Wow. I just read Eric’s post over at Curious Reasoning, and I have to say that I am all fired up to start blogging directly in LaTeX. Plus, I’m listed in his blogroll! If I don’t have to jump through all those crazy hoops to typeset LaTeX on the blog anymore, what excuse do I have not to write? I mean, besides the excuses of teaching Calculus, doing research, writing up results, going to seminars, developing my CV, keeping an eye out for job prospects…
So, what have I been up to teaching and research-wise lately? Our semester started two weeks ago today, though I was gone most of last week while attending the Joint Mathematics Meetings. I didn’t go to very many research talks (I even skipped my own advisor’s talk so I could go see Alcatraz with Katherine and Jimbo!), but it was probably the most productive week I’ve had in a long time when it comes to writing up research results.
I’m teaching Calculus II, and as I mentioned in a previous post, I’m doing my best to implement some new course policies this semester. The biggest change, and probably the single greatest increase in work for me, is that I’m going to have students rewrite homework solutions if they don’t meet my (high) standards. I just finished grading the second written homework assignment today, and a lot of students will be doing rewrites because of notational errors (e.g., writing an equals sign when two things aren’t really equal, or not changing the bounds of integration when integrating by substitution). I need to do a good job of selling this rewriting enterprise when I hand back the homework tomorrow. If I don’t, then I’m sure I’ll get slammed when it comes to teaching evaluations at the end of the semester.
Research-wise, my research advisor Dan Nakano is currently 9300 miles away (more or less) in Sydney, Australia, spending the semester on sabbatical at the University of Sydney. And you know what they say, “When the research advisor is away, the postdocs will play.” We’re in contact by email, but he’s only been there a few days, so I haven’t heard very much lately.
But seriously, I have a number of projects on my plate that I am or should be working on. One project is writing up the results from my thesis for publication. I artificially set a deadline of the end of this month to have a draft of that project completed, but that’s definitely not going to get done before the end of February.
The second research project I’m working on is joint work with Dan and his PhD student Nham Ngo. We’re looking at the ring structure of the cohomology ring
. Here
is the first Frobenius kernel of the unipotent radical of the Borel subgroup of a simple, simply-connected linear algebraic group
over an algebraically closed field
of characteristic
. Equivalently, we’re studying the cohomology ring for the restricted enveloping algebra of a certain nilpotent Lie algebra
(the nilradical of the Borel subalgebra of the Lie algebra of
).
Anyway, it has been known for something like 25 years (due to an observation of my advisor Brian Parshall) that there is a filtration on
such that the associated graded ring is isomorphic to
, the tensor product of a symmetric algebra (polynomial ring) with the ordinary Lie algebra cohomology of
. And in her 1983 PhD thesis, UVA graduate Ronny Crane proved that this lifts to an ungraded ring isomorphism in type
, assuming
is greater than the Coxeter number
of
. But nobody seems to have studied this problem very much since then.
So, Dan, Nham and I started looking at this problem before Dan left for down under. We’re pretty sure we can extend the ring isomorphism to all Lie types, provided
. We ran into some trouble trying to lower the bound to
, but I had some ideas last night that might enable us to do it. The tricky part is that my idea involves getting our hands dirty with explicit cocycle representatives, and arguing that certain coproducts are zero by explicitly showing that certain maps are coboundaries. It could get messy.
More as it develops…