2016 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 cohomology / Flag varieties / Elliptic Functions / Integrable Systems |
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| Outline of Annual Research Achievements |
During the 2016/17 Academic Year , I explored further the study of some quadratic algebras with applications to quantum Schubert and Grothendieck Calculi, Special Functions, Combinatorics of Painleve VI, Graph Theory and Integrable Systems.
More specifically, I have proved some new results which relate certain representations of algebras I'm looking for, with (A) generalized cohomology rings of type A flag varieties and quantum Schubert and Grothendieck Calculi; (B) the Orlik--Terao and Postnikov-Shapiro algebras associated with a graph arrangement;
(C) the Ehrhart polynomials of certain convex integral polytopes (for example the Chan--Robbins-Yuen one); (D) new identities between higher genus Riemann theta functions; (E) give new explicit expressions for integrals of motions for (some) Calogero--Moser-Gaudin type models.
Part of my results are summarizes in my papers.
During the year 2016/17, Noumi studied elliptic hypergeometric integrals of Selberg type from the viewpoint of q-difference de Rham theory and elliptic Lagrange interpolation functions. In particular, Noumi obtained (in collaboration with Masahiko Ito) a new determinant formula associated with elliptic hypergeometric integrals of type BC_n.
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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
By the common opinion of the members of the Project, we have obtained good progress in the study of combinatorial aspects of Painleve VI, since we had opportunity to visit each other for joint face-to-face research.
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| Strategy for Future Research Activity |
1. (with M. Noumi) Study in more details connections of representation 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;
2. (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.
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| Causes of Carryover |
Since I'm citizen of Russia, but have Japanese residency and live in Kyoto, I had problems to obtain visa to visit countries, namely, USA, Switzerland and France.
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 2016/2017 Academic Year in full.
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| Expenditure Plan for Carryover Budget |
In this Academic Year I have plans to visit Russia, Moscow State University and HES, South Korea (KIAS), and Australia (University of Brisbane and Sydney), as well as visit and invite my Japanese co-authors and collaborators from Tokyo, Kobe, Nagoya, Fukuoka, Okayama,Kofu.
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[Presentation] On some quadratic algebras2016
Author(s)
A.N.Kirillov
Organizer
Lecture course "On some quadratic algebras"
Place of Presentation
The Stockholm University, Sweden
Year and Date
2016-10-13
Int'l Joint Research / Invited
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