Rebecca E. Field
Assistant Professor of Mathematics
- Office: Roop Hall 114
- Phone: 568-4962
- E-mail: fieldre at math.jmu.edu
This semester (Spring 2009), I am teaching Math 232, Integrated
Calculus. The webpage for 232 is here.
I've linked a copy of my cv,
but a short summary is:
This
is my first year as a tenure track assistant professor at JMU. My
last jobs were visiting scholar positions at Reed College in Portland,
Oregon and Cambridge University in the UK. Before that I was at
Bowdoin College, UC Santa Cruz and University of
Wisconsin-Madison. I got my PhD from the University of Chicago in
August of 2000. My BA (in mathematics and studio art) is from
Bowdoin College.
My research is on the interactions between algebraic geometry and
algebraic topology, particularly actions of algebraic groups on
varieties. One tool to study group actions is the classifying
spaces of the group, which encodes all possible actions, and one way to
study these classifying spaces is to look at their invariants.
For example, if one is interested in characteristic classes of
principal G
bundles over smooth algebraic varieties, one would look at the Chow
ring of the classifying space BG in the sense of Totaro (this is a
limit of Chow rings of finite dimensional approximations of BG - the
Chow ring is the ring of algebraic cycles mod rational
equivalence).
I have an exciting new paper coming out joint with Ian Grojnowski
(preprints available on request) "BSO(2n) as an extension of BO(2n) by
BSp(2n)" in which we show that for any
cohomology theory, there is a copy of the cohomology of BSp(2n, C)
sitting inside the cohomology of BSO(2n,C). This is despite the
fact that there is no map between SO(2n) and Sp(2n). Moreover,
that copy of BSp(2n) encodes the difference between the cohomology of
BO(2n,C) and that of BSO(2n,C). This is particularly nice both
because BO(2n) and BSp(2n) are more thoroughly understood than BSO(2n)
and because this is a very strong generalization of the Langlands
transfer map from the representation ring of SO to the representation
ring of Sp (recall Sp(2n) and SO(2n+1) are Langlands dual; the map of
representation rings comes from SO(2n) contained in SO(2n+1)).
This transfer map gives a map from the K-theory of BSO to the K-theory
of BSp (since K theory is just the representation ring completed at the
augmentation ideal), but not only does it lift to all other
cohomology theories, but we have a map lifting it to the level of
classifying spaces, albeit in a highly non-geometric way.
The paper I am currently finishing up is an offshoot of my joint work
with Grojnowski in which we explicitly compute E*G2
for any complex oriented cohomology theory using the descent spectral
sequence.
In anyone is interested in a more detailed research summary, I wrote one.
Here are links to a few preprints of the arXiv.
Other
(non-math) Stuff
James Madison University -
Rebecca E. Field - 25 August,
2008