2016 Fiscal Year Research-status Report
Random Matrix Theory and its applications（国際共同研究強化）
| Project/Area Number |
15KK0162
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Project Period (FY) |
2016 – 2018
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| Keywords | Random Matrices / Quantum information |
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| Outline of Annual Research Achievements |
During the last year, I completed the following projects:
(1) together with Michael Brannan (Texas A&M University), we solved a conjecture of Vaughan Jones about the dual of the Temperley-Lieb basis. We proved that all coefficients of the dual basis in the original basis are non-zero and gave an explicit formula for them. This gave rise to a preprint that is submitted.
(2) together with Brannan, we finalized the first part of a long run joint project on quantum groups and quantum information and produced the preprint "Highly entangled, non-random subspaces of tensor products from quantum groups". This paper is now submitted. We use the notion of rapid decay in the theory of quantum groups in order to obtain highly entangled quantum channels that mimic to some extent in a non random way the behaviour of random quantum channels.
(3) together with Novak and Sniady we produced the preprint "Semiclassical asymptotics of GL_N(C) tensor products and quantum random matrices" and submitted it. Our result considerably extends previous results b Sniady and myself, and also results by Bufetov and Gorin. We establish the optimal rate for asymptotic freeness of random variables arising from irreducible representations of GL_N(C).
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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
1- regarding research, the progress is quite smooth. We achieved important milestones on three projects as mentioned above, and we also completed four other projects mentioned in another report, that are also related to random techniques and therefore to this project.
2- regarding the organization of research, we are working on the scientific program of the IHP trimester that I am co-organizing. I visited a few times co-organizers and IHP in order to have scientific discussions on the upcoming program, and our progress is according to the schedule.
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| Strategy for Future Research Activity |
Most of my research plan of this project is related to two events that take place at Institut Henri Poincare (IHP) in Paris. The first one was about asymptotic representation theory and finished late march and I participated to it twice (once indicated above, and once more between late march and early april, together with a conference in Lyon -- the return in april explains why it is not counted this year).
The second program is a trimester that takes place this fall. I am co-organizing it and it is about the mathematics of quantum information theory. I will spend a semester in Paris from this late summer on in this context, to hold this program, collaborate on quantum information, operator algebras and random matrices with co-organizers and participants, and I will extend my stay to continue collaboration with colleagues (Nechita, Aubrun, Belinschi, Male).
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