Vladimir Igorevich Arnold died two years ago, though I didn't know it until this morning. The Notices of the American Mathematical Society gave him a 22-page tribute, which I read with a growing sense of astonishment. My memories of him all revolve around one short but awe-inspiring book, which served as my ultimate reference on catastrophe theory. It's style was austere and its material was daunting, and I imagined that its author must be equally austere and distant Russian, who caused graduate students to tremble when he spoke.In reality, VI Arnold was as generous, open, and warm-hearted as it gets. Over his career he was the adviser for 52 doctoral students, a staggering number that must be close to a record. I wish I could have met him -- my professional life would probably have changed permanently. Here is what he had to say about the role of fashions in mathematics:Development of mathematics resembles a fast revolution of a wheel: sprinkles of water are flying in all directions. "Fashion" is the stream that leaves the main trajectory in the tangential direction. These streams of epigone works attract the most attention, and they constitute the main mass, but they inevitably disappear after a while because they parted from the wheel. To remain on the wheel, one must apply effort in the direction perpendicular to the main stream.I should have heard that thirty years ago.Arnold's impact on late 20th century mathematics is only now being added up and comprehended. It was immense, so much so that some have begun to compare him to Isaac Newton. For some mathematicians, like Euler, centuries must pass before the true scale and impact of their ideas becomes clear. This may be true of Arnold also.It's hard even to summarize his creative work, which deeply touched every facet of modern mathematics. He may well go down in history as the last "universal" mathematician, having had a profound knowledge of every major division of the discipline. But at present he is known best for the development of symplectic geometry and topology, with huge impact upon physics, for his work in the Riemannian geometry of infinite dimensional Lie groups, for a dazzling array of results that practically define singularity theory, for having developed and deepened the radically new notion of structural stability, for his solutions to Hilbert's 13th and 16th problems, for having invented Lagrangian and Legendrian cobordisms, for his contributions to the stability theory of Anosov systems, and for characterizing what is now known as "Arnold's Strange Duality". I'll have to stop there, but the list goes on and on. This was one hell of a mathematician.It's an honor just to have lived in his era, and to have worked in a very small way on some of the same problems that he solved with such panache. He may not have liked mathematical "fashions", but he certainly had style.Loren
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