Math 456: Spring 2010

Class Meeting : TuTh 1:00--2:15, Has add 113

Instructor : Luc Rey-Bellet

Office :  1423 J LGRT
Phone :  545-6020
E-Mail :   luc <at> math.umass.edu
Homepage:   http://www.math.umass.edu/~lr7q
Office Hours :   Tu 3:30--5:00,   Th 2:30--4:00,   or by appointment.

Syllabus: This course is an introduction to mathematical modeling. The goal of the class is to lear how to translate problems from "real-life" into a mathematical model and how to use mathematics to solve the problem. We will pick problems from a variety of source, from social sciences, biology, and natural sciences. The class cover three main topics:

• Modeling using game theory.

• Modeling using differential equations.

• Modeling using probability.

Text: As the class progresses I will post regularly some handouts. You should use as a COMPLEMENT of your class notes, not as a substitute. They will provide a summary of the class and a reference for definitions and main results.

• Lecture 1 Games
  
• Lecture 2 Dominated strategies
  
• Lecture 3 Nash equilibrium
  
• Lecture 4 Monopolies and duopolies
  
• Lecture 5 Mixed strategies and expected payoffs
  
• Lecture 6 Nash equilibria and reaction curves
  
• Lecture 7 Nash equilibria and equality of payoffs
  
• Lecture 8 Zero sum games and the minimax theorem (forthcoming)
  
• Lecture 9 Evolutionary stable states (forthcoming)
  
• Lecture 10 The sex ratio game
  
• Lecture 11 Probability models
  
• Lecture 12 Counting
  
• Lecture 13 Conditional probabilities
  
• Lecture 14 Bayes formula
  
• Lecture 15 Difference equations and the gambler's ruin
  
• Lecture 16 Expected value, variance and Chebyshev inequality
  
• Lecture 17 The law of large numbers and Monte-Carlo method
• Lecture 18 Backward induction and dynamic programming (forthcoming)
  

Course Web page: Please bookmark www.math.umass.edu/~lr7q/m456-spring2012/m456home.html The page will be updated regularly. Check it often.

Grading and assignments: Homework will be assigned regularly throughout the class. The homework will be graded.

The class will be divided in groups of three students. Every group will select a subject in consultation with the instructor. You have a lot of freedom to choose your subject as long as it is related to mathematical modeling. In particular game theory is used in a very wide variety of contexts and will give you plenty of options, depending on your tastes and backgrounds.

The group project as 2 parts:

• An in-class presentation of around 15 minutes (+ a few minutes for questions). This presentation is done as a group. The main goal of the presentation is to explain our project to your fellow students. Make sure you spend enough time to explain the problem and give a quick summary of your findings.
• The second part is a paper which is written individually by each member of the group. I DO NOT want three times the same paper. If, in your group, you have divided the work and considered different aspects, your paper may certainly reflect this, but each paper should present the whole project in its entirety.
  There is no lower or upper bound on the length of the paper since your topic may request more or less space. But being precise and concise are two necessary attributes of well-written mathematics.
  Your paper should adress the following points.
  • Motivation for your choice of topic.
  • Modeling issues. How hard is it to translate your model in mathematics?
  • What are your sources? Gives references to the literature.
  • A clean presentation of the mathematics involved. Make your results clear. If the computations are long, present your main finding and put the computation in an appendix.
  • A summary of your findings and open questions.

Homework:

• Homework 1: Do the exercise in Lectures 1--4. Due on Tuesday February 14.
  
• Homework 2: Do the exercise in Lectures 5--7. Due on March 1.
  
• Homework 3: Do the exercise in Lectures 11--14. Due on March 29.
  
• Homework 4: Do the exercise in Lectures 15--17. Due on May 1.