UGA VIGRE Research Groups, Spring 2007
Brian Boe, Lenny Chastkofsky, Dan NakanoPostdocs: Jonathan Kujawa, Emilie Wiesner
Graduate students: Irfan Bagci, Ben Connell, Bobbe Cooper, Mee Seong Im, Wenjing Li, Kenyon Platt, Carrie Wright, Ben Wyser
Undergraduate: 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 1
Faculty: Pete Clark
Postdocs: Victor Kreiman, Jiayuan Lin
Graduate students: Maxim Arap, Angel Brown, Lev Konstantinovskiy, Nathan Walters, Steven WinburnThe Hodge conjecture
This year 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.
Algebraic Geometry Group 2
Postdocs: Victor Kreiman, Jiayuan Lin
Graduate students: Michael Guy, Jerry Hower, Val Hower, Lev Konstantinovskiy, Peter Petrov, Joe RusinkoBranchvarieties
One traditional difficulty in algebraic geometry is the presence of nilpotents. For example, studying families of varieties in a projective space leads naturally to non-reduced schemes (example: the double line x2 =0) and to the notion of a Hilbert scheme (invented as the name suggests by Grothendieck).
One way to avoid nilpotents is to work with cycles, which then leads to the notion of Chow variety (invented as the name suggests by van der Warden). Both Hilbert scheme and Chow variety are very classical.
Another way to work with ordinary reduced varieties, without any nilpotents, is the theory of branchvarieties, which was invented only recently (December 2005 to be exact). The theory is brand new, and the number of open research problems is gazillion. Many classical results about Hilbert schemes and Chow varieties must certainly have analogs for branchvarieties, but they are waiting to be discovered. So that's what we will try to do.
Malcolm Adams, Caner Kazanci, Andrew Sornborger
Graduate students: Jennifer Belton, Aaron Caraher, Jacob Keenum, Leopold Matamba, Brandon Samples, Renee Zawistowski, Juhyung Yi
Undergraduates: Eric Cho, Ken YamamotoDynamical 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.
Graduate students: Ted Ashton, Aja Johnson, Steve Lane, Al LaPointe, Yang Liu, Matt Mastin, Amy Reeves
Undergraduates: 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
Faculty: Bob Anderson, Brad Bassler, David Edwards
Graduate students: Josh Hughes (physics), Tom Lanier (physics), Justin Manning, Emily Pritchett (physics)Quantum 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.