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Lafayette College’s Summer 2013 REU will run from Sunday, June 9, 2013 until Saturday, August 3, 2013. Participants are expected to be continuing their undergraduate education in the Fall of 2013, and NSF regulations require that only U.S. citizens and permanent residents are eligible to participate. Each participant will receive free housing and a $3,200 stipend.

There will be three projects, listed below. In order to apply to the program you will need to

- upload a personal statement about your mathematical interests and goals.
- upload a copy of a current transcript (unofficial transcripts are acceptable).
- arrange for two letters of recommendation to be submitted to the website.

The deadline for applications is March 1, 2013. For other questions about the REU, please send an email to mathreu@lafayette.edu

You can get to the application page through this link.

**Project 1**

General Area: Discrete Mathematics

Topic: The colored cubes puzzle

Advisor: Ethan Berkove

Given a palette of six colors, there are exactly 30 distinct ways to color a cube so that each face consists of one color. The colored cubes puzzle is easy to state: given an arbitrary collection of n^{3} colored cubes, when can these cubes be stacked into a larger n x n x n cube so that each n x n face is one color? It turns out that there is a solution to the colored cubes puzzle whenever n > 2. A proof of this result and a number of open problems can be found in this paper.

Based on the group’s interests, we’ll explore some of the open problems that come from the colored cubes puzzle. Some background in discrete mathematics/ combinatorics will be helpful. Programming experience might also be useful, but is not required.

**Project 2**

General Area: Combinatorics

Topic: Graphs, matroids, polynomials

Advisor: Gary Gordon

How many ways can you color the countries on a map so that no countries sharing a common border receive the same color? This classic counting problem is solved by a polynomial defined on a graph, the *chromatic* polynomial. We will study generalizations of this polynomial to matorids (which generalize graphs), and other combinatorial structures, like rooted graphs, and partially ordered sets.

The questions we will study will depend on the interests of the group, but a background in combinatorics or discrete math would help. So would some computer programming skills, but that is not a requirement. There are lots of open questions in areas that should be accessible, and you will learn something about matroids, probably from a new book.

**Project 3**

General Area: Abstract Algebra, Discrete Mathematics

Topic: SET®, Geometry and Groups

Advisor: Liz McMahon

The card game SET® is played with a special deck of 81 cards. The cards provide an excellent way to visualize finite geometry. In fact, the entire deck is a model for four-dimensional affine geometry, modulo 3. Studying the geometry of the game leads to some very interesting questions about the symmetry groups of maximum size collections of cards containing no sets. Building on previous work of REU students, we will study these symmetries using group theory.

There are many other connections between SET® and mathematics, including some interesting probability questions, other counting questions, and generalizations to higher dimensional geometry. A background in abstract algebra is a requirement, discrete math and/or geometry is a plus, and some experience with computer programming could be useful.