UGA VIGRE Research Groups, Fall 2004
Applied Mathematics
Mathematical aspects of electrical excitation and wave propagation in the heart
Andrew Sornborger
Abstract of introductory talk: This talk is meant to give a preview of the fall VIGRE group that I'll be running. I'll be talking about both the cellular basis for electrical excitation (on small scales): how a cell generates an electrical field, how the field can propagate from one cell to another. I'll also talk about how the electrical activity in cardiac tissue (on large scales) is put to use in the heart and ways in which it can fail.
Faculty: Malcolm Adams, Jerome Jungster, Andrew Sornborger
Graduate students: Jason Baldeau, Bree Ettinger, Dong Wook Kim, Samuel Obara, Jie Yu
Undergraduates: India Barber
Algebraic Geometry
Sheaves and quivers
Abstract: Recently, motivated and inspired by developments in Physics (string theory), there has been a lot of interest in the derived categories of coherent sheaves of algebraic varieties. Some recent breakthroughs include the facts that in some cases, a variety can be determined by its derived category, in other cases, this is far from being true. This is currently an extremely active and important area of research. In this VIGRE group our aim will be to gain a working knowledge of derived categories by computing them for concrete examples. For instance, the derived categories of Del Pezzo surfaces have beautiful and simple descriptions. We will attempt to generalize some of these descriptions to new cases of varieties.
For us in this VIGRE group, a coherent sheaf will be the cokernel of a map of vector bundles on a variety. Most varieties we will consider will be toric: these have simple (combinatorial) descriptions. We will start with the basic definitions of categories and derived categories and apply these definitions to our examples. I will keep the prerequisites to a minimum. If you would like more details or background on the subject please see me.
A possible continuation of the subject in the spring semester would be the investigation of what are called "stability conditions" on dervied categories. Stability is a classical algebro-geometric notion and recently it has been generalized to the context of derived categories. This has made possible some very interesting, surprising and surprisingly simple algebro-geometric constructions which we can explore.
Faculty: Elham Izadi, Mitch Rothstein
Postdocs: Jiayun Lin
Graduate students: Michael Guy, Sarah Mason (University of Pennsylvania), Peter Petrov, Joe Rusinko, Juhyung Yi.
Undergraduates: Zach Cochran
Mathematical Physics
Clifford algebra and spinors
Cal Burgoyne
Abstract of introductory talk: In physics the concept of Clifford algebras (often in disguise) play an essential role in the discussion of spin. Spin cannot be constructed by tensor methods. Pauli and Dirac Spinors can be studied in terms of Clifford algebras. Clifford algebras on Euclidean spaces of low dimension and on Minkowski space have important applications in physics. In the introductory talk we will construct the Clifford algebra Cl(2) and consider its relationship with the geometry of the Euclidean plane.
Faculty: Cal Burgoyne, Mitch Rothstein, Robert Varley
Postdocs: Nadia Mazza
Graduate students: Adam Fletcher, Sarah Hofmann, Emily Pritchett (physics), Sheree Sharp
Undergraduates: Josef Broder, Myles Akin
Number Theory
Rational points on curves
Abstract of introductory talk: A more down-to-earth version of the topic would be: Given a polynomial f(x,y) with rational coefficients, what can be said about the points (x,y) with rational coordinates x and y such that f(x,y)=0? I will give a short survey of this subject.
Faculty: Dino Lorenzini
Postdocs: Clay Petsche
Graduate students: Robert Brice, Sungkon Chang, Jerry Hower, Jacob Keenum, Nausheen Lotia, Daeshik Park, Charles Pooh, Dong Hoon Shin, Mitch Wyatt, Juhyung Yi
Algebra
Support varieties for symmetric groups and Lie algebras
David Benson, Brian Boe, Daniel Nakano
Abstract: Support varieties were developed about 25 years ago in the pioneering work of Alperin and Carlson. Since that time these module varieties have played a prominent role in the modern day development of modular representation theory. During the first month we will present introductory lectures on support varieties, the representation theory of symmetric groups/supergroups, and support varieties of Young/permutation modules (from work of Hemmer and Nakano). This will lead us to look at the following open problems:
1) calculation of support varieties for signed Young/permutation modules
2) calculation of support varieties for Specht modules
3) nucleus for symmetric groups (ordinary and block theoretic versions)
After the first month we will split the larger group into small working groups about half the time, interspersed with further development of theory as needed. Periodically we will have members of the group report on progress of the projects.
The prerequisites are a good understanding of linear algebra and group theory. Those who are interested in our previous work from last year can view our 2003-04 VIGRE Algebra Group webpage at
http://www.math.uga.edu/~nakano/vigre/vigre.html
Faculty: Dave Benson, Brian Boe, Lenny Chastkofsky, Brad Findell (Mathematics Education), Jerome Jungster, Dan Nakano
Postdocs: Jonathan Kujawa, Nadia Mazza
Graduate Students: Irfan Bagci, Phil Bergonio, Kenyon Platt, Stephen Winburn, Carrie Wright
Computational Differential Geometry
Faculty: Jason Cantarella
Graduate students: Ted Ashton
Undergraduates: Seth Dowling, John Foreman, Amy Reeves