The VIGRE Algebra Group was led by David Benson, Brian Boe, and Daniel Nakano. The other members
of the group are

Faculty:

Leonard Chastkofsky
Jerome Jungster

Postdocs:

Jo Jang Hyun
Nadia Mazza

Graduate Students:

Phil Bergonio
Bobbe Cooper
G. Michael Guy
Graham Matthews
Kenyon J. Platt

The main goal of the VIGRE Algebra Seminar was to study the structures of certain varieties
of nilpotent matrices associated to algebraic groups and their Lie algebras. These varieties are
generalizations of the ``restricted nullcone'' which manifests itself in the study of support varieties
for restricted Lie algebras. Carlson, Lin, Nakano and Parshall have recently described the restricted
nullcone for all types over fields of good characterstic. Their proof employed the verification of a
conjecture by Jantzen on the support varieties of Weyl modules which was due to Nakano, Parshall
and Vella.

During the semester material was presented on basic Lie theory and conjugacy classes along with
methods due to Nakano and Tanisaki on intersecting orbit closures. The material was presented by Boe
and Nakano. These methods were then adapted for the purpose of computing these varieties of nilpotent
matrices for faithful representations of the simple Lie algebra. For the exceptional Lie algebras, we broke
the large group into smaller groups led by Benson, Boe and Chastofsky. The work was then subdivided
and each group was in charge of certain computations. This was very successful because the entire group
was engaged and involved in the process, and led to the completion of the computation of these varieties
for exceptional groups over good characteristic for the minimal and adjoint representations. The graduate
students were also involved in writing this section of our work.

At first, we did not know how to handle the case when the groups were classical and the representation
was the adjoint representation. However, after several weeks of thinking about the problem, we were able
to solve it. Much of the credit for this part of the project is due to Boe with contributions from Benson and
Nakano.  We now have a paper which is almost complete in which we determine explicitly (as a union of
orbit closures), by elementary methods, the order-r nilpotent elements for the minimal and adjoint
representations, for all r and all simple Lie algebras in good characteristic.  As a corollary, we successfully
verified using more elementary methods the computation of the restricted nullcone (the case r=p).

The group has begun work on looking at these varieties over bad characteristics. Each graduate student went
to the library to look up pertinent literature on the topic and gave a short talk on their findings in the seminar.
These findings have led us to use available computer software to do some of these computations. Markus
Hunziker recently gave a talk in our seminar on the construction of Chevalley basis. Hunziker, Matthews, and
Mazza are looking at ways to implement the necessary calculation on MAGMA. Once this is done, we hope to
be able to compute the restricted nullcone for bad characterstics (an open problem of interest). In future work,
we hope to find new results on support varieties for Weyl modules in bad characteristic.