授業の目的 【日本語】
Goals of the Course(JPN) | | コンパクト群上のHaar測度は、左右不変な確率測度です。この測度は唯一無二です。その存在はもともと非構成的な方法で証明されました。しかし、この確率測度に対する期待値の計算が重要と見られれいます。
多項式の場合がWeingarten計算の目的です。 |
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授業の目的 【英語】
Goals of the Course | | | The Haar measure on a compact group is a probability measure that is left and right invariant. It is unique. Its existence was originally proved via non-constructive methods. However, it is important to be able to compute expectations with respect to this probability measure. In the case of polynomials, this is the goal of Weingarten calculus. |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | この講義では、専門知識がない高度な学部生でも理解できる形でWeingarten計算を説明します。参加者は、コンパクト行列群の任意のHaar測度に対する任意の多項式の積分を理論的に計算する方法を学びます。
その後、古典群の例に集中します。この場合、Weingarten計算の漸近性と応用について説明します。 |
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到達目標 【英語】
Objectives of the Course | | | We will describe the Weingarten calculus in a way that is achievable to advanced undergraduates students with no prior knowledge in the area. Successful participants will learn how to compute theoretically integrals of any polynomial with respect to any Haar measure of a compact matrix group. We will focus on the examples of classical groups. In this case, we will describe asymptotics and applications of Weingarten calculus. |
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授業の内容や構成
Course Content / Plan | | The plan is tentative and may very depending on the pace of the lecture and the interests of the audience.
Lecture 0: A short introduction to the Weingarten calculus and its recent developments. In this lecture, we will focus on ideas and history. Details and proofs will be given in subsequent lectures.
Lecture 1: Preliminaries. The Haar measure. Some elementary calculations of integrals against the Haar measure in finite dimension.
Lecture 2: The fundamental theorem of Weingarten calculus.
Lecture 3: The particular case of U_n, O_n. Elementary asymptotics.
Lecture 4: The Novak and the Collins-Matsumoto expansion. Advanced asymptoticss
Lecture 5: applications: asymptotic freeness of matrices (Voiculescu-type theorems) and asymptotic independence of entries (Borel-type theorems) |
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履修条件
Course Prerequisites | | 4年生対象科目。
この講義は英語で行います。
This course will be taught in English. |
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関連する科目
Related Courses | | |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | Submit a report. The topic will be given during the class. The mark is “pass" or “fail". |
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不可(F)と欠席(W)の基準
Criteria for "Fail (F)" & "Absent (W)" grades | | | To pass, you must attend at least five classes out of six (if the instructor is notified beforehand of an absence, negotiation on a case-by-case might be possible) . |
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参考書
Reference Book | | |
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教科書・テキスト
Textbook | | arXiv:2207.08418 Moment Methods on compact groups: Weingarten calculus and its applications
Benoit Collins Proceedings of the online ICM
arXiv:2109.14890 The Weingarten Calculus Benoit Collins, Sho Matsumoto, Jonathan Novak AMS |
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課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | |
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注意事項
Notice for Students | | |
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他学科聴講の可否
Propriety of Other department student's attendance | | |
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他学科聴講の条件
Conditions for Other department student's attendance | | | If there are any available seats. |
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レベル
Level | | |
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キーワード
Keyword | | |
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履修の際のアドバイス
Advice | | |
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授業開講形態等
Lecture format, etc. | | |
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遠隔授業（オンデマンド型）で行う場合の追加措置
Additional measures for remote class (on-demand class) | | |
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