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課程名稱 |
測度與機率模型
Measure and Probabilistic models |
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開課學期 |
109-1 |
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授課對象 |
理學院 數學系 |
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授課教師 |
黃建豪 |
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課號 |
MATH4011 |
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課程識別碼 |
201 49860 |
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班次 |
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學分 |
3.0 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期一4(11:20~12:10)星期三3,4(10:20~12:10) |
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上課地點 |
天數304天數304 |
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備註 |
總人數上限：40人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1091MATH4011_ |
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課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
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課程大綱
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課程概述 |
The goal of this course is to introduce problems arising from mathematical statistical physics, especially, Curie-Weiss model and random matrix theory. The first half of the course will discuss basic measure theory, integration, functional analysis and measure-theoretic probability. In the second half of the course, we will turn to methods of moments, large deviations and concentration inequalities. Students will see that how these tools apply to Curie-Weiss model and random matrix theory. Concepts of special random sequences such as exchangeable sequences show up sometimes. |
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課程目標 |
Providing enough backgrounds for undergraduate students who are interested in probability theory. Students will learn how modern probability theory interacts with statistical physics. The first correlated system beyond the concept of ‘i.i.d.’ is the exchangeability. In physics, there is no obvious reason to differ one particle to another. De finnetti’s theorem tells that exchangeable sequences are mixtures of i.i.d. sequences. An approach to analyze this problem is based on moments of random variables. Other correlated systems are random matrices and point processes. They could be used to understand real data. |
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課程要求 |
基本分析與機率 |
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
Leadbetter, A Basic Course in Measure and Probability_ Theory for Applications (2014)
Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). An introduction to random matrices |
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參考書目 |
https://en.wikipedia.org/wiki/Method_of_moments_(probability_theory)
https://en.wikipedia.org/wiki/De_Finetti%27s_theorem
https://en.wikipedia.org/wiki/Large_deviations_theory
https://en.wikipedia.org/wiki/Wigner_semicircle_distribution |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
Exams |
70% |
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2. |
Problem sets |
20% |
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3. |
Class participation |
10% |
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週次 |
日期 |
單元主題 |
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第1週 |
9/14,9/16 |
Introduction and measure theory; 第三周開始每周一小時影片課程, 每周第三堂改成Office hour; 上半學期為拓樸與泛函四堂課, Large deviations 兩堂課. 下半學期影片為Large deviations 三堂課, concentration inequalities二堂課; |
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第2週 |
9/21,9/23 |
Measure space and measurable functions |
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第3週 |
9/28,9/30 |
Integral // Integral // OH |
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第4週 |
10/05,10/07 |
Convergence theorem // Convergence theorem // OH |
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第5週 |
10/12,10/14 |
L^p spaces // Urysohn's lemma // OH |
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第6週 |
10/19,10/21 |
Riesz representation // Convergence of measures // OH |
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第7週 |
10/26,10/28 |
Tightness // // OH |
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第8週 |
11/02,11/04 |
Moments and weak convergence // Concept of measure-theoretic probability // OH |
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第9週 |
11/09,11/11 |
OH // Midterm Exam // Midterm Exam |
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第10週 |
11/16,11/18 |
de Finetti Theorem // // OH |
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第11週 |
11/23,11/25 |
Curie-Weiss model // // OH |
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第12週 |
11/30,12/02 |
// // OH |
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第13週 |
12/07,12/09 |
Random matrix theory // // OH |
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第14週 |
12/14,12/16 |
GOE // // OH |
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第15週 |
12/21,12/23 |
empirical spectral measure // // Efron-Stein inequality and examples |
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第16週 |
12/28,12/30 |
Concentration // // Guionnet 4.2.1 |
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第17週 |
1/04,1/06 |
// OH // |
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