2018 Fiscal Year Research-status Report
Combinatorics around Painleve VI
| Project/Area Number |
16K05057
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Keywords | Umemura polynomials / Cohomology theories / Flag varieties / Integrable systems |
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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
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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| Strategy for Future Research Activity |
I'm planning to continue the study of combinatorial and algebra-geometric
properties of the Umemura polynomials in connection with Schubert Calculi and Integrable Systems and noncommutative quadratic algebras. Natural elliptic version of the Umemura polynomials will be considered and studied.
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| Causes of Carryover |
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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