2019 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 – 2021-03-31
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| Keywords | Umemura polynomials / FK-algebras / Cohomology theories / Flag varieties / Integrable systems |
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| Outline of Annual Research Achievements |
I continue the study of combinatorial properties of certain polynomials which naturally appear in an investigation of some integrable systems such as Painleve V I, rational, trigonometric and elliptic Calogero-Moser type models and etc. I and S. Fomin discovered hidden algebraic structure behind these integrable systems (the so-called Fomin-Kirillov algebras(FK-algebras for short)) . The study algebraic and combinatorial properties of these algebras plays important role in the study of algebraic and combinatorial properties of the integrals of motion of the systems mentioned.The results of these investigations in the course of the preparation/writing books "Algebraic Bethe Ansatz"(jointly with R.Sakamoto), and "On some quadratic algebras".
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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 FK-algebras falls on the crossroad of many branches of Mathematics and Mathematical Physics such as Algebra,Geometry, Combinatorics, Special Functions,Low dimensional Topology,Integrable Systems and so on and so far.
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| Strategy for Future Research Activity |
I'm planning to continue my research concerning some quadratic algebras and related polynomials such as the Okamoto and Umemura ones, as well as the Schubert and Grtothendieck ones.
I'm planning to report on my results obtained in the body of the Project, during the Conference ECCOM 20, Bogota, Columbia, scheduled on June 15-26, 2021 instead of June 15-26, 2020, since coronavirus pandemics. By this reason I ask to extend my Kakenhi grant on the next Academic Year 2021.
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| Causes of Carryover |
My research of algebraic and combinatorial properties of the Okamoto, Umemura, Grothendfieck,.. polynomials touches a wide variety of fields in Mathematics such as Algebra, Algebraic Geometry and Combinatorics, Special Functions, Knot Theory, Integrable Systems and Probability Theory. To have further progress concerning the Project , I'm planning to visit (partly online) some leading specialists in the fields of Mathematics listed above in Japan, e.g. M.Noumi, Y.Yamada, S.Okada, M. Masuda, M. Yoshinaga, M. Yamazaki. M. Kapranov,..., as well as some oversea ones, including A. Borodin, M. Haiman, ....
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Research Products
(10 results)
