| Professor | Nobuaki Obata |
|---|---|
| Assistant Professor | Etsuo Segawa |
| M2 | Yumi Igarashi |
Probability theory, tracing back to the study of gambling, has developed along the contemporary mathematics into stochastic analysis (differential and integral calculus in infinitely many variables). While, algebraic and combinatorial structures in probability are motivating problems in quantum (noncommutative) probability. I am interested in exploring the origin of stochasticity in the noncommutative nature. My study is in principle driven by mathematical interests and is encouraged by communication with researchers in different research fields including life and social sciences.
Within the Kolmogovian probability theory a random variable isdefined as a measurable function on a probability measure space,so the random variables form a commutative *-algebra.The mean (expectation) of a random variable defines a positive linearfunction on the commutative *-algebra.Stochastic quantities of a random variable is obtained by means of the commutative *-algebra and the positive linear function.Observing this situation from an upper level, we obtain the concept of algebraic probability space, where the *-algebra is no longer assumed to be commutative.This paradigm is called ``quantum probability," tracing back to the probabilistic interpretation of quantum theory.Nowadays, quantum probability has developed purely mathematicallykeeping profound contact with many other disciplines.
Quantum probability allows various concepts of independence and sovarious central limit theorems.In this line, quantum stochastic processes are introduced and studied on the basis of different concepts of independence.Within quantum probability, a classical random variable is decomposedinto quatum components (quantum decomposition) andits noncommutative nature is revealed which is invisible in the classical scheme.Thereby combinatorial approaches and methods of orthogonal polynomials are introduced and work efficiently.
(references) Luigi Accardi and Nobuaki Obata: Foundation ofQuantum Probability, Makino Publ. 2003, in Japanese.Akihito Hora and Nobuaki Obata: Quantum Probability and Spectral Analysis of Graphs, Springer, 2007.
A mathematical object consisting of vertices and their adjacency relation is called a graph. A graph equipped with additional structures (e.g., orientation, edge weights, vertex weights) is generally referred to as a network. The ``complex network" is popular among many researchers in various fields of science and our study is motivated also by its interdisciplinary nature.
In the study of networks one of the key concepts is spectrum. Employing various methods developed along with quantum probability theory, we have investigated spectral structures of networks. Moreover, dynamics on networks (coupled oscillators, Markov chains, quantum walks, etc.) are studied by using spectral methods. We are also interested in asymptotic behaviors of spectra of growing and random graphs, and spectral reconstruction problems.
(references) Book in Japanese in preparation.
In the Ito theory a fundamental role is played by the stochastic integrals defined through Brownian motion. By means of white noise theory, the Ito integral is reformulated as a white noise integral with values in infinite dimensional distributions. Similarly in the quantum Ito theory, originally due to Hudson and Parthasarathy, a fundamental role is played by a quantum stochastic integrals defined through three basic quantum stochastic processes, annihilation, creation, and conservation processes. By means of quantum white noise theory, the quantum stochastic integrals are reformulated as white-noise-operator-valued integrals against quantum white noises. So quantum white noise is a key concept for unified approach to quantum stochastic processes.
The origin of dissipation in macro-scopic systems is understood through scaling limits (coarse graining) of micro-scopic equations of motion in statistical mechamics, or through central limit theorem for independent random variables in probability theory. Macro-scopic equations of motion thus obtained often become quantum stochastic differential equations, which is a major target of quantum stochastic analysis. In order to covering non-linear models, we hope to establish general theory of quantum white noise equations including higher powers of quantum white noises.
(references) Nobuaki Obata: White Noise Calculus and Fock Space, Lecture Notes in Mathematics Vol.1577, Springer, 1994. Un Cig Ji and Nobuaki Obata: Quantum white noise calculus and applications, in ``Real and Stochastic Analysis (Malempati M. Rao, Ed.)," Chapter 4, World Scientific, 2014.
We are enjoying active collaboration between mathematics and other fields of science and encouraging steady development of interdisciplinary researches. Since the foundation in September in 2007, the circle of our activities have expanded with the supports of many people. We organize workshops to promote encounters of researchers with different backgrounds.
http://www.dais.is.tohoku.ac.jp/~amf/
Probability and Mathematical Statistics
(Kazimierz Urbanik Center for Probability and Mathematical Statistics) Vol.31 (2011)-
http://www.math.uni.wroc.pl/~pms/index.php"
Complex Analysis and Operator Theory (CAOT) (Birkhauser) Vol.4 (2010)-
http://www.springer.com/birkhauser/mathematics/journal/11785"
Infinite Dimensional Analysis, Quantum Probability and Related Topics (IDAQP)(World Scientific) Vol.1 (1998)-
http://www.worldscinet.com/idaqp/idaqp.shtml
Stochastic Analysis and Applications (Taylor & Francis) Vol.20 (2002) -
http://www.tandf.co.uk/journals/titles/07362994.asp
Interdisciplinary Information Sciences(GSIS Tohoku University) Vol.12 (2006)- Editor; Vol.14 (2008)- Editor-in-Chief
http://www.is.tohoku.ac.jp/publication/IIS.html
I graduated from the Faculty of Science at Kyoto University in 1981 with BS in Natural Science (major in Mathematics), and from the Graduate School of Science in 1984 with MS in Mathematics under supervision of Professor Hisaaki Yoshizawa. There I studied unitary representation of the group of diffeomorphisms following the paper by Gelfand, Graev and Vershik.
After my Master degree I was employed as assistant professor in the Faculty of Science at Nagoya University in 1984. While working at Nagoya University I completed my Doctor Thesis (Ph. D) in 1989 under the supervision of Professor Takeyuki Hida. In 1991 I promoted to lecturer and, after reformation of the faculty in 1995, I served as associate professor in Graduate School of Mathematics at Nagoya University. In 2001 as professor I moved to Graduate School of Information Sciences at Tohoku University, where I am still lodged.
I spent two years in Mathematics Institute at University of Tuebingen during 1989-1991 working with Professor Herbert Heyer. My time in Germany was spent in visiting quite a few institutes in Europe, among others, encountering Professors von Waldenfels at Heidelberg, Meyer at Strasbourg, Hudson at Nottingham, Accardi at Roma motivated me to initiate the study of quantum white noise. The lecture notes "White Noise Calculus and Fock Space" (Springer, 1994) was prepared during my stay in Tuebingen and encouragement by Professor Heyer is gratefully acknowledged.
The newly reformed Graduate School of Mathematics at Nagoya University started accepting students with various back grounds beyond the usual mathematics. The collaboration with Professor Kenji Oosawa (Bio-Physics, currently in Gumma University) and Dr Tetsuhiko Yoshida (Chemistry, Toa Gosei Ltd.) was my first experience of interdisciplinary research. We took out a few patents and received L'Oreal Art and Science Promotion Prize in 1998.
From 2008 to 2019 I serve as the Editor-in-Chief of Interdisciplinary Information Sciences, an international journal published by Graduate School of Information Sciences at Tohoku University. I have served as a member of the editorial boards of some international journals as mentioned above.
I have written three books in English and five books in Japanese. I edited more than ten books or special issues in journals. I published more than 70 journal papers, 80 other papers (in proceedings and/or in Japanese), see Academic Publications
In Tohoku University I have been involved in quite a few administrative jobs. From 2014 to 2019 I served as vice-dean of the Graduate School and as the dean in 2020. Since 2022 up to now I am serving as special advisor to the President of Tohoku University for education of mathematics and data science.
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