My 2018-19 sabbatical: January - June

Summer in New Zealand, followed by Spring in France

In the summer of 2017, I decided to roll the dice and apply for a second Fulbright award, to support a period for research away from home, and make up some of the lost salary resulting from taking a full-year sabbatical (at 60% instead, of 100% for a one-semester leave). I had a Fulbright during my previous sabbatical, funding a four-month stay at Dublin City University in Ireland. The Fulbright Program funds academic exchange for university faculty (as well as graduate students, high school teachers, and other professionals), with the purpose of increasing good will toward the U.S. around the world. My previous Fulbright was pretty successful, and there seems to be encouragement for grantees to seek multiple awards, so I thought I might have a decent chance at getting one.

I couldn't decide on one place to go, and didn't want to be away from home for too long, so I decided to apply for a Global Scholar award. These allow the grantee to visit up to three countries, in at least two different hemispheres. I have always had in mind (since 1968) to escape to New Zealand, and it happens that two world-class matroid theorists work in Wellington. One of them, Geoff Whittle, I had possibly met in passing at a conference where we both spoke, in 1995, I don't really remember. But we have a common friend/coauthor, Joseph Kung, so I emailed Geoff and asked if I could come visit, and, surprisingly, he said yes. This is how I ended up living in Wellington for ten weeks. Geoff, Dillon Mayhew, and I spent many hours talking and drinking flat whites (invented in Wellington), and eventually Geoff and I proved a classification result for Orlik-Solomon algebras of graphic matroids, using an old result of his (with Dirk Vertigan) generalizing the Whitney 2-isomorphism theorem to polymatroids. Dillon and I also came close to a theorem about computation of a matroid from its lattice of cyclic flats, but there's still some work to do. Darcy came for a 2.5-week visit, and we got to take part in the Fulbright orientation, which included a night at a traditional Maori marae. We took a three-day trip to the Queen Charlotte Sound on the north tip of the South Island, and had a great time hiking and chilling. I saw the Southern Cross. I tried to learn the rules of cricket. I played frequently with local Irish, bluegrass, and old-time musicians - I was fortunate to be able to bring mandolin and mandola for the trip (two free checked bags for travel to NZ, and it happens my new mandola came with a flight case). There is an outstanding weekly Irish session, and the Welsh Dragon Pub, and they have a really great (award-winning) website: wellington.session.nz - I played with them almost every week. I visited with the Fulbright folks fairly regularly, for their morning tea, and got to know some very interesting kiwis otherwise, and learned a lot about the country and culture (part of the Fulbright deal).While in Wellington I also finished substantially revising a large paper (joint with Dan Cohen and Dick Randell), and submitted it to a special issue of European Journal of Mathematics dedicated to our dear departed colleague Stefan Papadima. I just had a wonderful, bang-up, good time in New Zealand, learned a bunch of matroid theory, and solved a problem I've carried around for a long time. Huzzah! I returned home on the Fall Equinox, March 21, arriving on the Spring equinox, March 21, for a two week visit before leaving for hemisphere #2.

The second half of the semester was spent at University of Montpellier, visiting Clément Dupont and Thomas Haettel. I had met Thomas at a workshop at MSRI on Berkeley, where he talked about his generalization (with two coauthors) of some work of Brady and McCammond on geometric structures on the beaid group. Then I found out that Clément was at the same place - he is a bright young mathematician who is working in arrangements theory, coming from the perspective of analytic number theory and sheaf cohomology. I had several projects related to the expertise of Thomas and Clément, and benefited a great deal from my talks with them. On the b=negative side, Clément helped me to understand where was the error in my argument concerning the homology of the Milnor fiber, and taught me much about sheaf cohomology. We started a related to some of my previous work in arrangements that relates to his current work on strong amplitudes in quantum field theory, but that work is barely begun. With Thomas we studied a construction from a paper of mine with Emanuele Delucchi, to try to endow it with a CAT(0) metric. Several clever ideas (of Thomas') failed to work, and we left with a difficult geometric computation left to do, which might lead to a proof of a big conjecture about Artin groups. This work remains ``in progress." During my time in Montpellier, I did my best to learn French, and played in the weekly Irish session there (Fridays at 11 pm ,,,), shopped at the le marche des Arceaux for cheese and bread, and walked the city. I had the opportunity to visit Paolo Belingeri and John Guaschi in Caen for a week. They also are thinking about graphic arrangement groups (GAGs) - and it seems like there could be some collaboration in the future. They've proved that GAGs are iterated "almost-direct products" of right-angled Artin groups (RAAGs), which should allow us to compute cohomology by generalizing an old paper of my coauthor Dan Cohe, (with Alex Suciu). I also was fortunate to spend three days and two nights with Henry Crapo (one of the early developers of matroid theory and combinatorics), who I had met briefly on a couple of occasions in the 1990's. We had a marvelous time together, touring and discussing his main project (with Gian-Carlo Rota), to make geometry truly coordinate-free through the use if combinatorial algebras. This is clearly related to questions about topology of arrangements - in fact I showed him one of my old papers (on formality) with the same picture (of a pathological example) as he used to illustrate "the matroid of circuits" of a matroid. He left me with a lot of stuff to thunk about, and a request to "prove something about the Whitney algebra." Shortly after that, on the summer solstice, I returned home to home and wife - it took us six hours to get home from the Phoenix airport because a brush fire closed the freeway. Two days later, we celebrated our 35th wedding anniversary (by cycling around the neighborhood ..).

Here's a timeline, for those who are still reading.