HOMEWORK PROBLEMS FROM THE TEXTBOOK AND ASIDES

HOMEWORK PROBLEMS FROM THE TEXTBOOK AND ASIDES: Solution key available here

Week 12: Nov 24, 2014:
- Read: Chapter 18
- Continue working on projects and exam. Do the first part by Wednesday 10am and submit!
Week 11: Nov 17 to 21, 2014:
- Read: Chapters 15 and 16
- Continue working on your project, do first part of exam by Wednesday 10am
- Here is a general survey on the history of Random Matrix Theory by myself and my physics advisor / mentor: Nuclei, primes and the random matrix connection
- General reading:
  - Hayes: The Spectrum of Riemannium: a light description of the connection between random matrix theory and number theory (there are a few minor errors in the presentation, basically to simplify the story). This is a quick read, and gives some of the history.
  - Conrey: L-functions and Random Matrix Theory: This is a high level description of the similarities between number theory and random matrix theory.
  - Katz-Sarnak: Zeros of Zeta Functions and Symmetry: Another high level article similar to the others.
  - Diaconis: Patterns in Eigenvalues: this is a bit more readable than the others, and is based on a distinguished lecture he delivered.
Week 10: Nov 3 to 7, 2014:
- Read Chapter 13, 14 and do take-home (must be emailed to me by 11:59pm on the 9th or handed in by 3pm on Thursday the 6th). No class on Friday or Monday.
- Lecture 28: Lecture 26: http://youtu.be/3n2BTQKdFfo Lecture 27:
- Homework: Read and discuss in groups of at least 2 and be prepared to present in class on Friday:
  - Read classics: Hardy and Littlewood: Partitio Numerorum III: http://fuchs-braun.com/media/8cdd73c813c342f8ffff80d1fffffff0.pdf
  - Tao: Odd numbers exceeding 1 are the sum of at most 5 primes: http://arxiv.org/pdf/1201.6656v4.pdf
  - Helfgott: Series of papers on Goldbach:
    - Major Arcs: http://arxiv.org/abs/1305.2897
    - Minor Arcs: http://arxiv.org/abs/1205.5252/
    - Ternary True: http://arxiv.org/abs/1312.7748
Week 9: Oct 27 to Oct 31, 2014:
- Read Chapter 3, 13.
- Lecture 21: http://youtu.be/hg1xKs7WMQQ Lecture 22: http://youtu.be/Wavl_-DsdWw Lecture 23: http://youtu.be/jzSt1Uepv-c
- Homework: Due Friday October 31: Exercise 3.1.18. Exercise 3.1.24 (calculate the zeroth, first,... up to and including the seventh). Exercise 3.1.28. Exercise 3.3.4. Exercise 3.3.8. Exercise 3.3.16. Exercise 3.3.19.
Week 8: Oct 20 to Oct 24, 2014:
- Read Chapter 3, 13.
- Lecture 18: http://youtu.be/zCbm7hZUY9Q Lecture 19: http://youtu.be/EGDPdOuK3Jg Lecture 20: http://youtu.be/DfSCUElVr6s
- Homework: Due Friday October 31: Exercise 3.1.18. Exercise 3.1.24 (calculate the zeroth, first,... up to and including the seventh). Exercise 3.1.28. Exercise 3.3.4. Exercise 3.3.8. Exercise 3.3.16. Exercise 3.3.19.
Week 7: Oct 13 to Oct 17, 2014:
- Read Chapters 2 and 3
- Lecture 16: http://youtu.be/XhVHeawbLPc Lecture 17: http://youtu.be/z8WstIYV3Xc
- Homework: Due Friday, October 17: From my book: Exercises 2.1.4, 2.1.6, 2.1.14, 3.1.5, 3.1.8, 3.1.9 (very important).
Week 6: Oct 6 to Oct 10, 2014:
- Read Chapters 2 and 3
- Lecture 13: http://youtu.be/BFVuimP8ZLE Lecture 14: http://youtu.be/I_10ADutXD8 Lecture 15: http://youtu.be/ejbZw-tIN9w
- Homework: Due Friday, October 17: From my book: Exercises 2.1.4, 2.1.6, 2.1.14, 3.1.5, 3.1.8, 3.1.9 (very important).
Week 5: Sept 29 to Oct 3, 2014:
- Read Chapter 12.
- Lecture 11: http://youtu.be/BFVuimP8ZLE Lecture 12: http://youtu.be/I_10ADutXD8
- Homework: Due Friday, October 3: 9.1.1, 9.3.2, 9.3.3. Also calculate the Fourier transform of \(f(x) = \exp(-\pi x^2 / N)\), which is needed in using Poisson Summation. Exercises 12.1.8, 12.1.10, 12.3.8. You should also be working on a research project, and have done a MathSciNet / Pi Mu Epsilon problem.
Week 4: Sept 22 to 26, 2014:
- Read: Chapter 9, Section 9.1, 9.2, 9.3, 9.4; Sections 9.5 and 9.6 are optional and worth at least skimming. Read my paper on the Modulo 1 Central Limit Theorem (http://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/Mod1CLT_BenfProd10.pdf), and on the Weibull distribution and Benford's law (http://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/WeibullBenf80.pdf), and start reading Chapter 12.
- Lecture 8: http://youtu.be/8dPhdo98Gk8 Lecture 9: http://youtu.be/AA_GHlM6sU4 Lecture 10: http://youtu.be/tg47OJkNkcQ
- Homework: Due Friday, October 3: 9.1.1, 9.3.2, 9.3.3. Also calculate the Fourier transform of \(f(x) = \exp(-\pi x^2 / N)\), which is needed in using Poisson Summation. Exercises 12.1.8, 12.1.10, 12.3.8. You should also be working on a research project, and have done a MathSciNet / Pi Mu Epsilon problem.
Week 3: Sept 15 to 19, 2014:
- Read Chapter 11, read Chapter 9.
- Lecture 5: http://youtu.be/Utf4esln9e0 Lecture 6: http://youtu.be/XuqIurfW_fs Lecture 7: http://youtu.be/v9eoWGQkoeM
- Homework: Due Friday, September 19 (though you are strongly urged to start working on them earlier): #1: Consider the function \(g(x) = \exp(-1/x^2)\) for \(x \neq 0\) and \(0\) otherwise. Prove that all the derivatives of this function are zero at zero. Note this is problem A.2.7 in our book. #2: Let \(f(x) := \sum_{n=0}^\infty a_n x^n\) have radius of convergence \(\rho > 0\). Prove \(f'(x) = \sum_{n=0}^\infty n a_n x^{n-1}\). From the book: #3: Do 11.2.2 (important), #4: Do 11.2.5, #5: Do 11.2.7, #6: Do 11.2.10, #7: Do 11.3.6.
Week 2: Sept 8 to 12, 2014:
- Read Chapter 11, start reading Chapter 9.
- Lecture 2: http://youtu.be/MonfQXBshnI Lecture 3: http://youtu.be/vVjOYrsHGqM
- No class on Friday (I'll be in Boston for my uncle's funeral): please watch https://www.youtube.com/watch?v=XwnzWOc3_-0&feature=youtu.be if you want to see math, and keep working on the problems and reading the Fourier analysis chapter. I'm going through the difficult theory behidn a lot of the calcualtions, and leaving the more algebraic arguments for you to read. If you ever want more detail in class and less emphasis on the general mechanics, LET ME KNOW! I'm trying to use class time productively and do things in class different than you can do at home.
- Homework: Due Friday, September 19 (though you are strongly urged to start working on them earlier): #1: Consider the function \(g(x) = \exp(-1/x^2)\) for \(x \neq 0\) and \(0\) otherwise. Prove that all the derivatives of this function are zero at zero. Note this is problem A.2.7 in our book. #2: Let \(f(x) := \sum_{n=0}^\infty a_n x^n\) have radius of convergence \(\rho > 0\). Prove \(f'(x) = \sum_{n=0}^\infty n a_n x^{n-1}\). From the book: #3: Do 11.2.2 (important), #4: Do 11.2.5, #5: Do 11.2.7, #6: Do 11.2.10, #7: Do 11.3.6.
Week 1: Sept 1 to 5, 2014: First Class: http://youtu.be/7b-6F4qJ4Ls
- Welcome slides for first class.
- Handout for first class.
- tart exploring mathematical software (R, Mathematica). Learn LaTeX. I have handouts and videos: http://web.williams.edu/Mathematics/sjmiller/public_html/math/handouts/latex.htm
- Homework: To be emailed to me by 8pm on Sunday, September 7: Email me a short note (a paragraph suffices) on what you want to get out of this course, and what lesson you learned from the graduation speech by Uslan (http://www.graduationwisdom.com/speeches/0018-uslan.htm). Full credit, 20/20, so long as you answer both questions on time.
- Homework: To be done by start of class on Monday, September 8: Read the handout on Benford’s law (Chapter 1 of the book I’m editing). If you do not know probability, read Chapter 8 of the class textbook (Invitation to Modern Number Theory); note this is how grad school operates: if you don’t know something, you need to pick it up quickly. You should make sure you get up to page 206 at least, and the rest can wait till Wednesday. Also read Sections 11.1 and 11.2 of the class textbook (though feel free to keep reading Chapter 11, as that will be our first unit).