The sixth volume of ‘Collected works’ by M. F. Atiyah contains papers published in the period between 1987 and 2004. It includes four longer contributions: the monograph on geometry and dynamics of magnetic monopoles (written together with N. Hitchin); a beautiful description of the Jones theory and its Witten’s reformulation in terms of TQFT (topological quantum field theory) described in the Lincei Lectures on the geometry and physics of knots (together with three shorter related papers); cohomological and arithmetical properties of the Dedekind η-function and relations to index theorems discussed in the Rademacher Lectures in 1987; a long paper (with E. Witten) on M-theory on a manifold with G2-holonomy (together with two companion papers on related topics and two papers on twisted K-theory and its relation to physics). The whole volume contains 49 papers. Shorter contributions fall into several different groups.

A series of six papers is devoted to unexpected relations between the spin statistic theorem, configurations of different points in the space, equivariant cohomology, representations of the symmetric group and the Nahm equation. There are two papers on Skyrmions written with N. Manton. There is a paper (with L. Jeffrey) on the equivariant Euler class in infinite dimension and a paper (with G. Segal) indicating a possible role of K-theory in string theory. There is a series of papers describing the work of outstanding mathematicians and physicists (S. Donaldson, E. Witten, V. Jones, F. Hirzebruch, R. Bott, R. Penrose, J. A. Todd, K. Kodaira, H. Weyl and I. M. Gel’fand). The turn of the century was an occasion to evaluate the evolution of mathematics. This volume includes the survey looking back to mathematics of the 20th century, which has been translated and reproduced many times. The book contains more than a thousand pages of elegant ideas, deep insights and extraordinary mathematics. It should be on the shelf of any mathematical library.

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