by Martin Scharlemann
Abigail Thompson was born on 30 June 1958 in Norwalk, Connecticut, the youngest of four children. Her father was an industrial chemist; her mother was a long-time editor of the journal Current History, the oldest American publication exclusively devoted to international affairs. Her two brothers both became lawyers, the younger after an earlier career as violinist for the Utah Symphony. Her sister became a medical doctor.
Abby grew up in the rural countryside, with lots of time on her hands for reading. The World Book Encyclopedia, aimed at younger readers and targeting its articles at the reading level of the expected audience, became a window onto a different world. She always liked math and remembers, at age twelve, first encountering in the World Book a description of the Möbius band. She was particularly amazed that she could cut it down the middle and it remained a single piece. And so began her life with topology.
Abby attended Philadelphia’s Germantown Friends School through the 11th grade, leaving a year early to attend Wellesley College. Her mother had graduated from Wellesley and thought it was a great place; Abby was attracted by the beauty of the campus. Although the liberal arts college did not particularly focus on mathematics, Abby treasures the recollection of a four-student topology class taught by Ann Stehney, using the Moore method (she still has the notes from the class).
As she approached graduation from Wellesley, Abby was torn between music (she is a cellist) and math; she had majored in both. Eventually she decided to begin graduate school in mathematics at New York University (NYU). These were the days before REUs, and Abby found herself underprepared for graduate study; unhappy at NYU she left after two years and moved on to Rutgers. She found the faculty at Rutgers very supportive, but wandered a bit between advisors, bothered that the algebraic topology and high-dimensional topology she encountered in her courses seemed to have little connection to what had originally attracted her to the subject. That all changed when she took a low-dimensional topology course from Bill Menasco, who was at Rutgers on a postdoc. Abby began very happily to work with Bill, but his appointment at Rutgers was temporary, and it was not clear what Abby’s next step should be.
Rob Kirby had, for several years, encouraged his students, former students, and his research colleagues to visit Cambridge University during summer academic breaks. In support, Cambridge’s Raymond Lickorish made extensive arrangements for housing and ongoing mathematical activities. In 1983, as Bill Menasco prepared for his last year at Rutgers, he approached Rob about bringing Abby along for a summer in Cambridge. Rob enthusiastically agreed, providing support not only for her summer visit, but also for her participation in the following special year in low-dimensional topology at the Mathematical Sciences Research Institute (MSRI). It was a pivotal year for Abby: beyond all the mathematical activities in Cambridge and at MSRI, she made substantive advances towards her PhD dissertation in knot theory. And she met her future husband, Joel Hass.
Whether it was math or soccer that brought Abby and Joel together is lost to history. Frequent soccer had become a part of the Cambridge experience for Kirby students, and this tradition continued into MSRI’s special year. Abby and Joel were both enthusiastic participants, though Abby’s background in field sports was shaky: she recalls that at Germantown Friends School, mandated to participate in sports, she was relegated to playing goalie in lacrosse, on Team 7 (out of 7). In any case, Abby and Joel met at MSRI, and married two years later.
In the meantime, Abby decided to finish her Rutgers PhD work at UC Santa Barbara, after I recruited her for a special year in topology there. Among the luminaries (and future luminaries) who also came were Raymond Lickorish, Erica Flapan, Jim Van Buskirk, and Steve Bleiler. These were augmented by frequent short term visitors, including Joel Hass and Francis Bonahon. In January 1986, Abby passed her required French Exam, just in time for her dissertation defense, both in the same week, and was awarded her Rutgers PhD. Julius Shaneson was her official Rutgers advisor.
Following her PhD, Dr. Thompson was awarded a series of prestigious fellowships: a Lady Davis Fellowship to Hebrew University in Jerusalem, a University of California President’s Fellowship to Berkeley, then Davis, and an National Science Foundation Postdoctoral Fellowship. Following her appointment as an Assistant Professor at University of California, Davis, she was further awarded an Alfred P. Sloan Foundation Research Fellowship, and then three separate years as a member of the Institute for Advanced Study in Princeton.
Among the highlights of her academic career was her contribution to the solution to the problem of recognizing the 3-sphere, a problem that had engaged low-dimensional topologists for several decades. In recognition of this and other work in low-dimensional topology, in 2003 the American Mathematical Society awarded her the Ruth Lyttle Satter Prize in Mathematics. The Satter prize is awarded every two years “to recognize an outstanding contribution to mathematics research by a woman in the previous five years”.1
I’m proud to have been able to collaborate with Abby for several decades following her PhD. Probably the most accessible theorem to come out of that work was our paper “Detecting unknotted graphs in 3-space” (J. Differential Geom. 34 [1991], pp. 539–560), in which we showed that there is an algorithm to decide whether a graph embedded in 3-space could be isotoped into the plane. I first heard of the problem from Fico González Acuña, quoting Iowa’s Jonathan Simon, who had conjectured a solution: the graph is planar if and only if its complement has free fundamental group and every proper subgraph can be moved into the plane. Abby learned of the conjecture from Simon himself, on a visit to Iowa, and suggested we work on it. Our first thought was that, because the question is about the 3-sphere, a new idea we could try was the use of “thin position”. This was an idea introduced by Gabai for knots in the 3-sphere and one we were trying to understand. In the end, thin position wasn’t needed; the proof we found could have been found much earlier. I think what we did bring to the table was a different viewpoint on the problem. In any case, we were pleased to learn that our solution of the “graph planarity problem” settled a \$20 bet between Cameron Gordon and Jon Simon on its eventual solution.
Another joint paper that should be mentioned is our “Thin position for 3-manifolds” (pp. 231–238 in Geometric Topology, Contemporary Math. 164, Providence, RI: Amer. Math. Soc., 1994), currently the most cited publication for each of us. (Math Reviews counts 106 citations as of March 2024.) As I recall the history: we were at a geometric topology conference in Haifa and, finding ourselves in the same place at the same time, thought we should work on something together. One idea that came up, rather casually, was whether thin position had anything to say about 3-manifolds in general, not just about knots in the 3-sphere. Without a lot of effort we realized that it could indeed be made to say useful things about Heegaard splittings. The output of the process fit in well with contemporary concerns: incompressible surfaces in 3-manifolds, and weak reducibility of Heegaard splittings. Rather than elation over what has turned out to be one of our most useful ideas, I think at the time we felt a bit disappointed about how easy it was, and so it became a rather hastily written contribution to the conference proceedings.
Beyond her scientific impact, Abby’s work has extended in many directions. She has been vigorously engaged in campus and professional work, serving, for example, as a member of the Executive Council of the Academic Senate of UC Davis, and as Chair of the UC Davis mathematics department. Most recently she has served as Vice President of the American Mathematical Society and as Secretary of the Association for Mathematical Research. For many years she ran the COSMOS program at UC Davis, a residential summer program for highly talented high school students. She discovered, in the tide pools of Dillon Beach, what may be a new species of marine worm (the absence of a corpse has muddied the waters around whether the little critter can be certified as a new species by the scientific community.) In further marine biology research, she was cited in a marine biology publication for her northernmost (at the time) sighting of a species of sea slug, a troubling harbinger of climate change. In a completely different direction, she was awarded the title “Hero of Intellectual Freedom” for 2020 by the American Council of Trustees and Alumni (ACTA) and was feted with a breakfast in her honor at ACTA’s 25th anniversary celebration in Washington, DC. The award recognized her work exposing efforts by the UC administration to undercut the primacy of academic oversight in the hiring of new academic appointments.
Abby and Joel have three children: Ellie and her husband, Eddie, live in nearby Marin, raising baby Jacob; Ben works in IT for various start-ups in Brooklyn; Lucy is a student in Tel Aviv.
Martin Scharlemann received his Ph.D. in mathematics from UC Berkeley in 1974. Following a year at the Institute for Advanced Studies and a year at the University of Georgia, he moved to UC Santa Barbara, where he is now a Distinguished Professor Emeritus.
