Operations Research Toolkit on HIV/AIDS. (2012). Retrieved January 3, 2012, from Population Council: http://www.popcouncil.org/Horizons/ORToolkit/index.htm
We will use supercharacter theory, a powerful new algebraic mechanism recently pioneered by Diaconis-Isaacs and André, to study certain exponential sums which arise in number theory (e.g., Gauss, Kloosterman, and Ramanujan sums). From this new perspective, revisiting even the most elementary groups can yield interesting results. For instance, many of the fascinating arithmetic properties of Ramanujan sums
follow immediately from basic supercharacter theory and elementary linear algebra applied to .
Students will learn the rudiments of supercharacter theory and begin working on problems right away. We expect that a careful survey of familiar groups will reveal supercharacter tables which contain exponential sums whose properties are of general interest. Moreover, the powerful machinery of supercharacter theory should automatically yield simple proofs of many difficult algebraic formulas.
Our project revolves around several open-ended questions and features many points of departure. If time permits, we will also study some deeper foundational issues. Being a new area, supercharacter theory is fertile ground for student research and there is plenty of low-hanging fruit to be found on this frontier.
Prerequisites. The ideal candidate has a strong background in Abstract Algebra and Linear Algebra. Some knowledge of Number Theory and familiarity with Mathematica (or comparable software) is also desirable.
Given a word whose letters are the generators of a group , is there an algorithm to determine if the word represents the identity in ? If so, how many steps will it take? In this project we will study a method to answer these questions by building a metric space that encodes the structure of the group.
One of the main ideas of geometric group theory is that you can recover algebraic properties of a group by encoding the structure of the group in a metric space and analyzing the geometric properties of the space. This method has proved fruitful for answering when many interesting questions about groups. One example is Maxwell Dehn's classical word problem. "Let be a finitely generated group, and be a word whose letters are generators of . Is there an algorithm to determine if equals the identity in ?''
In this project we will discuss a common method used by geometric group theorist to solve the word problem for a finitely generated group by relating words in the group to paths in a metric space representing the group's structure. Our main focus will be Dehn functions. A Dehn function is a measure of the computational complexity of the solution to the word problem. Given a word of length representing the identity, a Dehn function bounds the number of steps needed to reduce to the identity. This turns out to be equivalent to studying the relation between perimeter and area in the encoding metric space. Our main source of examples will be spaces built out of squares and equilateral triangles satisfying a combinatorial nonpositive curvature condition that mimics the geometry of the Euclidean and hyperbolic planes.
The ideal students for this project will have taken linear algebra and a course in which they have written rigorous mathematical proofs. Coursework in group theory or topology will also be helpful but not necessary.