Rebecca E. Field

Associate Professor of Mathematics

Office: Roop Hall 114
Phone: 568-4962
E-mail: fieldre at jmu.edu

I've linked a copy of my cv, but a short summary is:
I got tenure, so am now an illustrious Associate Professor!  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 recently started thinking about the motivic version of these types of computations. 


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 very old at this point).

Here are links to my preprints of the arXiv.

Since 2022 I have been participating in and help organize the Southeast Regional Workshops in Topology.  We held one on algebraic topology in July of 2022 (organized with Julie Bergner and Nick Kuhn with a NSF-RTG grant to UVA) and one in July of 2025 (organized with J.D. Quigley with a 4VA grant that I was a co-PI on).   Both of these workshops resulted in joint papers.  J.D. and I plan on applying for funding to host more workshops, starting in summer of 2027.
 
I am also working on several other projects away from my main research area.   The main one lately is in differential geometry with David Duncan and Ravi Shankar.  We've got a preprint showing that every vector bundle over every homotopy 7-sphere admits a complete metric of non-negative sectional curvature.  This extends previous results of Goette, Kerin and Shankar.  We're also working on a continuation of that paper exploring some uses of the t-invariant.  Ilarion Melnikov and I have also been investigating the cohomology of Bieberback groups using spectral sequence techniques.  We also have a preprint (with Bryce Weaver) on the arXiv computing the geodesic flow on the conifold (a relatively simple singular space) and on its small resolution.   

Other topics include a combinatorics project that I'm working on with Timothy Tarter, an undergraduate at JMU, 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 by McGuire, Tugemann and Civario).  Another is (sometimes joint with Brant Jones) is on hash algorithims (a branch of cryptography).  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 finally finished the first draft of my paper on physical properties of hyperbolic space in relation to the history of clothing and armor.  I am working on a draft that has more math in it!  The next installment was "Stumbling towards a pattern: how to make pants", was given also at the joint meetings.  I also have a one-off paper that I wrote for Bridges on something called a ribbon geodesic which is an entire tubular neighborhood of a geodesic made out of geodesics where each point in the neighborhood lies on a unique geodesic.  For surfaces, arbitrary length ribbon geodesics correspond to being able to wrap a ribbon around the surface smoothly.  Unsurprisingly, positive curvature is an obstruction to their existence in two dimensions, but the Hopf fibration gives ribbon geodesics on the three sphere.  This paper was an outgrowth of my crazy-pandemic-art-project which was to dress my dishes to go the opera.  Since no one was going to the opera during the pandemic, dressing my dishes to go was more reasonable than usual.




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 really hard to not think).  It ended up taking about six months to mostly recover (I went back to teaching just a bit too early).  It's been over eleven years now, and the last of the physical symptoms (exhaustion and nystagmus) have receded, and I am feeling fine!  I ended up spending almost two years away from math (between the extreme need for sleep that lasted over a year and the impossibility - for me - of getting real work done during the school year), but did manage to get some math done while on sabbatical.  I am now active in research once again and attending Topology Seminar at UVA as usual. Since then, I've also started doing art again (the semester program at ICERM on Illustrating Mathematics was a huge help/inspiration here).  I have an artist statement that I'm working on False eyelash Hilbert curve: iterated that's about gender expression in mathematics.

Other (non-math) Stuff

                                                                                                                                                                                                                                         I am nerdier than 98% of all people. Are you a nerd? Click here to find out!

James Madison University - Rebecca E. Field - November 1, 2025