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現在位置: ホーム ja シラバス(2020年度) 理学研究科 数学・数理解析専攻 解析学特論F

解析学特論F

JA | EN

科目ナンバリング
  • G-SCI11 90288 LJ55
開講年度・開講期 2020・前期
単位数 2 単位
配当学年 修士
対象学生 大学院生
使用言語 日本語
教員
  • COLLINS Benoit Vincent Pierre(理学研究科 准教授)
授業の概要・目的 The purpose of the course is to study some mainstream topics of Random Matrix theory. This field was initiated in statistics and mathematical physics; nowadays it uses various tools of mathematics, including analysis, probability, combinatorics and functional analysis. The student who successfully takes this class will become familiar with recent results in Random Matrix theory.
到達目標 The first goal of the course is to become familiar with the fundamental results of Random Matrix Theory (Wigner's semi-circle theorem, Marchenko-Pastur's theorems, etc) and with the tools needed by this theory (Stieltjes transform, moment methods, concentration of measure, etc).
The second goal is to become aware of the many research trends in Random Matrix Theory.
授業計画と内容 We expect to cover the following topics:
1- preliminary analysis tools: concentration of measure, matrix and eigenvalues inequalities, moment methods, Lindeberg method (3-4 classes)
2- operator norm of random matrices and semicircular law; Wigner's theorem (3-4 classes)
3- multimatrix models and free probability (3-4 classes)
4- Gaussian ensembles, extremal singular values and determinant point processes; other topics (3-4 classes)
The total amount of classes is 15, including feedback.

The instructor expects to write in English on the blackboard and teach in Japanese (unless there is a need -- e. g. foreign student -- and unanimous preference for the class to be taught in English).
履修要件 A familiarity with matrix theory, advanced linear algebra, and measure theory is necessary.
Basic knowledge in probability theory is preferable, although not absolutely necessary (cf preparation section).
授業外学習(予習・復習)等 In order to prepare oneself to the class, it is desirable to read in advance sections 1.1 and 1.2 of Tao's book. Students are expected to devote a fair amount of work outside the class to fill in details of some proofs, and prepare reports on side topics.
参考書等
  • Topics in Random Matrix Theory, Terence Tao, (AMS, GSM 132),
  • An Introduction to Random Matrices, Anderson, Guionnet, Zeitouni, (Cambridge),
  • Free Probability and Random Matrices, Mingo, Speicher, (Springer),