日本語
College of Science and Engineering
/
Department of Mathematical Sciences
(Male)
DAISUKE
TARAMA
Associate Professor
■Graduate school/University/other
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03/2007 Kyoto University Faculty of Engineering Undergraduate School of Engineering Science Graduated
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03/2009 Kyoto University Graduate School of Informatics Department of Applied Mathematics and Physics Master's course Completed
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03/2012 Kyoto University Graduate School of Informatics Department of Applied Mathematics and Physics Doctoral course Completed
■Academic degrees
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Doctor of Informatics (03/2012 Kyoto University)
■Career history
04/01/2012-09/30/2012 Graduate School of Informatics, Kyoto University/Research Fellow
10/01/2012-03/31/2013 Laboratoire Jacques-Louis Lions, UMR 7598 CNRS-Paris 6, Université Pierre et Marie Curie/Chercheur Post-doctorant
04/01/2013-03/31/2016 Department of Mathematics, Kyoto University/JSPS Research Fellow (PD)
04/01/2016-03/31/2018 Department of Mathematical Sciences, Ritsumeikan University/Assistant Professor
04/01/2018- Department of Mathematical Sciences, Ritsumeikan University/Associate Professor
■Academic society memberships
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Mathematical Society of Japan
■Subject of research
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Geometric studies on integrable systems
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Dynamical properties of integrable systems
■Research summary
Theory of integrable systems and geometry
The research interests of Daisuke Tarama focus on geometric dynamical systems theory of finite-dimensional completely integrable Hamiltonian systems appearing from analytical mechanics and on related geometries, such as symplectic geometry, Poisson geometry, and algebraic and complex analytic geometry.
So far, he has mainly studied mathematics related to free rigid body dynamics describing the motion of rigid body under no external force, as well as their mathematical extensions.
The free rigid body dynamics is formulated as a Hamiltonian system on the cotangent bundle over the rotation group of dimension three, whose integrability and the stability of the equilibria are well known since long time ago. Further, the free rigid body dynamics has been extended to completely integrable systems on more general Lie groups from 1970’s.
1. Viewpoint of algebraic and complex analytic geometry:
The integrability of the free rigid body dynamics is characterized by the Lax equation and the elliptic curves appearing as spectral curve and the integral curve. In the previous studies, he has studied the geometry and the monodromy of the elliptic fibrations obtained by varying the natural parameters for the free rigid body dynamics in relation to the bifurcation phenomena and the Birkhoff normal forms around the equilibria. He has also studied the relation between the eigenvector mapping for the Lax equation and some Kummer surfaces.
2. Viewpoint of dynamical systems theory:
The integrability of the free rigid body dynamics on semi-simple Lie groups has been studied well since 1970’s. However, the stability of its equilibria has been analysed only from 2000’s. He is analysing such stability properties using the techniques in Lie algebra theory and Poisson geometry.
Besides, Daisuke Tarama deals with the spectral problem of the differential operators for the quantum systems corresponding to the generalized free rigid body dynamics using the representation theory. He also treats the construction of integrable systems or, more generally, Lagrangian fibrations based on the techniques of complex analytic geometry, such as the theory of elliptic surfaces.
He will push forward to researches on the interplay between dynamical systems, including integrable systems, and geometry.
■Research keywords
Theory of Integrable Systems, Geometric Mechanics, Dynamical Systems Theory, Algebraic and Complex Analytic Geometry
■Research activities (Even top three results are displayed. In View details, all results for public presentation are displayed.)
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Papers
On the complete integrability of the geodesic flow of pseudo-H-type Lie groups Wolfram Bauer, Daisuke Tarama Analysis and Mathematical Physics Online First, 28 pages 10/2018 1664-2368
10.1007/s13324-018-0250-8
Analytic extension of the Birkhoff normal forms for the free rigid body dynamics on SO(3) J.-P. Françoise, D. Tarama Nonlinearity 28/ 5, 1194-1216 2015
The U(n) free rigid body: Integrability and stability analysis for the equilibria T. S. Ratiu, D. Tarama Journal of Differential Equations 259/ 12, 7284-7331 2015
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Grants-in-Aid for Scientific Research (KAKENHI)
Link to Grants-in-Aid for Scientific Research -KAKENHI-
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Competitive grants, etc. (exc. KAKENHI)
Geometric and Analytic Study on Integrable Systems Research in Paris 2012 10/2012 03/2013 Main representative
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■Teaching experience (Even top three results are displayed. In View details, all results for public presentation are displayed.)
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Courses taught
2017 Thesis Seminar
2017 Geometry Ⅱ Lecture
2017 Introductory and Geometry Ⅱ Lecture
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■Message from researcher
Research area enriched by geometry and physics
The free rigid body dynamics, a typical example of solvable systems in analytical mechanics, is related with many areas in mathematics.
To formulate the free rigid body dynamics, techniques in differential geometry are used, such as the theory of Lie group and Lie algebra, symplectic geometry, and Poisson geometry. Since the integral curve and the spectral curve of the Lax equation are elliptic curves, there is also a relation to algebraic and complex analytic geometry. Moreover, the corresponding quantum mechanics requires the methods in representation theory, PDE theory, and functional analysis.
The researches on integrable systems connect many different areas of mathematics, particularly of geometry, in a systematic way. I hope to contribute to understanding better this fascinating story woven by geometry and physics.
■Research keywords(on a multiple-choice system)
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Geometry
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Basic analysis