**Associate Professor of Mathematics**

- Office: Roop Hall 114

- Phone: 568-4962

- E-mail: fieldre at jmu.edu

This semester (Fall 2019), I'm teaching two large sections of Calculus I as well as spending some time at the Illustrating Mathematics semester program at ICERM.

I've linked a copy of my cv, but a short summary is:

I now have tenure am an illustrious Associate Professor! I spent the 2017-2018 school year on sabbatical as a visiting scholar at University of Virginia. 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 main area of 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 paper 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 we are about to submit is an offshoot of this joint work in which we explicitly compute E*G2 for any complex oriented cohomology theory using the descent spectral sequence. I gave a talk on this paper at Topology Seminar at UVA a while ago.

In anyone is interested in a more detailed summary of this research, I wrote one (summary is quite old at this point).

Here are links to a few preprints of the arXiv.

For a while, I ran a Faculty Research Seminar. It meets on Fridays 3:30-5:00pm (when there is no department meeting), in Roop 105.

I am also working on several other projects away from my main research area. The main one is with Bryce Weaver and Ilarion Melnikov on string theory (an outgrowth of Mathematical Physics Coffee Hour). We've been working on computing the geodesic flow on the conifold (a relatively simple singular space) and on its small resolution. Our preprint is now available! For a while Ilarion and I were working on a continuation of this work to the quantum setting (with the canonical model), but that ended up with differential equations too complex to make any predictions. Now Ilarion and I are talking about gerbs and starting string t heory project with David Duncan.

Other topics include a project on prosthetics and 3D printing with Greg Janson at Piedmont Technical College, Roshna Wunderlich, Klebert Fietosa, Callie Miller, and Heather McLeod, a combinatorics project left over from grad school that I need to write up one of these days and a project on sudoku (joint with Laura Taalman, Beth Arnold, Steve Lucus and sometimes John Lorch). We were thinking about the 18 symmetric clue problem (sibling to the 17 clue problem solved a few years ago December by McGuire, Tugemann and Civario). Another is (sometimes joint with Brant Jones) is on hash algorithims (a branch of cryptography). I've got a paper in progress on 'looped lightening diagrams'. In addition to this, I was working with a (temporarily on haitus) community organization called Transportation for the Public (as the group mathematician) which is trying to improve public transportation in Harrisonburg. Another is on 3D printing fabric. Mathematically, chain mail is a co-knit (its default state is fully stretched out and can be compressed, unlike knits whose default is compressed and can be stretched), so my current designs are based on this idea. I spent time training as an painter and metal fabricator, and along those lines, I'm writing a paper based on a talk I gave at the joint meetings a few years ago titled "Physical properties of hyperbolic space in relation to the history of clothing and armor." The next installment "Stumbling towards a pattern: how to make pants", was given at the next-to-latest joint meetings. There's also a project on gentle dent minimization in metal tubes, and a mild obsession with football helmets. David J. Stroll (anthropologist at Colby College in Maine) and I are vaguely thinking about stuff related to football.

In March of 2014 I was in a bicycle accident and landed on my head (I was wearing a helmet, but it was over four years old, so didn't help as much as it could have). The accident resulted in MTBI (Mild Traumatic Brain Injury) with a small subdural hematoma and a fairly bad concussion. The original injury was to the back of my head and the bleed was in the front, so for the quite a while after I had trouble translating between words and mental images. I spent the following four months crocheting doilies (I was supposed to be on complete cognitive rest, and it is

Completely unrelated to my bike injury, I turn out to have a mild form of hip displasia where the head of my femurs are slightly oblong. The main symptom of this is the fact that cartilidge in both of my hips wore out in my mid 40s. The condition is rare, and 90% of the people with tit are female and it is pretty much impossible to predict. In May of 2018 (at the end of my sabbatical), I had my first hip resurfacing (note: not a hip replacement). There are a couple of reasons I chose resurfacing over replacement. It's only really possible to get two hip replacements in a lifetime because each replacement cuts off a chunck of bone in ones femur (the second one especially). As I'm so young and my family lives such a long time, it seemed safer to postpone this process (you can still get your two replacements after a resurfacing). The other main reason was flexability and movement. According to my doctor Dr Dennis Gross of Columbia, South Carolina, I will loose only about 10% of my flexibility long term, which means I'll be able to dance (ballet) and really do I did before the surgery except perhaps running (which I already hated). I had the second one done this past December (note: three weeks is not a long enough recovery period for a job like teaching - the semester was a bit of a disaster), and now have recovered sufficiently in both hips to do anything I did before surgery (and also anything I did before my hips started to discintegrate!) include riding my bike. If anyone wants to talk about hip surgery or hip resurfacing, I'd be happy to do so.

Other (non-math) Stuff

James Madison University - Rebecca E. Field - August 28, 2016