Math 210: Tentative Syllabus	Spring 2019

This course focuses on the mathematics applications in everyday life. There will be an emphesis on discovery, understanding, and communication. The impact of the Internet cannot be underestimated.

Instead of building a machine, we will primarily investigate some significant examples. Here are some possibilities, each of which are huge areas of study.

• Caution: Interpreting misleading data.   "Air Travel Surges by 123%!"   [New York Times]

• Computer Stuff

• Basic probability and statistics, Bayes' Rule, Markov Chains,

  Specific applications such as to how Google search works, and how YouTube and NetFlix make their suggestions, and how to rank college football teams when most teams have not played other.

  Some specific probability and statistics questions
  As part of this we will introduce and use some topics from Linear Algebra.

  • Some Notes on Linear Algebra
  • Eigenvalues and Eigenvectors (Strang)
  • Vectors and Least Squares
  • Some Basic Maple Examples
  • Distance from a point to a line

• Voting and Social Choice   What is a fair election? Arrow's impossibility theorem.
  • Voting (Lecture Notes),
  • COMAP, For All Practical Purposes: Mathematical Literacy in Today's World, 9th edition (2013) [earlier editions are fine], W.H. Freeman & Co. [See Part III, Voting and Social Choice, p. 325-450]
  • Ellenberg, Jordan, How Not To Be Wrong, Penguin Press, New York, 2014, pages 376--392, 418--420.
  • Book: Volic, Ismar, Making Democracy Count: How Mathematics Improves Voting, Electoral Maps, and Representation, Princeton Univ. Press (2024)
  • A Better Electoral System in Maine (2018),     Ranked Choice Voting in Maine, 2018,   Maine Nov. 20, 2018
  • Ranked Choice Voting (New Yorker, Jan 2020)
  • How does Ranked-Choice Work? (New York City, 2021
  • Ranked-Choice: Alaska 2022
  • Arrow's Impossibility Theorem
  • Rabin Fairness Model
  • Courts, Commissions, and Consultations: How Mathematicians Are Working to End Gerrymandering 2022
    Gerrymandering-and-Math-2025

• Quantitative guidance in making decisions.
  Saaty: Hard Mathematics Applied to Soft Decisions
  [also: The Analytic Hierarchy Process: A Methodology for Win-Win Management
  and Saaty: On the Measurement of Intangibles]
  Markov Chain: A^k (reqires Maple)
  A numerical example (requires Maple)

• Dynamical Systems. Conflicts between competing populations. mathematical models for spreading of rumors, diseases, and effective vaccination procedures. The Kermack-McKendrick model for an epidemic.
  Measles Outbreak (NYT Feb. 2019)    
  Antivaxxers (Feb 2019)
  NY measles: Unvaccinated barred from public spaces (March 2019)
  Measles: Confusing clusters of new infections (Apr 2019)

  Can You Stop an Outbreak of a Contagious Disease? [2025, Interactive]
  Stop a Measles Outbreak (video NY Times 2025)
  Ebola (March 26, 2019)

  Modeling an epidemic
  Brauer, Mathematical epidemiology (2017)
  Understanding Epidemiology Models (May 2020)   The point of these models is not really to predict the future but to understand the problem. For the COVID-19 virus, the current data is way too skimpy and unreliable to predict anything -- and the virus keeps evolving.
  Chasing the Elusive Numbers That Define Epidemics
  Wuhan coronavirus, Feb. 10, 2020,   data?, and update, Feb 28, 2020
  The Coronavirus, by the Numbers, March 2020

  The reproductive number, R₀, of an infectious disease (video)
  R₀ :   Researchers still aren't sure how contagious COVID-19 is compared to other viruses, but it appears to spread more rapidly than the flu. "In public health we measure something called the R₀ (pronounced 'R-naught') which is the average number of people infected by a person with the virus," says Dr Brown. He goes on to explain that the R₀ for influenza is about 1.3, meaning about 1.3 people get the virus from every ten who have the infection. Measles, on the other hand, is extremely contagious, with an R₀ of 12-18.
  Jan 2021: Hard Lessons of Modeling the Coronavirus Pandemic
  Dec 2022: Covid-19 Models (American Math. Soc.)

  Predator-Prey Murray,   Predator-Prey, Jones

  Biodiversity Viscous fluid in a long pipe Hagen-Poiseuille equation
  Blood Flow, Blood Pressure, and Resistance
  tectonic plates

• Coping with large data.

  SVD Face Recogntion [Nathan Kutz]: Lecture 1,   Lecture 2,   Lecture 3

  Biology [Bingni Wen Brunton]: PCA 1,  PCA 2,  PCA 3, 

• Secure encryption of data.
  Secure Communications
  Caesar's Encryption
  NY Times 1998 Article
  RSA (Martin Gardner),
  Public Key Encryption (Wikipedia)
  RSA Cryptosystem (wikipedia)
  RSA Encryption (Math 210)
  RSA cryptosystem
  RSA Polynomial (Quanta)
  RSA Labs FAQ
  SOME REFERENCES:
  • Chapter 8 in Elementary Number Theory and Its Application, 6th Edition 6th Edition (2011), by K. Rosen, Pearson
  • An Introduction to Mathematical Cryptography, Hoffstein, Pipher, & Silverman, 2nd edition (2014), Springer, Undergraduate Texts in Mathematics.
  • RSA Labs FAQ
  • Breaking RSA (1999)
  • S. Landau: A valuable review of topica and textbooks (2004)
  • RSA-Polynomial-Quanta

• Linear Programming: Effective allocation of resources.
  A Simple Example
  Two Examples (Strang)

  Simplex Method

To be successful in this course, you should be present for all class meetings and plan to take good notes.

Some References