Consortium Schools:

Summer 2013 Research Projects:
Matroids at Willamette University. A unipancyclic (UPC) graph is a graph containing exactly one cycle of every possible size. Only a handful of these are known to exist, although searches have been performed through all graphs with 56 or fewer vertices. We generalize this problem by seeking to find and characterize UPC matroids. There are UPC matroids that are not graphic, so this does result in a larger family. We also plan to consider problems concerning algebraic matroids. Applicants should have had a course in Linear Algebra; exposure to some Graph Theory is helpful but not required. The REUT will include introductory material on matroids; there is no assumption that participants would have seen matroids prior to the REUT. Four students and one teacher will be invited to join this research group. Tile Invariants and the Mathematics of Tiling at Linfield College. Can a finite region in the plane be tiled by a given set of tiles? If so, can we say something about how different tilings of a region must be related? In the last twenty years combinatorial group theoretic and topological techniques have been applied with great success to the study of tiling problems in the plane. This program will begin with a study of recent results, focusing on methods for determining tile invariants, linear combinations of the tiles that must be present in any tiling of a given region. We will then apply these techniques to the solution of new tiling problems involving polyomino tile sets. Four students and one teacher will be invited to join this research group. The Game of Go: Statistical Approaches to Artificial Intelligence at Lewis and Clark College. The Asian game of Go has simpler rules than Chess, but writing a Goplaying program that can compete with strong human players has proven exceedingly difficult. In fact, Go is considered one of the "grand challenges" of artificial intelligence. We will explore various statistical/machine learning approaches to the problem, including Monte Carlo methods and learning from recorded games. This summer's work will focus on decomposing the board into regions, so that the program can perform and combine several local searches (with the work possibly spread over several machines) rather than performing a single global search. Desired skills:
Four students and one teacher will be invited to join this research group. Groups and Geometry at the University of Portland. There are many different interactions between the spatial discipline of geometry and the much more abstract discipline of group theory. For example, a cube, which is a geometric object, admits many different rotational symmetries. The set of all such symmetries can be made into a group, where composition of symmetries defines the group operation. This summer, we will be studying the interactions between particular classes of geometric spaces (Riemann surfaces and cube complexes, neither of which we attempt to define here) and the groups that can be naturally associated to these spaces. By studying the types of groups that "act nicely" on particular spaces, we stand to gain information about the structure and properties of both the groups and the spaces they act on. Though most of the definitions and major results required for this project will be taught during the initial weeks, applicants will need to know some basic group theory. Though not required, a little knowledge of geometry and topology would also be handy, and basic computer programming skills are a plus. Four students and one teacher will be invited to join this research group. Research topics from summer 2012. Research topics from summer 2009. Research topics from summer 2008. Research topics from summer 2007. 