- name: Junya Takahashi
- title: assistant professor

Division of Mathematics, Graduate School of Information Sciences, Tohoku University

- e-mail: t-junya [at] tohoku.ac.jp (please replace "at" with @)
- postal address: Division of Mathematics, Graduate School of Information Sciences, Tôhoku University, 6-3-09 Aoba, Sendai 980-8579, Japan.

Now, when a Riemannian manifold collapses or degenerates, how is its spectrum influenced ?

I am studying the behavior and the limit of the spectrum under such deformations, and also the relationship between manifolds and differential operators.

This research field is very interesting and has a lot of interaction with differential geometry, topology, singularity theory, and analysis of partial differential equations, differential operators and elliptic boundary value problems.

- Spectral Geometry (the eigenvalues of the Hodge-Laplacian acting on
*p*-forms) - Spectrum and Collapsing of Riemannian manifolds (small and large eigenvalues)
- Geometry, topology and analysis of differential forms (L^2-Stokes theorem, L^2-harmonic forms and L^2-cohomology)
- Analysis on manifolds and singular spaces (elliptic boundary value problems, resolution of singularities)

- géométrie spectrale (les valeurs propres du laplacien agissant sur les
*p*-formes différentielles) - spectre et effondrements de variétés riemanniennes (des petites et des grands valeurs propres quand les variétés s'effondre)
- géométrie, topologie et analyse de formes différentielles (théorème de L^2-Stokes et des formes harmoniques L^2)
- analyse sur des variétés et sur des espaces singularitiés (problèmes aux limites elliptiques, resolution de singularités)

- L^2-harmonic forms on incomplete Riemannian manifolds with positive Ricci curvature,
pdf ,

Mathematics 6 (5), 75 (2018); doi: 10.3390/math6050075.

- with Colette Anné,
Partial collapsing and the spectrum of the Hodge-de Rham operator,
pdf ,

Analysis & PDE. 8 (2015), 1025-1050.

arXiv:1007.2949[math.DG], hal-00503230, v2 [HAL]

- with Colette Anné,
*p*-spectrum and collapsing of connected sums, pdf ,

Trans. Amer. Math. Soc. 364 (2012), 1711-1735. - Collapsing to Riemannian manifolds with boundary and the convergence of
the eigenvalues of the Laplacian,

Manuscripta Math. 121 (2006), 191-200. - The gap of the eigenvalues for
*p*-forms and harmonic p-forms of constant length,

J. Geom. Phys. 54 (2005), 476-484. - Vanishing of cohomology groups and large eigenvalues of the Laplacian on p-forms,

Math. Zeit. 250 (2005), 43-57. - On the gap between the first eigenvalues of the Laplacian on functions and p-forms,

Ann. Global Anal. Geom. 23 (2003), 13-27. - Small eigenvalues on p-forms for collapsings of the even-dimensional spheres,

Manuscripta Math. 109 (2002), 63-71. - Collapsing of connected sums and the eigenvalues of the Laplacian,

J. Geom. Phys. 40 (2002), 201-208. - On the gap between the first eigenvalues of the Laplacian on functions and 1-forms,
pdf ,

J. Math. Soc. Japan 53 (2001), 307-320. - Upper bounds for the eigenvalues of the Laplacian on forms on certain Riemannian manifolds,

J. Math. Sci. Univ. Tokyo 6 (1999), 87-99. - The first eigenvalue of the Laplacian on p-forms and metric deformations,

J. Math. Sci. Univ. Tokyo 5 (1998), 333-344.

- Second Russian-German Geometry Meeting dedicated to 90-anniversary of
A.D. Alexandrov,

Euler Institute, Sankt Petersburg, Russia, June 2002. - Univ. de Neuchâtel, Switzerland, August-September 2004.
- Univ. de Tours, France, September 2004.
- Analytic aspects of problem in Riemannian geometry, L'Aber-Wrac'h, Brest, France, mai 2005.
- Univ. de Nantes, France, mai 2005.
- École Polytechnique, juin 2005.

- Division of Math.
Graduate School of Information Sciences (GSIS), Tohoku University

- Graduate School of Information Sciences,
Tohoku University

- Geometry and Analysis Seminar (GAS)
at GSIS

- Mathematical
Institute, Tohoku University

last update: 9 May 2018

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