COMMUNICATION Remembering Mark Mahowald 1931–2013

Douglas C. Ravenel with contributions by Martin C. Tangora, Stewart B. Priddy, Donald M. Davis, and Mark J. Behrens

Introduction Mark Mahowald was a dominant figure in algebraic topol- ogy for decades. After being trained as an analyst he became a self-trained homotopy theorist in the early 1960s. His 1967 AMS Memoir The Metastable Homotopy of 푆푛, with its dozens of charts packed with arcane in- formation, established him as a master of computations related to the homotopy groups of spheres. It came to be known in the field simply as “the red book.” It was the unanimous opinion of experts that he knew this subject far better than anyone else. His insight and intuition were legendary. Countless coworkers benefited from his ideas and advice. Mark first attended college at Carnegie Tech in Pitts- burgh and planned to major in chemical engineering. After two years, knowing that his father could not afford Photo courtesy of Martin Tangora. to support him any longer, he joined the naval ROTC so Mahowald sailing on Lake Michigan with colleague he could support himself with the four-year scholarship Carl Verhey in 1968. that they offered. At the same time he transferred to the University of Minnesota and changed his major to mathematics “because I could get a degree faster that hired four more algebraic topologists and became one way.” of the leading centers of homotopy theory in the world. In 1963 Northwestern hired him as a full professor Mark was the prime mover in the creation of the Midwest at the age of thirty-one. Within a decade Northwestern Topology Seminar, which meets three weekends a year to this day. Doug Ravenel is Fayerweather Professor of Mathematics at He had a lifelong passion for sailing. He built himself a the University of Rochester. His email address is doug@math. small sailboat as a teenager. After moving to Chicago he rochester.edu. became involved in racing on Lake Michigan. He named Martin C. Tangora is retired from the Department of Mathemat- three succesive sailboats Thetajay, a reference to the ics, Statistics, and Computer Science at the University of Illinois at notation 휃 for certain hypothetical elements in the stable Chicago. His email address is [email protected]. 푗 homotopy groups of spheres. Stewart B. Priddy is professor emeritus in the Department of Math- ematics at Northwestern University. His email address is priddy@ math.northwestern.edu. Recollections of Students and Colleagues Donald M. Davis is professor of mathematics at Lehigh University. His email address is [email protected]. Martin C. Tangora Mark Behrens is John and Margaret McAndrews Professor at the When Mark Mahowald came to Northwestern in 1963, I University of Notre Dame. His email address is [email protected]. was very discouraged. It was not at all clear how I could For permission to reprint this article, please contact: finish my PhD before the time limit ran out. Around that [email protected]. time Peter May came to the Chicago area to talk about DOI: http://dx.doi.org/10.1090/noti1391 his thesis, and Mahowald and Bob Williams told me that

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Photo courtesy of Donald M. Davis. twenty that had already fallen did not deter Mark from off amazing theorems to young mathematicians whose driving me and the rest of the NU topology group to the task was to provide a proper proof. Midwest Topology Seminar at the University of Chicago. Mark’s mathematics is best described as nontraditional. We were about the only car on the road. This was Mark; Arrows in his diagrams often lacked heads, for he was nothing could stop him. inevitably going to get the direction wrong. Once when he was talking about duality, I noticed he subconsciously References started to write “표푑” instead of “푏표.” Rob Thompson [BDM02] Robert R. Bruner, Donald M. Davis, and Mark recently told me that Mark said he wasn’t concerned that Mahowald, Nonimmersions of real projective spaces the 푏표-based Adams spectral sequence didn’t have an implied by 푡푚푓. In Recent Progress in Homotopy Theory 퐸2-term expressible as an Ext group, because he “never (Baltimore, MD, 2000), volume 293 of Contemp. Math., understood homological algebra anyways.” Indeed, for Amer. Math. Soc., Providence, RI, 2002, pp. 45–68. Mark everything could be thought of as a CW complex MR 1887527 and drawn with his idiosyncratic “cell diagrams.” [DM82] Donald M. Davis and Mark Mahowald, The nonreal- Mark’s talks were always difficult for folks to under- izability of the quotient 퐴//퐴2 of the Steenrod algebra, Amer. J. Math. 104(6):1211–1216, 1982. MR 681734 stand. A geometer once told me about a talk Mark gave which began “In this talk, all spaces are mod 2.…” He told me he didn’t understand this or anything that followed. Mark J. Behrens His writing was equally difficult to grasp at times. Perhaps I first met Mark Mahowald as a prospective graduate he was gifted with an extraordinary vision and cursed with student visiting Northwestern University. As an under- an inability to put it into words. I, however, believe that graduate I had been studying homotopy theory with there are no words adequate to describe his incredible Norihiko Minami, a graduate from the Northwestern world-view of computational homotopy theory. school of topology, and Mahowald met with me for a full For Mark, mathematics was not so much about the hour to talk about his current research. I was immediately big theorem as understanding the bigger picture. The bewildered with some weird mixture of the Steenrod homotopy groups of spheres were full of mysteries, and Algebra, certain polynomial equations over 픽4, and the he was the oracle. His standards of proof were different random appearance of the group 푆퐿2(픽3). Frankly, I had than is typical in our field—at times it felt he was no idea what was going on but courteously pretended more like a physicist with brilliant intuition than an that I did. uptight mathematician bound by the restraints of rigor. Only later did I discover that he was trying to tell Mark told me that the Kervaire invariant elements were me about the supersingular elliptic curve at the prime 2 “simultaneously 푣2-periodic and 푣푛-periodic for all 푛”—a and his latest gadget, Topological Modular Forms (tmf for paradoxical situation that somehow strengthened Mark’s short, but back then it was called 푒표2). I remember as a hope to prove they all existed, but as Hill-Hopkins-Ravenel first-year graduate student at Peter May’s sixtieth birthday later discovered, forced their eventual nonexistence. For conference joking with my fellow graduate students about Mark, a theorem was proven by understanding why it the prospects of “푒표2 resolutions” after hearing a similarly was true, by placing it in a philosophical setting where incomprehensible talk Mahowald gave at the conference. its truth was so self-evident he could come up with a Little did I know that in a few short years I would be half dozen different arguments for its validity. He always hanging onto every word that this man said and shaping told me that when doing a computation, arrive at the my career around the stuff he was talking about. answer through two independent means so as to make Indeed, I was bitten by the computation bug the sure you don’t make a mistake. I think Adams summed following year, and my advisor, Peter May, suggested I go up Mahowald’s enigmatic brilliance the best in his review up to Northwestern to talk with Mahowald about possible of his paper “The primary 푣2-family”: thesis projects. It was not long until I was commuting This is mathematics as it is lived; some of it seems up to Northwestern every week to talk to Mark to learn like work in progress. May it continue to progress. about his unique perspectives on the stable homotopy groups of spheres. Amazingly, he would devote his entire afternoons to talking with me. Talking with Mark was like Douglas C. Ravenel learning a language by the immersion technique. At first, Mark Mahowald was an inspirational mathematician. I I had no idea what he was trying to tell me. Each week, I met him in the mid-1970s, early in my career. He was would ask him the same questions, and he would politely hugely encouraging and made me feel like I had just met pretend I hadn’t asked him the same question the week a rich uncle I never knew I had. Like everyone else, I was before and give a different explanation, until one took. impressed by the depth of his intuition and insight. I got His vision, passion, and mathematics will dominate my the impression he could do the Adams spectral sequence own work for the rest of my life. in his sleep. I would later learn that he regularly took graduate If you do not know what the Adams spectral sequence students and postdocs under his gentle wing in this is, you will not learn it here. Suffice it to say it is an manner, initiating them in an oral tradition of homotopy exquisite mystery that hard-core homotopy theorists like theory that cannot be learned from any paper or book. Mahowald have a long-term love-hate relationship with. Mark was generous with his ideas and would freely hand He could draw charts of it with abandon, explaining

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