Collected works. Volume 6.

*(English)*Zbl 1099.01024
Oxford Science Publications. Oxford: Clarendon Press (ISBN 0-19-853099-4/hbk). xxv, 1030 p. (2004).

Publisher’s description: “Professor Atiyah is one of the greatest living mathematicians and is well known throughout the mathematical world. He is a recipient of the Fields Medal, the mathematical equivalent of the Nobel Prize, and is still at the peak of his career. His huge number of published papers, focusing on the areas of algebraic geometry and topology, have here been collected into six volumes, divided thematically for easy reference by individuals interested in a particular subject. This sixth volume in Michael Atiyah’s collected works contains a selection of his publications since 1987, including his work on skyrmions, ‘Atiyah’s axioms’ for topological quantum field theories, monopoles, knots, K-theory, equivariant problems, point particles, and M-theory.”

Quoting from the Preface: “In 1988 the Oxford University Press published five volumes of my collected works, consisting of papers that had appeared by that date. In the sixteen years since, further papers were produced and it seemed appropriate to collect these into a sixth volume. This includes the two small monographs on knots and monopoles. Inevitably, with increasing administrative roles and with advancing years, a greater proportion of my papers have been surveys of one kind or another, rather than technical mathematical papers. …not all my papers, for the period 1988-2004, are reprinted here. I have been somewhat selective and only included papers which contain new results or at least a new perspective. There are in addition a few general articles which might be of interest to the mathematical community, including a number of obituaries, or celebratory articles.”

In his excellent review on [M. Atiyah’s Collected Works. Volume 5: Gauge theories. Oxford Science Publications. Oxford: Clarendon Press. (1988; Zbl 0691.53003)] Hitchin wrote: “This, the fifth volume of Michael Atiyah’s Collected works (so far!), is in many respects the most valuable. This is not a question of the mathematics contained in it being more up-to-date – the index theorem will never be outmoded – but is more concerned with the much wider audience which the author’s work in gauge theory attracted.” And, at the end of this review Hitchin noted: “He has continued to be in the forefront of the evolution of these new methods, but we need to wait a few years for volume 6 of the current work to gain the perspective which this book offers.”

Indeed, the sixth volume is in a way a continuation of the fifth volume, but it is more than that. It could have the subtitle Geometry and Physics (just look at the titles of this volume !) and offers a fantastic perspective on the links between geometry (including topology) and physics, that appeared in the last 30 years.

We quote from the paper 143 of this volume: “What we are now witnessing on the geometry/physics frontier is, in my opinion, one of the most refreshing events in the mathematics of the 20th century. The ramifications are vast and the ultimate nature and scope of what is being developed can barely be glimpsed. It might well come to dominate the mathematics of the 21st century.”

Let us quote also the type (D) of reaction by mathematicians towards these developments, which Michael Atiyah recommends in the paper 144 of this volume: “(D) Finally, and most ambitious of all, we may try to understand the deeper meaning of the physics-mathematics connection. Rather than view mathematics as a tool to establish physical theories, or physics as a way of pointing to mathematical truths, we can try to dig more deeply into the relation between them. This may led us into the perennial problem of deciding whether mathematical results are invented or discovered. This investigation may only have philosophical or theoretical interest but it could lead to better understanding and even to new insights and genuine progress.”

Quoting from the Preface: “In 1988 the Oxford University Press published five volumes of my collected works, consisting of papers that had appeared by that date. In the sixteen years since, further papers were produced and it seemed appropriate to collect these into a sixth volume. This includes the two small monographs on knots and monopoles. Inevitably, with increasing administrative roles and with advancing years, a greater proportion of my papers have been surveys of one kind or another, rather than technical mathematical papers. …not all my papers, for the period 1988-2004, are reprinted here. I have been somewhat selective and only included papers which contain new results or at least a new perspective. There are in addition a few general articles which might be of interest to the mathematical community, including a number of obituaries, or celebratory articles.”

In his excellent review on [M. Atiyah’s Collected Works. Volume 5: Gauge theories. Oxford Science Publications. Oxford: Clarendon Press. (1988; Zbl 0691.53003)] Hitchin wrote: “This, the fifth volume of Michael Atiyah’s Collected works (so far!), is in many respects the most valuable. This is not a question of the mathematics contained in it being more up-to-date – the index theorem will never be outmoded – but is more concerned with the much wider audience which the author’s work in gauge theory attracted.” And, at the end of this review Hitchin noted: “He has continued to be in the forefront of the evolution of these new methods, but we need to wait a few years for volume 6 of the current work to gain the perspective which this book offers.”

Indeed, the sixth volume is in a way a continuation of the fifth volume, but it is more than that. It could have the subtitle Geometry and Physics (just look at the titles of this volume !) and offers a fantastic perspective on the links between geometry (including topology) and physics, that appeared in the last 30 years.

We quote from the paper 143 of this volume: “What we are now witnessing on the geometry/physics frontier is, in my opinion, one of the most refreshing events in the mathematics of the 20th century. The ramifications are vast and the ultimate nature and scope of what is being developed can barely be glimpsed. It might well come to dominate the mathematics of the 21st century.”

Let us quote also the type (D) of reaction by mathematicians towards these developments, which Michael Atiyah recommends in the paper 144 of this volume: “(D) Finally, and most ambitious of all, we may try to understand the deeper meaning of the physics-mathematics connection. Rather than view mathematics as a tool to establish physical theories, or physics as a way of pointing to mathematical truths, we can try to dig more deeply into the relation between them. This may led us into the perennial problem of deciding whether mathematical results are invented or discovered. This investigation may only have philosophical or theoretical interest but it could lead to better understanding and even to new insights and genuine progress.”

Reviewer: Vasile Brînzănescu (Bucureşti)

##### MSC:

01A75 | Collected or selected works; reprintings or translations of classics |

01A60 | History of mathematics in the 20th century |

01A70 | Biographies, obituaries, personalia, bibliographies |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C80 | Applications of global differential geometry to the sciences |

53C05 | Connections, general theory |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |