Weighted Voting and Indices of Power

Suppose I own 3/7 of the stock in a particular company and each of two other investors own 2/7. How are we to make decisions? The first thing that occurs to most folks is weighted voting. I get three votes and each of the others gets two. If you think about it for just a minute this solution is less than satisfactory. Any coalition of two people can pass or defeat any measure. That is, I don’t have any more influence, or power, than either of the others. Perhaps if we insist that five votes are needed to pass or defeat a measure that would help? A minute or two of thought should convince you that I have entirely too much power now (e.g., the other two, 4/7 of the ownership, cannot pass anything without my approval). In a way, there really is no good solution to this problem. There are fundamental problems with weighted voting; nevertheless people persist in using it. We will analyze some of the ways that mathematicians and political scientists have proposed to measure voting power. We will discover that all such measures known suffer from flaws. We will ask ourselves what, if anything, can be salvaged from this distressing state.

Written by:  Charles Noneman, Chrissy Donovan, and Elissa Brown

Geometry with Differential Forms

The subject of Differential Forms, also called the Exterior Calculus, is an amalgam of ideas from calculus and linear algebra. Differential forms are particularly well suited to the expression of fundamental theorems in vector calculus like Green’s and Stokes’ Theorems and, especially, to the study of curvature. This project will use differential forms and tensors to explore the geometry of surfaces and of space-time. Particular topics may include connection forms, parallel transport, geodesics, general relativity, and Riemannian geometry. Math 344 is strongly recommended for this project.

Written by:  Emma Turetsky, David Guild, Michael Feinberg, and Kyle Drake

Challenge Math Curriculum

Students in this comps group will work with children in the Northfield Public Schools weekly throughout the year, developing a curriculum for either Challenge Math pull-outs with elementary children, or a Math Club with middle school or high school children. During fall term, students will read math education literature on Discovery-Based Learning, psychological development, along with books and articles about running Math Circles and the like. During winter term, students will do independent reading in several areas of mathematics (like Graph Theory, Combinatorics, Number Theory), and devise lesson plans based on these areas. During the spring term, students will write up a book for use by future parent-volunteers to run Challenge Math or Math Club groups.

Written by:  Hannah Breckbill, Aparna Dua, Luke Hankins, and Robert Trettin

Traffic Flow

Have you ever found yourself stuck in traffic, inching along at best, when you suddenly arrive at a point where for some unknown reason cars around you are suddenly free to increase speed? The source of the slowdown often is a mystery. There are mathematical models that explore this phenomenon. The goals of this project are first to understand the differential equations that can be used to describe traffic flow in simplified situations and then go on to visualize and improve these models.

This project will begin with directed reading on the subject. One possible topic might be the wave phenomena of automobile brake lights. Another might be the effect of a red light on traffic. Although it is unlikely that we will be able to solve all of the world’s traffic problems, at least we will better understand the mathematics governing them.

It would be helpful if some members of the group had experience with Differential Equations (although this is not a requirement). Another useful tool might be Mathematica.

Written by:  Haggai Nuchi, Tyler Mitchell, and Peter Jamieson

CMS Direct

Students in this project will be working with data provided by CMS Direct (www.cmsdirect.com), a catalog marketing services firm in Minneapolis. The students will apply statistics to purchase data to determine trends and do market basket analysis. The students are expected to give a final formal presentation to representatives of the company. (There will be a more complete description of project at Thursday’s presentation.)

Written by:  Joseph Lindner, Samantha Morin, Khahn Nguyen, and Max Olivier

Group Representations in Probability and Statistics

A group representation assigns an invertible matrix to each element of a group in a way that preserves group structure. That is, the matrix assigned to the product of two group elements is the product of the matrices assigned to each element.

Group representations are a powerful way to study random walks.

Random walks on groups arise naturally in many problems. Here are 3 examples:

(1) Think of the group Zn (the integers mod n) as n points wrapped around a discrete circle. A frog hops left or right on the circle, each with probability ½. How many hops does it take the frog to reach a given point? To reach every point? After how many hops is the position of the frog close to random?

(2) Consider a deck of n cards. Two cards are picked at random and their position in the deck is switched. This is called the random transposition shuffle. How many such shuffles does it take to mix up the deck of cards? Here the group structure is the symmetric group Sn. By looking at the representations of the symmetric group one can show that it takes about (1/2) n log n shuffles for the deck to get close to random.

(3) Consumers are asked to rank 5 brands of cookies. The rankings can be treated as permutations of 5 objects, leading to a function on the group S5. For data that is represented as permutations there is a wealth of statistical problems that can be addressed with representation theory.

For this project students will study the monograph Group Representations in Probability and Statistics by Persi Diaconis. The author ran away from home at 14 to join the circus and become a magician. He eventually wound up at Harvard graduate school in mathematics. Diaconis received the prestigious MacArthur Fellowship after establishing rigorous results that it takes about seven riffle shuffles to mix up a deck of cards. This comps project requires Probability and Abstract Algebra I.

Written by:  Jeremiah Chung and Heung Jin Kwon

Bayesian Modeling of Spatial Data

Bayesian inference methods offer an alternative approach to the classical, or “frequentist”, methods for data analysis that are covered in any introductory statistics course (e.g. Math 215, or even Math 245 and 275). Both approaches start with a model for your data that depends on some unknown parameters, but from here things diverge. The Bayesian approach treats all unknown parameters as random variables which are modeled with a prior distribution. This prior is then updated (using Bayes Theorem) by conditioning on the data that you observed. Inference for unknown parameters is based on this updated distribution. In the last 10-15 years Bayesian methods have gained in popularity mainly due to an increase in computing power (they rely heavily on computers for estimation and prediction) and because of their usefulness in situations where classical inference methods are cumbersome or impossible (such as problems with few observations and lots of unknowns).

This project will begin with directed reading on Bayesian inference. Once a good foundation for Bayesian data analysis is established, we will focus on modeling spatial data and explore Bayesian models used to describe spatial dependence. We will apply these methods to air pollution data collected by the MN Pollution Control Agency and hopefully construct a sensible model to explain how local ozone levels depend on other air pollutant levels and on weather conditions such as temperature and wind speed. If time permits we may also investigate spatiotemporal models which describe dependence through time and space.

Group members are strongly encouraged to take Math 265 (Probability) and to have some familiarity with the command line side S-plus or R (although this is not required).

Written by:  Bassirou Sarr, Christina Knudson, and Edward Kuhn

Directed reading in analytic number theory: You can help pick the topic!

This is an opportunity for one or two (preferably two) people to delve deeply into material that uses functions of a complex variable to investigate and solve problems in number theory, and to get substantial (weekly) experience in presenting that material at the blackboard. Prerequisites are functions of a complex variable and at least one of number theory and abstract algebra I. Other courses that may prove relevant include the recent algebraic geometry seminar and abstract algebra II. One or more specific topics will be chosen, according to the participants’ background and interests, from possibilities ranging from elliptic functions and modular forms to Dirichlet’s theorem on primes in an arithmetic progression (which says that if a and b are any two positive integers with gcd(a,b) = 1, then the sequence a, a+b, a+2b, a+3b, … contains infinitely many primes).

Written by:  Ryan Smith and Yuan Tian

In search of bijections and other combinatorial arguments

As an example of the sort of problem we might consider, suppose we write down the numbers 1 through n in order and then provide them with signs. For instance, if n = 7, we might end up with

1 + 2 – 3 – 4 + 5 + 6 – 7 (= 0) or

1 – 2 + 3 – 4 + 5 + 6 – 7 (= 2) .

In general, if Z(n) is the number of ways that we can get such an expression that equals 0 and W(n) is the number of ways that we can get 2 , it seems that for small values of n , Z(n) and W(n) are really close together. Now one way of showing that two finite sets are the same size is to come up with a bijection (1-1 correspondence) between them. Can we find something like a bijection between the set counted by Z(n) and the set counted by W(n) ?

Ever since Franklin found a beautiful “bijective proof” of Euler’s pentagonal number theorem in the 1880’s, and more especially in recent years, many such proofs have been found, but there should be some more out there! There is no mandatory prerequisite for this project, but some knowledge of combinatorics (in particular, generating functions) is desirable, and programming experience (so we can get experimental data) should also be helpful.

Written by:  David Lonoff and Daniel McDonald