UGA VIGRE Research Groups, Fall 2006
Brian Boe, Lenny Chastkofsky, Dan NakanoVictor Kreiman, Jonathan Kujawa, Emilie Wiesner
Irfan Bagci, Ben Connell, Bobbe Cooper, Mee Seong Im, Wenjing Li, Kenyon Platt, Carrie Wright, Ben Wyser
Tyler KellyCohomology of Lie algebras
We are investigating the cohomology of nilpotent Lie algebras in (small) prime characteristic. There is a very elegant theorem (due to Kostant) describing the answer in characteristic zero, and it is known that the answer is the same in characteristic p if p is big enough. For small p there can be "extra" cohomology, but it is unknown exactly when and to what extent this happens. The group has formulated a number of conjectures about this over the past year, and we plan to continue refining and proving these. We use computer programs that the group has written (and continues to modify) to formulate and test conjectures, and we have some new ideas about proving them, using an inductive technique involving chains of nilpotent subalgebras. On the more theoretical side, we will introduce the group to spectral sequence techniques and the use of representation theory to compute the cohomology groups.
Algebraic Geometry Group
Pete Clark, Victor Kreiman, Jiayuan Lin
Maxim Arap, Angel Brown, Lev Konstantinovskiy, Joe Rusinko, Nathan Walters, Steven WinburnThe Hodge conjecture
This semester the Algebraic Geometry VIGRE group will be studying problems related to the Hodge conjecture. We will start with some basic facts about the cohomology of algebraic varieties and try to understand what the Hodge conjecture means. We will then move to some elementary classical examples involving abelian varieties and K3 surfaces, understand what the Hodge conjecture means in those cases and possibly attempt to prove it for those.
Applied Mathematics Group
Malcolm Adams, Caner Kazanci, Andrew Sornborger
Aaron Caraher, Jacob Keenum, Leopold Matamba, Brandon Samples
Dynamical systems and epidemiology
We will be using Maple, Matlab, and qualitative analytic methods to study models of epidemiology. We will begin by reviewing some basic classical results in the field but our goal will be to understand the onset of limit cycles in a variation of a model for parvo virus.
Dino LorenziniSybilla Beckmann
Michael Guy, Jerry Hower, Val Hower, Al LaPointe, Nathan Walters, Juhyung Yi, Renee Zawistowski
Grant FiddymentAlgebraic graph theory
One usualy associates to a graph G on n vertices two (n x n)-matrices, the adjacency matrix A and the Laplacian matrix L. Both A and L have a set of eigenvalues, and a Smith normal form over the integers.
Much has been written on the relationships between the eigenvalues and the combinatorics/topology of the graph. In this seminar, we will start by investigating the information encoded in the Smith normal form of the Laplacian of a graph.
Equivalent to the Smith normal form of a graph is a finite abelian group that has 'appeared' independently in several different fields, and is known under several names, such as the component group, the critical group, or the sandpile group. This interesting group is the main motivation for studying the Smith normal form of the laplacian. Its order is the number of spanning trees of the graph.
Ted Ashton, Aja Johnson, Steve Lane, Yang Liu, Matt Mastin, Amy Reeves
James Dabbs, Rachel Whitaker, Meredith PerrieRopelength of Composite Links
The minimum length of unit-diameter rope required to tie a knot is called its ropelength. Minimum ropelengths have been numerically approximated for prime knots and links with up to 10 crossings (see this paper). But nobody has computed ropelengths for composite knots and links in an organized way. This group will continue last year's project of generating a definitive computerized table of composite knots and links with 12 or fewer crossings and finding the minimum ropelength shape of these knots and links.
Mathematical Physics Group
Robert Varley, Cal Burgoyne
Bob Anderson, Brad Bassler, David Edwards
Josh Hughes, Tom Lanier, Justin Manning, Emily PritchettQuantum mechanics
This group will study Feynman's path integral formulation of quantum mechanics and the extent to which the path integral has rigorous mathematical interpretations. The key ingredient in the path integral is the value of the action associated to each path in the configuration space of the physical system. The group will fill in the necessary background in physics and mathematics, including the standard formulation of quantum mechanics in terms of the energy operator and its eigenvalues.
Prerequisites: several variable calculus and linear algebra. (Previous experience with quantum mechanics is not assumed.)
Workload: Intermediate between attending a weekly seminar and taking a 3 credit hour class with no tests. The participants will prepare and present material as the group progresses through some planned topics and works on some problems. An average of 1-2 hours per week outside the seminar should suffice to keep up; the demands of preparing the presentations can be shared and distributed, so that they might average out to another 1-2 hours per week.