2005 Fiscal Year Final Research Report Summary
Comprehensive Research of Hypergeometric and Painleve Systems
| Project/Area Number |
14204009
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Kobe University |
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Principal Investigator |
NOUMI Masatoshi Kobe University, Graduate School of Science and Technology, Professor, 大学院・自然科学研究科, 教授 (80164672)
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| Co-Investigator(Kenkyū-buntansha) |
OHTA Yasuhiro Kobe University, Graduate School of Science and Technology, Associate Professor, 大学院・自然科学研究科, 助教授 (10213745)
MASUDA Tetsu Kobe University, Graduate School of Science and Technology, Assistant, 大学院・自然科学研究科, 助手 (00335457)
TAKANO Kyoichi Kobe University, Faculty of Science, Professor, 理学部, 教授 (10011678)
SAITO Masahiko Kobe University, Faculty of Science, Professor, 理学部, 教授 (80183044)
YAMADA Yasuhiko Kobe University, Faculty of Science, Professor, 理学部, 教授 (00202383)
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| Project Period (FY) |
2002 – 2005
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| Keywords | Painleve equation / discrete Painleve equation / elliptic hypergeometric functions / monodromy manifold / affine Weyl group symmetry / special solutions of Riccati type |
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| Research Abstract |
1.Representation theoretic study of integrable systems and Painleve systems : (1) Construction of a new Lax pair associated with the affine Lie algebra <so>^^^^(8) for the Painleve equation P_<VI> which reveals a group-theoretic origin of the affine Wey1 group symmetry. (2) Determinant representation of special solutions of hypergeometric and algebraic type to the Painleve equation P_<VI>.
2.Geometric study of moduli spaces : (1) Geometric investigation of the monodromy manifold for the Painelve equation P_<VI>, and the Riemann-Hilbert correspondence : (2) Foundation of higher dimensional discrete systems based on the Gremona transformations and the birational Weyl group actions on the configuration spaces of points in projective spaces. (3) Study of moduli spaces of vector bundles and Fourir-Mukai transformations. Their applications to instanton counting.
3.Discrete integrable systems and discrete Painleve systems: (1) Introduction of higher order q-Painleve systems with affine Weyl group symmetry of type A. (2) Description of the elliptic Painleve equation of type E_8 in terms of γ-functions. Construction of special solutions of Riccati type, expressed in terms of elliptic hypergeoemetric functions. (3) Construction of q-hypergeometric solutions to the hierarchy of q-Painleve equations in Sakai's list. (4) Theoretic study of totally positive Weyl group actions and totally positive R-matrices, together with applications to combinatorics and solvable lattice modeles.
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