2018 Fiscal Year Final Research Report
Research on free probability and its applications to probability, combinatorics and representation theories
| Project/Area Number |
15K17549
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Basic analysis
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| Research Institution | Hokkaido University |
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Principal Investigator |
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| Research Collaborator |
Yoshinaga Masahiko
Miyatani Toshinori
Tsujie Shuhei
Ueda Yuki
Asai Nobuhiro
Sakuma Noriyoshi
Collins Benoit
Lehner Franz
Franz Uwe
Schleissinger Sebastian
Arizmendi Octavio
Huang Hao-Wei
Wang Jiun-Chau
Szpojankowski Kamil
Bozejko Marek
Ejsmont Wiktor
Simon Thomas
Wang Min
Thorbjornsen Steen
Skoufranis Paul
Gu Yinzheng
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Keywords | free probability / infinite divisibility / Levy processes / unimodal distributions / combinatorics / cumulants / Markov processes / stable distributions |
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| Outline of Final Research Achievements |
Free probability is a field strongly motivated by mathematics of quantum physics (functional analysis and operator algebras). It focuses on "non-commutative random variables". This theory is constructed in analogy with probability theory, and there are surprising correspondences with probability theory. This research project has investigated various aspects of free probability, including limit theorems in comparison with probability theory, unified theory of cumulants, and applications to asymptotic representation theory of symmetric groups. In addition to finding analogy with probability theory, this project created some feedback to probability theory by finding a new aspect of Markov processes and probability distributions.
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| Free Research Field |
確率論，関数解析学
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| Academic Significance and Societal Importance of the Research Achievements |
近年の数学においては様々な分野の相互関係が見つかっている．本研究でも分野横断的な側面が強く，確率論，関数解析，組合せ論，複素関数論などの分野を活用して研究が進み，逆にこういった分野への新たな視点を提供することもできた．特にランダム行列やマルコフ過程などのように応用範囲の広い分野に対しても新たな視点が得られた．これによって学術的な交流が活発になり，かつ将来的に成果が社会に還元されるためのポテンシャルとなった．また後継者に研究を伝えていくことで，将来を担う世代の教育にも貢献できる．
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