$ associated with
    $\mathbb{HF}_*^{(\bullet,\bullet]}$. We observe that this rectangle barcode
    contains information about Usher's boundary depth and spectral invariants
    developed by Oh, Schwarz, and Viterbo. Moreover, we introduce a novel invariant
    extracted from $\mathcal{B}_*^{(\bullet,\bullet]}$, which proves instrumental
    in detecting periodic solutions of Hamiltonian systems. Additionally, we
    establish relevant stability results, particularly concerning the bottleneck
    distance and Hofer's distance.

    DOI: 10.48550/ARXIV.2312.07847

    arXiv

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    Other Link: http://arxiv.org/pdf/2312.07847v3

  • Topological complexity of monotone symplectic manifolds

    Ryuma Orita

    2023.11

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    Language:English   Publisher:arXiv  

    We study Farber's topological complexity for monotone symplectic manifolds.
    More precisely, we estimate the topological complexity of 4-dimensional
    spherically monotone manifolds whose Kodaira dimension is not $-\infty$.

    DOI: 10.48550/ARXIV.2311.14989

    arXiv

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    Other Link: http://arxiv.org/pdf/2311.14989v2

Research Projects

  • Robot motion planning and symplectic geometry

    Grant number:21K13787

    2021.4 - 2026.3

    System name:Grants-in-Aid for Scientific Research Grant-in-Aid for Early-Career Scientists

    Research category:Grant-in-Aid for Early-Career Scientists

    Awarding organization:Japan Society for the Promotion of Science

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    Authorship:Principal investigator  Grant type:Competitive

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

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  • Applications of periodic orbits in Hamiltonian dynamics and persistence modules

    Grant number:20K22302

    2020.9 - 2022.3

    System name:Grants-in-Aid for Scientific Research Grant-in-Aid for Research Activity start-up

    Research category:Grant-in-Aid for Research Activity start-up

    Awarding organization:Japan Society for the Promotion of Science

    Orita Ryuma

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    Authorship:Principal investigator  Grant type:Competitive

    Grant amount:\2860000 ( Direct Cost: \2200000 、 Indirect Cost:\660000 )

    In this research I dealt with Ginzburg-Gurel conjecture which states that "every Hamiltonian diffeomorphism of closed symplectic manifolds has infinitely many non-contractible periodic orbits, provided that the diffeomorphism has one orbit". Here a manifold is said to be symplectic if it admits a non-degenerate closed two-form. I investigated the problem by assuming some conditions on the fundamental group of the manifold and the symplectic form. Actually, I proved that the conjecture is true for spherically monotone symplectic manifolds whose fundamental group is assumed to be virtually abelian or an R-group.
    During the period, I proved that the equivalence between R-groups and torsion-free group of type N. Accordingly, since torsion-free groups of type N are principal, I could apply the theory of Bredon cohomology for them.

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  • Research on applications of Floer theory and persistence modules

    Grant number:18J00335

    2018.4 - 2021.3

    System name:Grants-in-Aid for Scientific Research Grant-in-Aid for JSPS Fellows

    Research category:Grant-in-Aid for JSPS Fellows

    Awarding organization:Japan Society for the Promotion of Science

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    Authorship:Principal investigator  Grant type:Competitive

    Grant amount:\3640000 ( Direct Cost: \2800000 、 Indirect Cost:\840000 )

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  • Morse理論の様々な応用に関する研究

    Grant number:14J07057

    2014.4 - 2017.3

    System name:科学研究費助成事業 特別研究員奨励費

    Research category:特別研究員奨励費

    Awarding organization:日本学術振興会

    折田 龍馬

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    Authorship:Principal investigator  Grant type:Competitive

    Grant amount:\2500000 ( Direct Cost: \2500000 )

    本研究の最終年度は昨年度に引き続き、閉シンプレクティック多様体上のハミルトン力学系における非可縮な周期軌道の存在、特にGurel予想について研究した。Gurel予想はConley予想の変種であり、「閉シンプレクティック多様体上の任意のハミルトニアンは、非可縮周期軌道を少なくとも1つ持てば、無限に多くの非可縮周期軌道を持つ」ということを主張している。Gurel予想はまずGurel自身によって多様体がsymplectically atoroidalな場合に示され、その後GinzburgとGurelによってtoroidally monotoneな場合にまで拡張されている。昨年度私は、両者に当てはまらない多様体である2n次元トーラスについてGurel予想が正しいことを示した。
    そこで本年度は、以上の先行研究の拡張を行った。すなわち、基本群がvirtually abelian群かR群である場合に、symplectically asphericalな多様体についてGurel予想が正しいことを示した。さらに、同様の条件を満たす基本群を持つmonotoneな多様体に対しても成り立つことを示した。ここでR群とは、方程式の根が存在すれば一意であるような群である。以上の性質を満たすシンプレクティック多様体としてはトーラスの他に、小平・サーストン多様体や、それらと複素射影空間の直積などが挙げられる。
    証明にはフレアー・ノビコフ理論を用いた。フレアー・ノビコフホモロジーでは、その生成元として周期軌道のノビコフ作用を考慮する必要がある。しかし、基本群が上述のようにアーベル群に「近い」性質を持てば、有理ホモトピー論を援用することで、ハミルトニアンのイテレーションによる生成元の変容を記述することが出来た。

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Teaching Experience (researchmap)

  • Geometry IIA

    2020.10

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    Level:Undergraduate (specialized)  Country:Japan

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  • Geometry IIB

    2020.10

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    Level:Undergraduate (specialized)  Country:Japan

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  • Geometry IA

    2020.4

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    Level:Undergraduate (specialized)  Country:Japan

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  • Geometry IB

    2020.4

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    Level:Undergraduate (specialized)  Country:Japan

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  • Mathematics Exercise B

    2024.10

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    Level:Undergraduate (liberal arts)  Country:Japan

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  • Mathematics Exercise A

    2024.10

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    Level:Undergraduate (liberal arts)  Country:Japan

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  • Introduction to Study "Science, Technology and Society"

    2024.4

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    Level:Undergraduate (liberal arts)  Country:Japan

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  • General Natural Sciences Ⅰ

    2023.10
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    2024.3

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    Level:Graduate (liberal arts)  Country:Japan

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  • Primary Seminar in Science

    2022.4
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    2023.9

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    Level:Undergraduate (liberal arts)  Country:Japan

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  • Advanced Differential Topology

    2021.10

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    Level:Postgraduate  Country:Japan

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  • The World of Mathematics

    2021.10
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    2023.3

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    Level:Undergraduate (liberal arts)  Country:Japan

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  • Seminar in Mathematics

    2021.4

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    Level:Undergraduate (specialized)  Country:Japan

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  • Differential Topology

    2020.4

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    Level:Postgraduate  Country:Japan

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  • Basic Practice on Mathematics a

    2020.4
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    2023.9

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    Level:Undergraduate (liberal arts)  Country:Japan

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  • Basic Practice on Mathematics b

    2020.4
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    2023.9

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Teaching Experience

  • 自然科学総論I

    2023
    Institution name:新潟大学

  • 理学基礎演習

    2022
    Institution name:新潟大学

  • 理学スタディ・スキルズ

    2022
    Institution name:新潟大学

  • 先端科学技術総論

    2022
    Institution name:新潟大学

  • 数学講究

    2021
    Institution name:新潟大学

  • 微分位相幾何学

    2021
    Institution name:新潟大学

  • 数学の世界

    2021
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    2022
    Institution name:新潟大学

  • 数学基礎演習a

    2020
    Institution name:新潟大学

  • 幾何学IIA

    2020
    Institution name:新潟大学

  • 幾何学IIB

    2020
    Institution name:新潟大学

  • 幾何学IA

    2020
    Institution name:新潟大学

  • 幾何学IB

    2020
    Institution name:新潟大学

  • 微分位相幾何学特論

    2020
    Institution name:新潟大学

  • 数学基礎演習b

    2020
    Institution name:新潟大学

  • 理学スタディ・スキルズ

    2020
    -
    2022
    Institution name:新潟大学

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