| 研究実績の概要 |
I continue the study of combinatorial properties of two parameters Umemura polynomials Un,m(t) associated with some special solutions to PV I equations. I'm looking for cohomological interpretation of polynomials Un,m(t), namely, I expect that Un,m(t) =?Hilb(Vn,m,t) for some graph variety Vn,m.In this direction I computed generalized cohomology theory rings of varieties associated with multipartite and Hessenberg graphs, i.e. partial flag and Hessenberg varieties.Some comnbinatorial results in this connection are published in ASIM vol. 76 (2018), pp. 303-346,and in preparation.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
It is really very surprising/mysterious fact that some special appropriately normalized solutions to some integrable systems such as Painleve, Toda,Airy, KdV,etc, are formal power series/polynomials with positive integer coefficients. One explanation of this fact comes from an identifications of these positive integer numbers with intersections numbers (aka GW-invariants) of certain Calabi-Yau varieties. However, varieties which naturally appear in Schubert Calculi (and Okamoto and Umemura polynomials) are not Calabi- Yau in general, but Fano ones. I want to apply the technical developed in-depth in Mathematical Phys ice to Fano's graph varieties/
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| 次年度使用額が生じた理由 |
My research concerning properties of the Okamoto and Umemura polynomials touches many many fields of Mathematics, Including Analysis, Algebra, Geometry,
Special Functions, Integrable Systems and Combinatorics. To have further progress
concerning Project, I'm planning to visit several leading specialist in that fields of Mathematics in Japan, e.g., M.Noumi Y,Yamada,M.Masuda, H.Terao,S.Okada,H.Naruse,M.Yoshinaga, as well as some oversea Professors which will come to Japan.
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