Remembering Sir Christopher Zeeman

Renowned British mathematician Sir Christopher Zeeman passed away in February 2016 at the age of 91. He is best known for his contributions to the fields of geometric topology, singularity theory, and catastrophe theory. One year later, Tim Poston honors his memory. 

I did not meet Chris Zeeman again after completing my Ph.D. at the University of Warwick under his guidance—I left for Rio, and have wandered widely ever since—but he was always a vivid presence to me. Ian Stewart and I dedicated our book, Catastrophe Theory and its Applications, to him, “At whose feet we sit, on whose shoulders we stand;” we still do. Chris was the first faculty member appointed at the University of Warwick. He also founded the Warwick Mathematics Institute in 1965, which remains one of the glories of British mathematics and is now housed in a building that bears his name.

Chris began his research when topology was intensely algebraic, and achieved mastery in ‘spectral sequences’ (infinite systems of groups developed by Jean Leray in a prisoner-of-war camp) that topology maps into. But Chris’s geometric core quickly emerged, with new and deep results on piecewise-linear (PL) topology. To general surprise, this turned out to have essential differences from differential (curved) topology; for example, PL $n$ spheres can always unknot in $(n+3)$ dimensional space, like a curve in dimension 4. Chris also invented ‘tolerance spaces’—independently of Henri Poincaré, who had called them ‘physical continua’—and applied them to geography and the brain. He was eager to apply 20th century mathematics to all fields, as most ‘applied’ mathematics at the time occurred in pre-1900 areas.

This desire blossomed when Chris learned René Thom’s catastrophe theory, a mixture of deep mathematics and almost metaphysical ideas, from Thom himself. He set out to demystify it for a wide audience and apply it in many fields. The resulting impression that the ‘seven elementary catastrophes’ were single descriptors caused some controversy in the social sciences; for instance, many of  Chris’s social and biological models used one cusp catastrophe, which his readers often took as a limit. Even with a system truly governed by the bifurcations of a scalar field, where Thom’s theorem definitely says the only stably-possible bifurcations for $n$ internal variables with (for instance) two-dimensional control are folds and cusps, there is no limit on their number. Indeed, Chris’s famous ‘catastrophe machine’ had four cusps, and there is no reason cellular dynamics should not have four thousand. Thus, his models often showed fruitful and previously-unimagined possibilities for switching behavior, not inevitabilities; however, this was not well understood in the ‘soft sciences.’ In optics, buckling, laser physics, etc., such confusion did not arise, and the mathematics has continued to yield applications such as efficient simulation.

Even in $n$ dimensions—certainly in four or five—Chris had an almost tactile feeling for shapes and their changes, and an amazing gift for sharing his findings with both students and colleagues and in his famous public lectures. As his Ph.D. student and teaching assistant, I had the privilege of attending his undergraduate class in topology, which covered a variety of topics. Chris always engaged more intensely than any lecturer I have ever seen, but one day he came in without a greeting, turned his back on the class, and silently began drawing. At the top of the enormous roller blackboard he had previously drawn a row of figures like this one,identical except for different crossings, drawn as or . Below the original drawings, Chris reduced each figure step-by-step to simpler forms. For example, if the bottom two crossings showed the small loop to be on one side of the big loop, he pulled it upward, and then untwisted the remaining crossings one by one. Some figures reduced all the way to a circle; some could not. The class watched in silent fascination, until he made a slip while reducing the fourth figure. The whole class loudly objected.

Chris turned to face us. “So you agree there’s a subject!” he said triumphantly. Nothing could have conveyed more clearly that mathematics is not deduction from arbitrary axioms; axioms serve to capture objects (mental or physical), about which we have real intuitions.

My ears will never again hear him say, “What a lovely geometric argument!” But whenever I find one the phrase echoes in my mind. The news of his death made me realize how much of him, and yet how little, lives on in me and in others he taught.

A profound loss.