2017 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 | Quadratic algebras / Quantum K-theory / Integrable Systems / Painleve & Bethe Ansatz |
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| Outline of Annual Research Achievements |
Duering 2017 Academic Year I continue study combinatorial and algebraic properties of some quadratic algebras with applications to classical and quantum Schubert and Grothendieck Calculi, noncommutative deformations of Orlik-Terao and Zonotopal algebras (partly with G. Nenashev); representation theory of quadratic algebras in question to a[[lications to the theory elliptic hypergeometric functions and Integrable Systems and combinatorics of classical and elliptic Painleve IV. Exposition of aforementioned results will be summarize in my book "On some quadratic Algebras"( work in progress).
Jointly with R. Sakamoto we continue work on our book "Bethe Ansatz and Combinatorics". We already prepare the main body of book, but need to include a big amount of auxiliary material.
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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
We have progress with construction of representations of the quadratic algebras in questions, with applications to Combinatorics of Schubert and Grothendieck Calculi, Graphical Arrangements, and Elliptic Calculus applications to tegrable Systems, mainly in venue Painleve VI.
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| Strategy for Future Research Activity |
a. (with M. Noumi) To continue study in more details connections of rpresentation theory of quadratic algebras in question with the theory of elliptic hypergeometric functions, as well as connections of the latter with solutions to Painleve VI. To construct a quantum version of RSK;
b. (with T. Maeno and H. Naruse) To continue the study of algebraic and combinatorial properties of quadratic algebras in question with application to the quantum Schubert Calculus for flag varieties of classical types.
c. (with R. Sakamoto) To continue work and preparation for publication of our book (temporary title) "Bethe Ansatz and Combinatorics".
d. Preparation for publication of my book "On some quadratic Algebras".
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| Causes of Carryover |
Since I'm citizen of Russia, but have Japanese residency and live in Kyoto, I have problems to obtain visa and visit some countries, namely, SA,Switzerland, France and Australia. By this reasons I have cancelled my invited invitations to some Universities of that countries, and don't have enough time to use kakenhi fund on 2017/2018 Academic Year in full.
In this Academic Year I have plans to visit Russia, Moscow State University and HES, South Korea (KIAS), and hopefully Australia (University of risbane ,as
well as visit and invite my Japanese co-authors and collaborators from Tokyo, Kobe, Nagoya, Okayama, and invite my collaborators, M.Noumi, T.Maeno, Sakamoto, K. Hikami among others to Kyoto for joint research.
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