the part played by periodicity in K-theory. Bott made important contributions towards the index theorem, especially in formulating related fixed-point theorems, in particular the so-called 'Woods Hole fixed-point theorem', a combination of the Riemann–Roch theorem and Lefschetz fixed-point theorem (it is named after Woods Hole, Massachusetts, the site of a conference at which collective discussion formulated it). The major Atiyah–Bott papers on what is now the Atiyah–Bott fixed-point theorem were written in the years up to 1968; they collaborated further in recovering in contemporary language Ivan Petrovsky on Petrovsky lacunas of hyperbolic partial differential equations, prompted by Lars Gårding. In the 1980s, Atiyah and Bott investigated gauge theory, using the Yang–Mills equations on a Riemann surface to obtain topological information about the moduli spaces of stable bundles on Riemann surfaces. In 1983 he spoke to the Canadian Mathematical Society in a talk he called "A topologist marvels at Physics". He is also well known in connection with the Borel–Bott–Weil theorem on representation theory of Lie groups via holomorphic sheaves and their cohomology groups; and for work on foliations. With Chern he worked on Nevanlinna theory, studied holomorphic vector bundles over complex analytic manifolds and introduced the Bott-Chern classes, useful in the theory of Arakelov geometry and also to algebraic number theory. He introduced Bott–Samelson varieties and the Bott residue formula for complex manifolds and the Bott cannibalistic class. Awards In 1964, he was awarded the Oswald Veblen Prize in Geometry by the American Mathematical Society. In 1983, he was awarded the Jeffery–Williams Prize by the Canadian Mathematical Society. In 1987, he was awarded the National Medal of Science. In 2000, he received the Wolf Prize. In 2005, he was elected an Overseas Fellow of the Royal Society of London. Students Bott had 35 PhD students, including Stephen Smale, Lawrence Conlon, Daniel Quillen, Peter Landweber, Robert MacPherson, Robert W. Brooks, Robin Forman, Rama Kocherlakota, Susan Tolman, András Szenes, Kevin Corlette, and Eric Weinstein. Smale and Quillen won Fields Medals in 1966 and 1978 respectively. Publications 1995: Collected Papers. Vol. 4. Mathematics Related to Physics. Edited by Robert MacPherson. Contemporary Mathematicians. Birkhäuser Boston, xx+485 pp. 1995: Collected Papers. Vol. 3. Foliations. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xxxii+610 pp. 1994: Collected Papers. Vol. 2. Differential Operators. Edited by Robert D. MacPherson. Contemporary Mathematicians. Birkhäuser, xxxiv+802 pp. 1994: Collected Papers. Vol. 1. Topology and Lie Groups. Edited by Robert D. MacPherson.