The VIGRE Algebra Group was led by Brian Boe and Leonard Chastkofsky.
The other members of the group are:

Postdocs:

Jonathan Kujawa
Nadia Mazza
Emilie Wiesner

Graduate Students:

Irfan Bagci
Bobbe Cooper
Sarah Hofmann
Kenyon J. Platt
Sheree Sharpe
Caroline Wright

Undergraduate Student:

James Dabbs


The main goal of the VIGRE Algebra Seminar this term was to continue the investigation, begun
last fall, of the nilpotent cohomology of Lie algebras. In the first three weeks, Boe presented a proof of
Kostant's Theorem on nilpotent cohomology in characteristic zero, the result we are trying to extend to
(small) prime characteristic. Then Chastkofsky reviewed our conjectures from last semester regarding
where "extra" cohomology can appear in characteristic p, focusing on Lie algebras of type A and small
rank. These conjectures arose by examining calculations using the MAGMA program Boe has been writing.
By examining some of these calculations more closely, Chastkofsky was able to predict some general formulas
for specific cocycles which should induce new cohomology.

The students have proved some of these formulas. Working in groups of 2 or 3, they have all become involved
in running the MAGMA program to generate new data. They have presented their results during the
seminar. They have actively been working to refine the conjectures and produce completely explicit formulas
for new cohomology in arbitrary rank.

In particular, Cooper and Wright gave a joint hour-long presentation, followed by an additional half-hour presentation
the following week by Cooper. A few weeks later Cooper and Hofmann gave a similar joint presentation. Wiesner took on the
task of proving some of the conjectures, in particular the exhaustion of extra cohomology, using the theory of Weyl groups
and combinatorics of weights. She presented her results near the end of the semester.

As more of these calculations and results have been done, a better understanding of the combinatorics involved in the
problem has been obtained. Boe has been continually rewriting the program as the needs for what computations need to
be done has become clearer. We can now understand how extra cohomology arises in a large number of cases.
The goal is to see if by continuing this examination, we can account for ALL the extra cohomology in all cases. We have
already made good progress. We also hope to get the students more involved in the actual programming if the department
acquires a Mathematica license. We plan to continue this investigation in Summer of 2006.