Course title:               Math 21-499: Introduction to Undergraduate Research (2016 Spring Version)

Instructor:                 Steven J. Miller (sjm1@williams.edu)

Lectures:

• Lecture 1: Introduction to Zeckendorf Decompositions, Random Matrix Theory, L-Functions, and More Sums Than Difference Sets

Subpages:

• L-functions:
• MSTD:
• RMT:
• Zeckendorf:

Links for All:

• arxiv: http://arxiv.org/list/math/new
• OEIS: https://oeis.org/
• MathSciNet: http://www.ams.org/mathscinet/ (under tools can play the Erdos game)
• Googledocs for class: https://docs.google.com/spreadsheets/d/10lzna25c2-U8774AhF0O3E9P5H3ZCQi_eeMaqckY7Z8/edit#gid=0
   

TOPICS NOT DOING....

• Benford's law of digit bias: These projects involve probability and some Fourier analysis. One of the main questions concerns how rapidly Benfordness sets in, which is frequently related to the algebraic structure of numbers. There are also applications to financial data and models. I have tried in previous years to look at copulas with students, and that could be worth trying again.
  • Hill (general theory)
  • Raimi (survey)
  • Miller (introduction to Benford's law)
  • Kontorovich-Miller (connection to L-functions, random matrix theory and the 3x+1 problem).
  • Miller-Nigrini:
    • Products and the Mod 1 CLT (products of random variables tend to Benford)    
    • Almost Benford Behavior
    • Benford's law applied to Hydrology data
    • Useful paper: handout on order statistics.
  • Brown University Independent Study Group: Chains (applications to Bayesian theory, related to products) (very important read!)
  • Fewster (simple explanation of Benford's law)
  • Wikipedia page on copulas
  • For a few years I've been looking at fragmentation problems arising from this paper by Lemons. The general idea is quite interesting, but the paper has some leaps and needs to be put on a rigorous foundation. We made much progress in a continuous version last summer.

  • Here is another recent work with Corey Manack: Leading Digit Laws on Linear Lie Groups

   

• Zeros of Random Polynomials: These projects are heavy in probability and complex analysis. I haven't guided any students in this area yet, but would be open to seeing if something is possible.
  • Aldous - Fyodorov: Real Roots of Random Polynomials....
  • Edelman - Kostlan: How Many Zeros of a Random Polynomial are Real
  • Fairley: Number of Real Roots of Random Polynomials of Small Degree
  • Hughes - Nikeghbali: Zeros of Random Poly Cluster Uniformly Near Unit Circle (paper)
  • Hughes - Nikeghbali: Zeros of Random Poly Cluster Uniformly Near Unit Circle (preprint)
  • Ibragimov - Zeitouni: Roots Random Polynomials
  • Nezaka - Farahmand: Real Zeros Random Alg Poly with Binomial Elements
  • Schehr - Majumdar: Real Roots Random Poly Zero Crossing Prop Diffusion Eqs
  • Shepp - Vanderbei: Complex Zeros of Random Poly

• Video Introductions (subitems are research / colloquia talks)
  • Introduction to Benford's Law (9 minutes): https://youtu.be/GXRCoHXanjU
    • Longer Benford introductions:
      • Carnegie Mellon:  http://youtu.be/z2Rx1w5UDnk (slides here, technical talk: L-fns, 3x+1, Decompositions)
  • Tutorials:
    • Mathematica tutorial: https://www.youtube.com/watch?v=g1oj7CIqGM8 (webpage with templates here)
    • LaTeX tutorial: http://www.youtube.com/watch?v=dKUtJpG4Rt0 (webpage with templates here)

Googledocs for class: https://docs.google.com/spreadsheets/d/10lzna25c2-U8774AhF0O3E9P5H3ZCQi_eeMaqckY7Z8/edit#gid=0