2014 Fiscal Year Research-status Report
Random Matrix Theory and its applications
| Project/Area Number |
26800048
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| Research Institution | Kyoto University |
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Principal Investigator |
COLLINS Benoit 京都大学, 理学(系)研究科(研究院), 准教授 (20721418)
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| Project Period (FY) |
2014-04-01 – 2017-03-31
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| Keywords | Ramdom matrices / positive maps / Connes Problem / free quantum groups / free probability / MOE additivity / Weingarten formula |
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| Outline of Annual Research Achievements |
During last year, I completed a number of research projects.
(1) Together with Michael Brannan (who visited me in Kyoto) and Roland Vergnioux, we finished a paper where we proved that the Connes Embedding property holds for any free orthogonal group of parameter n>2. This paper is about to be accepted in Transactions of the AMS.
(2) Together with Ion Nechita (who visited me in Kyoto last spring) and Patrick Hayden, we completed a result about typical threshold for entanglement and PPT property for unbalanced states. We also discovered that random maps appropriately chosen can give examples of positive but indecomposable maps.
(3) In collaboration with Ping Zhong and Motohisa Fukuda (who both visited me in Kyoto), we obtained a simple and conceptual proof of the violation of the minimum output entropy (MOE) thanks to Bercovici and Voiculescu's notion of superconvergence. This resulted in a paper that was published in International Journal of Mathematics.
(4) Together with Todd Kemp and Antoine Dahlqvist we proved the strong convergence of matrix valued unitary Brownian motion towards the free unitary Brownian motion. This paper is also submitted.
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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
This year I completed many works on which I had been working for many years, so I would call my progress very smooth with respect to these projects. I did not complete some questions related to Weingarten calculus, but the research on this area is still on track too.
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| Strategy for Future Research Activity |
I still have many projects in collaboration
(1) an outstanding project with Marius Junge (on the classification of non-commutative Brownian motion). We need to write down some details and a paper. Some further consultation might be necessary.
(2) a project with Michael Brannan about the use of planar algebras and orthogonal quantum groups for quantum information purpose. We have achieved the basic results we were after, and we now need to work on the presentation, exposition and applications.
(3) I still have a project with Giordano and Al Nuwairan (to classify the extremal points of exchangeable separable sets). We already finished the equivariant part, and discovered that we can achieve the non equivariant part. It happens to have a relation to quantum groups, which we are exploring right now. We expect to have a paper rather soon.
(4) I will work with Cioppa on application of Weingarten calculus to the understanding of norms of operators. Buteau will also be included in this project.
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| Causes of Carryover |
I spent less money than I asked for for the first year, because some of my business travels were sponsored by other sources to an extent higher than I had anticipated (e.g. invitations covered my travel expenses whereas it had not been expected, etc)
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| Expenditure Plan for Carryover Budget |
I plan to use kakenhi according to my proposal, i.e. for the purpose of travelling to conferences, visiting coworkers, inviting coauthors working on a project with me, and buy some research furniture.
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| Remarks |
This page contains links to my publications, CV, research and academic activities.
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