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Though the environment was one of tranquil academic scholarship, I lived charged with anticipated performance anxiety about the seminars on the brain as a dynamical system I was scheduled to present to these (I feared) ready-to-be-disdainful, prize-winning, pure mathematicians and theoretical physicists.

My dorm-style sleeping room at IHES was, in winter, painfully cold and drafty; the narrow iron bed’s thin mattress contained lumps of persistently disturbing dreams, the small scratched table for work shim-irreparably wobbled. A faded poster of Van Gogh’s garden was tacked crookedly on the door facing the toilet in the dank, dimly lit small bathroom. A dwelling for distracted young mathematicians. A retired but still famous Parisian chef cooked many course, elegant meals every afternoon. The food was accompanied by so many liters of unlabeled red wine and peer pressure to be French and socially drink it that it became a choice between dulled, blunted, sleepy post-prandial afternoons or living on bread, many cheeses, apples and Perrier water, alone in my room. I chose the latter.

Thom’s gifts to us theoretically oriented non-mathematicians were diagrammatic, easy-to-visualize pictures that allow the intuitive capture of counter-intuitive discontinuities in functions. How we might imagine that a smooth and continuous change in a cause of something can lead to a big, discontinuous change in the results. His system of topological (shape not size) diagrams was useful when considering up to four causal variables and one to two dependent variables that described how things behaved.

For an important real life example, in modern clinical pharmacology, the smooth dose-response curve consistent with the physician’s intuition that if a little drug didn’t work, a little more may do so, should become an up and down search for the dose-region for the desired effect which may involve a lower amount than a previously ineffective drug dose. The therapeutic effect may occur in the middle of a narrow dose range with too much or no effect occurring out of this span. In many physical systems, sudden and global transitions in state, from incoherent light rays to coherent lasing and from laminar flow of fluids to turbulence, emerge unexpectedly when causal parameter are moved into what some call the critical region of the values of control parameters. Outside this region, cause and result

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