MATH 4011 Fall 2020 (大學部)

(Mon 4 11:20-12:10/ Wed 34 10:20-12:10)

ROOM : 304 Astro-Math

Description : 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.

Prerequisites : 分析導論一 and 機率導論

Student's duty : Problem sets, midterm, final

Measure-theoretic probability

Part I : Measure and integration - 5 weeks

Basic topology, Functional analysis

Part II : Concept of measure-theoretic probability - 2 weeks

Convergence of probability measures

Methods of moments

Mathematical statistical physics

Part III : Methods - 3 weeks

De Finetti's theorem for exchangeable sequences

Large deviations

Concentration inequalities

Part IV : Models - 4 weeks

Curie-Weiss model

Random matrix theory and Wigner semicircle law

Reference

1. Leadbetter, A Basic Course in Measure and Probability_ Theory for Applications (2014)

2. https://en.wikipedia.org/wiki/Exchangeable_random_variables

3. Indistinguishable Classical Particles Springer

4. Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). An introduction to random matrices. ; https://en.wikipedia.org/wiki/Random_matrix

W01 Sep 14

W09 Nov 11 Midterm

W17 Jan 06