Mathematics REU at Lafayette College

Projects for Summer 2008

Exploration of Polytopes
Prof. Jed Mihalisin

Graphs

Overview: This REU will be a computer aided exploration of polytopes. One can define a polytope as the convex hull of a finite set of points. In the plane, polytopes are simply polygons. In three dimensions, some examples of polytopes are cubes, pyramids, tetrahedra, etc. Four dimensional polytopes possess many surprising properties and even three dimensional polytopes provide many interesting, open research questions. These fascinating objects arise naturally in a variety of areas of mathematics and computer science. We will most likely investigate polytopes in dimensions five through ten.

Background and some references: Our project will have the flavor of "Experimental Mathematics." As researchers, you will use a search program I have developed to explore the unknown terrain of these higher dimensions, with a variety of possible goals. One possible goal would be to use intuition gleaned from the computer to develop your own conjectures concerning polytopes. Another subsequent goal would be design computer experiments to test these conjectures. A third goal would be the discovery or construction of polytopes with particularly beautiful or interesting structure.

Student information: Knowledge of linear algebra is essential. Familiarity with Maple and/or MATLAB are a definite plus but simply possessing a reasonable level of programming savvy should be sufficient. (In other words, either you should be familiar with one of those languages or capable of quickly learning how to use them.) More information can be found on my web page.

Episodic Cellular Automata
Prof. Cliff Reiter

Overview: Cellular Automata have been used to model many physical phenomena yet there are relatively few models that exhibit complex self organizing behavior. For example, a system modeling the interaction of multiple populations where episodes of rising and falling communities are expected should exhibit complex behavior. Some pictures from previous work I've done with students can be seen here. This project will study known examples of periodic and self organizing automata and quantitative measures associated with them. We will examine nonlinear differential equations associated with periodic behaviors and seek new cellular models exhibiting episodic behavior.

Background: Participating students need not be experienced at programming, but computer experimentation is likely to be a valuable tool used by most group members.

Dice Games and Voting Theory
Prof. Lorenzo Traldi

Overview: The simplest kind of dice game involves two players who roll identical dice - the winner is the player whose die has rolled a higher value. People have played such games (and more complicated ones too) for millennia, but the idea that the two players might use different dice seems to have been considered only recently -- it was popularized about 50 years ago in Martin Gardner's widely-read Scientific American column on mathematics. Two basic observations are:

Even if the possible rolls of two dice have the same mean, they don't necessarily tend to be tied in the long run. For instance, among 6-sided dice (1,1,4,5,5,5) and (1,2,3,3,6,6) are not tied, though both are tied with the standard die (1,2,3,4,5,6).
The relative strengths of dice are not necessarily transitive. For instance, consider the 3-sided dice (1,4,5), (2,2,6) and (3,3,4): the second is stronger than the first, the third is stronger than the second and the first is stronger than the third.

The second observation naturally brings to mind Arrow's impossibility theorem - there is no best one among the three dice mentioned, though there is a best one of any two of them. In fact, dice games give an interesting representation of multicandidate elections: if n voters give numerical assessments of several candidates, then each candidate is represented by an n-sided die which lists the voters' assessments of that candidate. But in Wikipedia's article on voting systems do you know what? You won't find dice mentioned!

Dice families are sets of dice whose rolls have the same mean value. These families have surprisingly varied structures; many families are completely transitive, for instance. This 4-page paper gives a quick summary of known results, with many questions that haven't been answered yet. Here's my favorite question: is there more than one family of n-sided dice such that n > 2 and most pairs of dice are tied? (The one example I know is hiding in that paper. Try to find it!)

In sum, there are two kinds of issues we'll investigate. The first involves learning about voting theory, and looking for axioms that characterize dice among voting systems. (We'll study an actual book about voting theory - not just Wikipedia!) The second involves thinking of dice families as discrete structures, and trying to prove theorems about them.

Background: This project does not require a particular background, though students who have already studied discrete mathematics or the theory of voting might be especially sure that they're interested. What is required is success in studying advanced mathematics, as we will be conducting genuine mathematical research - formulating conjectures based on examples and established theory, and trying to prove that the conjectures are correct.

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Feedback to RobRoot@Lafayette.edu