PHOTO

Junya Takahashi
Section
Graduate School of Information Sciences
Job title
Assistant Professor
Degree
  • 博士(数理科学)(東京大学)

  • 修士(数理科学)(東京大学)

Professional Memberships 1

  • Mathematical Society of Japan

Research Interests 7

  • harmonic forms

  • L^2 Stokes theorem

  • elliptic boundary value problem

  • collapsing of Riemannian manifolds

  • differential forms

  • eigenvalues

  • Hodge-Laplacian

Research Areas 1

  • Natural sciences / Geometry / Differential Geometry

Papers 13

  1. Small eigenvalues of the rough and Hodge Laplacians under fixed volume Peer-reviewed

    Colette Anné, Junya Takahashi

    to appear in Ann. Fac. Sci Toulouse 2022

  2. L2-harmonic forms on incomplete riemannian manifolds with positive Ricci curvature Invited Peer-reviewed

    Junya Takahashi

    Mathematics 6 (5) 1-11 2018/05/09

    Publisher: MDPI AG

    DOI: 10.3390/math6050075  

    ISSN:2227-7390

    More details Close

    We construct an incomplete Riemannian manifold with positive Ricci curvature that has non-trivial L2-harmonic forms and on which the L2-Stokes theorem does not hold. Therefore, a Bochner-type vanishing theorem does not hold for incomplete Riemannian manifolds.

  3. Partial collapsing and the spectrum of the Hodge-de Rham operator Peer-reviewed

    Colette Anné, Junya Takahashi

    Analysis & PDE 8 (5) 1025-1050 2015

    Publisher: MATHEMATICAL SCIENCE PUBL

    DOI: 10.2140/apde.2015.8.1025  

    ISSN:1948-206X

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    Our goal is to calculate the limit spectrum of the Hodge-de Rham operator under the perturbation of collapsing one part of a manifold obtained by gluing together two manifolds with the same boundary. It appears to take place in the general problem of blowing up conical singularities as introduced by Mazzeo and Rowlett.

  4. p-spectrum and collapsing of connected sums Peer-reviewed

    Colette Anné, Junya Takahashi

    Transactions of the American Mathematical Society 364 (4) 1711-1735 2012/04

    Publisher: AMER MATHEMATICAL SOC

    DOI: 10.1090/S0002-9947-2011-05351-1  

    ISSN:0002-9947

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    The goal of the present paper is to calculate the limit of the spectrum of the Hodge-de Rham operator under the perturbation of collapse of one part of a connected sum. It takes place in the general problem of blowing up conical singularities introduced by R. Mazzeo and J. Rowlett.

  5. Collapsing to Riemannian manifolds with boundary and the convergence of the eigenvalues of the Laplacian Peer-reviewed

    Junya Takahashi

    Manuscripta Mathematica 121 (2) 191-200 2006/10

    Publisher: SPRINGER

    DOI: 10.1007/s00229-006-0035-5  

    ISSN:0025-2611

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    We prove that the eigenvalues of the Laplacian acting on functions converge to those of the limit manifold for a special collapsing family of closed Riemannian manifolds without curvature bounds. The proof uses L-2-analysis.

  6. The gap of the eigenvalues for p-forms and harmonic p-forms of constant length Peer-reviewed

    Junya Takahashi

    Journal of Geometry and Physics 54 (4) 476-484 2005/08

    Publisher: ELSEVIER SCIENCE BV

    DOI: 10.1016/j.geomphys.2004.11.007  

    ISSN:0393-0440

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    We study the kth positive eigenvalue lambda(k)((p)) (M, g) of the Laplacian on p-forms for a connected oriented closed Riemannian manifold (M, g). If all non-trivial harmonic p-forms on (M, g) have constant length, then it follows that lambda(k)((p))(M, g) <= lambda(k)((0))(M, g) for all k >= 1. (c) 2004 Elsevier B.V. All rights reserved.

  7. Vanishing of cohomology groups and large eigenvalues of the Laplacian on p-forms Peer-reviewed

    Junya Takahashi

    Mathematische Zeitschrift 250 (1) 43-57 2005/05

    Publisher: SPRINGER

    DOI: 10.1007/s00209-004-0735-z  

    ISSN:0025-5874

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    For collapsing of closed Riemannian manifolds, the first positive eigenvalue of the Laplacian on p-forms may or may not tend to infinity. In special cases, we show that the existence of the first positive eigenvalue tending to infinity is related to vanishing of cohomology groups of generic fibers.

  8. On the gap between the first eigenvalues of the Laplacian on functions and p-forms Peer-reviewed

    Junya Takahashi

    Annals of Global Analysis and Geometry 23 (1) 13-27 2003/03

    DOI: 10.1023/A:1021294732338  

    ISSN:0232-704X

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    We study the first positive eigenvalue λ 1(p) (g) of the Laplacian on p-forms for a connected oriented closed Riemannian manifold (M, g) of dimension m. We show that for 2 ≤ p ≤ m - 2 a connected oriented closed manifold M admits three metrics gi (i = 1, 2, 3) such that λ1(p) (g1) &gt λ 1(0) (g1), λ1(p) (g2) &lt λ1(0) (g2) and λ1(p) (g3) = λ1(0) (g3). Furthermore, if (M, g) admits a nontrivial parallel p-form, then λ1(p) ≤ λ1(0) always holds.

  9. Collapsing of connected sums and the eigenvalues of the Laplacian Peer-reviewed

    Junya Takahashi

    Journal of Geometry and Physics 40 (3-4) 201-208 2002/01

    Publisher: ELSEVIER SCIENCE BV

    ISSN:0393-0440

    More details Close

    We study the behavior of the eigenvalues of the Laplacian acting on functions when one side of a connected sum of two closed Riemannian manifolds collapses to a point. We prove that the eigenvalues converge to those of the limit space, by using the method of Anne and Colbois. From this, we obtain a gluing theorem for the eigenvalues. (C) 2002 Elsevier Science B.V. All rights reserved.

  10. Small eigenvalues on p-forms for collapsings of the even-dimensional spheres Peer-reviewed

    Junya Takahashi

    Manuscripta Mathematica 109 (1) 63-71 2002

    DOI: 10.1007/s002290200288  

    ISSN:0025-2611

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    We prove that for some p there exist small eigenvalues of the Laplacian on p-forms for collapsings of the even dimensional spheres with curvature bounded below. These collapsings were constructed by T. Yamaguchi.

  11. On the gap between the first eigenvalues of the Laplacian on functions and 1-forms Peer-reviewed

    Junya Takahashi

    Journal of the Mathematical Society of Japan 53 (2) 307-320 2001/04

    Publisher: MATH SOC JAPAN

    ISSN:0025-5645

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    We study the first positive eigenvalue lambda ((p))(1) of the Laplacian on p-forms for oriented closed Riemannian manifolds. It is known that the inequality lambda ((1))(1) less than or equal to lambda ((0))(1) holds in general. m the present paper, a Riemannian manifold is said to have the gap if the strict inequality lambda ((1))(1) < lambda ((0))(1) holds. We show that any oriented closed manifold M with the first Betti number b(1)(M)= 0 whose dimension is bigger than two, admits two Riemannian metrics, the one with the gap and the other without the gap.

  12. Upper bounds for the eigenvalues of the Laplacian on forms on certain Riemannian manifolds Peer-reviewed

    Junya Takahashi

    Journal of the Mathematical Sciences, the University of Tokyo 6 (1) 87-99 1999/01

    Publisher: The University of Tokyo

    ISSN:1340-5705

  13. The first eigenvalue of the Laplacian on p-forms and metric deformations Peer-reviewed

    Junya Takahashi

    Journal of the Mathematical Sciences, the University of Tokyo 5 (2) 333-344 1998/04/01

    Publisher: The University of Tokyo

    ISSN:1340-5705

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Presentations 47

  1. Small eigenvalues of the Hodge-Laplacian with sectional curvature bouded below Invited

    Junya Takahashi

    2024/02/21

  2. Small eigenvalues of the Hodge-Laplacian with sectional curvature bouded below Invited

    Junya Takahashi

    Global Analysis and Geometry 2023 at Osaka, Osaka Metropolitan Univ. 2023/10/21

  3. Small eigenvalues of the Hodge-Laplacian with sectional curvature bouded below

    Junya Takahashi, Colette Anné

    2023/09/22

  4. Small eigenvalues of the rough and Hodge Laplacians under fixed volume Invited

    Junya Takahashi

    Geometry and Analysis Seminar -- Mini Workshop 2023, Tohoku Univ. 2023/01/25

  5. Small eigenvalues of the rough and Hodge Laplacians under fixed volume Invited

    Junya Takahashi

    2022/06/21

  6. Small eigenvalues of the rough and Hodge Laplacians under fixed volume

    Junya Takahashi, Colette Anné

    2022/03/31

  7. Partial collapsing and the spectrum of the Hodge-Laplacian International-coauthorship Invited

    Junya Takahashi

    2016/11/19

  8. Partial collapsing and the spectrum of the Hodge-Laplacian Invited

    Junya Takahashi

    2014/03/07

  9. Partial collapsing and the spectrum of the Hodge-Laplacian Invited

    Junya Takahashi

    2013/10/22

  10. 一般化された連結和の崩壊と Hodge-Laplacian の固有値の収束 Invited

    Junya Takahashi

    2013/08/25

  11. Partial collapsing and the spectrum of the Hodge-de Rham operator Invited

    Junya Takahashi

    Geometry and Probability, Kyoto University 2013/08/08

  12. Collapsing to Riemannian manifolds to the intervals and the convergence of the eigenvalues of the Laplacian Invited

    Junya Takahashi

    2006/04/18

  13. Vanishing of cohomology groups and large eigenvalues of the Laplacian on p-forms Invited

    Junya Takahashi

    International Workshop on Differential Geometry, Saga University 2005/12/24

  14. Riemann 多様体の崩壊と微分形式の固有値 Invited

    高橋淳也

    広島幾何学研究集会, 広島大学 2005/10/07

  15. Collapsing, surgery and the first eigenvalues of the Laplacian on $p$-forms Invited

    Junya Takahashi

    Geometry and Probability, Tohoku University 2005/09/06

  16. Riemann 多様体の崩壊と p-form の大きい固有値 Invited

    2005/08/20

  17. Collapsing of Riemannian manifolds and large eigenvalues of the Laplacian on $p$-forms Invited

    Junya Takahashi

    Kunitachi One-Day Symposium on Geometric Analysis, 一橋大学 2005/07/09

  18. Vanishing of cohomology groups and large eigenvalues of the Laplacian on p-forms in collapsing International-presentation

    Junya Takahashi

    Séminaire de Géométrie, Université de Nantes, France 2005/05/27

  19. Vanishing of cohomology groups and large eigenvalues of the Laplacian on $p$-forms in collapsing International-presentation Invited

    Junya Takahashi

    Analytic aspects of problem in Riemannian geometry, l'Aber Wrac'h, France 2005/05/09

  20. The gap between the first eigenvalues of the Laplacian on function and p-forms International-presentation

    Junya Takahashi

    Séminaire de Géométrie, Université de Tours, France 2004/09/10

  21. The gap between the first eigenvalues of the Laplacian on function and p-forms International-presentation

    Junya Takahashi

    Séminaire de théorie spectrale et géométrie, Université de Neuchâtel, Suisse 2004/08/24

  22. Vanishing of cohomology groups and large eigenvalues of the Laplacian on p-forms International-presentation

    Junya Takahashi

    Séminaire de théorie spectrale et géométrie, Université de Neuchâtel, Suisse 2004/08/20

  23. Small eigenvalues on p-forms of collapsing of the even dimensional spheres International-presentation International-coauthorship

    Junya Takahashi

    Séminaire de théorie spectrale et géométrie, Université de Neuchâtel, Suisse 2004/08/18

  24. Vanishing of cohomology groups and large eigenvalues of the Laplacian on $p$-forms Invited

    Junya Takahashi

    2004/06/29

  25. 崩壊における p-form の大きい固有値とコホモロジー群の消滅 Invited

    高橋淳也

    微分幾何学火曜セミナー, 筑波大学 2004/06/08

  26. Vanishing of cohomology groups and large eigenvalues of the Laplacian on $p$-forms Invited

    Junya Takahashi

    Spectral Geometry, Asymptotic Analysis and Related Topics, Keio University 2004/02/03

  27. Riemann 多様体の崩壊と微分形式の Laplacian の固有値 Invited

    高橋淳也

    談話会, 東北大学理学部 2003/12/22

  28. コホモロジー群の消滅と $p$-form の大きい固有値 Invited

    高橋淳也

    幾何学セミナー, 九州大学 2003/11/28

  29. コホモロジー群の消滅と $p$-form の大きい固有値

    高橋 淳也

    日本数学会幾何学分科会 千葉大学 2003/09/26

  30. Vanishing of cohomology groups and large eigenvalues of the Laplacian on $p$-forms

    Junya Takahashi

    Probability and Geometric Analysis, Yokohama City University 2003/09/17

  31. 球面の崩壊と $p$-form の Laplacian の固有値 Invited

    高橋淳也

    談話会, 立教大学 2003/06/16

  32. 球面の崩壊と $p$-form の Laplacian の固有値 Invited

    高橋 淳也

    情報数理談話会, 東北大学情報科学研究科 2003/06/10

  33. Riemann 多様体の崩壊と微分形式の Laplacian の第 1 固有値 Invited

    高橋 淳也

    大岡山談話会,東京工業大学 2003/01/06

  34. 球面の崩壊における小さい固有値と大きい固有値 Invited

    高橋淳也

    微分幾何学, 九重研究集会 2002/09/14

  35. 偶数次元球面の $p$-form の小さい固有値と gap 問題 Invited

    高橋 淳也

    第49回幾何学シンポジウム,大阪大学 2002/07/28

  36. Small eigenvalues on $p$-forms on the even dimensional spheres and the gap problem International-presentation

    Junya Takahashi

    Second Russian-German Geometry Meeting dedicated to 90-anniversary of A.D. Alexandrov, Euler International Mathematical Institute, Sankt Petersburg, Russia 2002/06/17

  37. 多様体の崩壊と微分形式に作用する Laplacian の第 $1$ 固有値 Invited

    高橋淳也

    幾何学セミナー, 東北大学 2002/05/07

  38. 偶数次元球面の崩壊における $p$-forms の小さい固有値

    高橋淳也

    日本数学会幾何学分科会, 明治大学 2002/03/28

  39. Riemann 多様体の崩壊と微分形式に作用する Laplacian の固有値

    高橋淳也

    トポロジーセミナー, 東京大学 2002/01/22

  40. 偶数次元球面の崩壊における $p$-form の小さい固有値 Invited

    高橋淳也

    多様体上の微分方程式, 金沢大学 2001/12/13

  41. Small eigenvalues on $p$-forms for collapsing of the even dimensional spheres

    Junya Takahashi

    Geometry and Probability, Tohoku University 2001/11/21

  42. 連結和の崩壊における Laplacian の固有値の収束とその応用 Invited

    高橋淳也

    微分幾何学火曜セミナー, 筑波大学 2001/10/23

  43. Collapsing of connected sums and the eigenvalues of the Laplacian

    2001/10/04

  44. 連結和の崩壊と Laplacian の固有値の収束 Invited

    高橋 淳也

    第48回幾何学シンポジウム, 茨城大学 2001/08/28

  45. 連結和の崩壊と Laplacian の固有値の収束 Invited

    高橋淳也

    幾何学と物理学セミナー, 早稲田大学 2001/06/01

  46. 連結和の崩壊と Laplacian の固有値の収束 Invited

    高橋淳也

    Lie 群と幾何セミナー, 上智大学 2001/05/16

  47. 連結和の崩壊と Laplacian の固有値の収束 Invited

    高橋淳也

    微分幾何セミナー, 東京都立大学 2001/04/20

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Research Projects 7

  1. Analysis of elliptic operators and its applications to Geometric Function Theory

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (B)

    Institution: Tohoku University

    2017/04/01 - 2022/03/31

  2. 特異点の幾何とホッジ・ラプラシアンの固有値の研究

    高橋 淳也

    System: 基盤研究(C)

    Institution: 東北大学

    2016/04 - 2021/03

  3. 多様体の崩壊に対するホッジ・ラプラシアンの固有値の極限の研究

    高橋 淳也

    System: 若手研究(B)

    Institution: 東北大学大学院情報科学研究科

    2012/04 - 2016/03

  4. Convergence theory of metric measure spaces and its development

    KASUE Atsushi, URAKAWA Hajime, NAYATANI Shin, KATO Shin, KUMURA Hironori, TAKAHASHI Junnya

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (A)

    Institution: Kanazawa University

    2007 - 2009

  5. 微分形式における小さい固有値の位相的性質の研究

    高橋 淳也

    System: 若手研究(B)

    Institution: 東北大学大学院情報科学研究科

    2007/04 - 2008/03

  6. リーマン多様体の崩壊と微分形式のラプラシアンの固有値の研究

    高橋 淳也

    System: 若手研究(B)

    Category: 若手研究(B)

    Institution: 東北大学大学院情報科学研究科

    2004/04 - 2007/03

  7. リーマン多様体の崩壊における微分形式のラプラシアンの固有値の研究

    高橋 淳也

    System: 特別研究員奨励費

    Category: 特別研究員奨励費

    Institution: 東北大学理学研究科学振特別研究員

    2002/04 - 2003/03

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