Postdocs:

Jonathan Kujawa
Emilie Wiesner

Graduate Students:

Irfan Bagci 
Ben Connell
Bobbe Cooper
Mee Seong Im
Wenjing Li
Kenyon J. Platt
Caroline Wright
Ben Wyser

Undergraduate Student:

Tyler Kelly

The  group continued its investigation of the cohomology of nilpotent Lie algebras
in prime characteristic. The cohomology in characteristic 0 has been known for
a while by Kostant's Theorem. The goal is to determine when extra cohomology can
occur in characteristic p.

The group had realized that a result obtained the previous year: that any extra cohomology that
occurs must have weight of the form $w.0+p\nu$, was very useful. This was the basis of Boe's
outline of a proof during the previous semester that there was no extra cohomology when p>h-2.
He had given an inductive argument to show that if any extra cohomology occurs, then there must
be an element $\nu$ in the weight lattice such that $p\nu$ is equal to a sum of distinct positive roots.
The proof would be finished if we could show that the only way this could happen was when p=h-1
and $\nu=\alpha_0$.

Nakano noted that an argument of Jantzen shows that there are strong restrictions on how large such
a $\nu$ can be. Chastkofsky then produced an argument which showed that if one could prove certain
inequalities for certain root sums, one could get the desired results. He proposed proving these by actually
proving explicit formulae for these sums, which he had conjectured by computer calculations. These were to
be proved case by case and students were divided into 3 groups, each group assigned one of the cases of type B,
C or D. (Exceptional cases could be covered by explicit computer calculations.)

The groups had varying degrees of success at completing their assignment, with at least one group
coming up with a more or less complete prooffor their case. However, Chastkofsky eventually came up with a
general proof which made the case by case analysis unnecessary.

The group also worked on the following problem: If p<h-1 is there always extra cohomology?
The group had answered this question affirmatvely last year in type A. The group began exploring other cases
using computer calculations, again dividing into smaller groups. Some groups were able to come up with
conjectures as to where this extra cohomology occurs, but some cases proved more challenging.
The exceptional types also proved a challenge here, with the computations taxing available computing resources 
in some cases.

One example that was worked on was $F_4, p=7$. We  first tried using Magma to guess at weights giving extra
cohomology, but our first guesses were unsuccessful. Chastkofsky then used Mathematica to come up with a list of
9 possiblities which would give extra cohomology, and the 9 students were assigned one each to
test further conditions if these could produce the desired weights. Wright however did all the homework
for the students by coming up with a computer program doing the calculation for all 9 cases, and verifying
that some of these are possible. Magma then confirmed that one does get extra cohomology in these cases.

Again, these computations became unnecessary for the proof when Nakano produced representation theoretic ideas
that showed that when $p<h-1$, additional cohomology must occur. However the proof is non-constructive and the
explicitcomputations that the group has done may still be of interest.

Towards the end of the semester, Chastkofsky noted that Boe's argument actually shows that if a weight occurs giving
extra cohomology then it must be of the form $w.p\mu$ where $p\mu$ is a sum of distinct negative roots. Cooper and Wright
had shown the previous year that in type $A_n$ at least, there is always extra cohomology with weight $p\alpha_0$,
by coming up with explicit elements $v$ such that $dv=pz$ for some $z$. (Wright presented this argument
again for the benefit of those who had not been with the group last year.) This leads to the following possibility: perhaps
one can extend the Cooper-Wright argument to get other elements $v$ with weight $p\mu$ such that $dv=pz$ and such that 
$dwv=wdv$ for some $w$ in the Weyl group, and perhaps one can get many more examples of extra cohomology this way.
Looking at some examples of known extra cohomology shows that they doindeed occur this way. One can even hope that all
extra cohomology occurs this way. This will be an area for further investigation.

Nakano presented the results from the VIGRE group at the AMS Sectional Meeting in Davidson, NC in March .
Boe has started writing up the results obtained by the group in TeX. Wright has drafted a version of her result with Cooper.