#### Commentary by M. Atiyah

In the detailed commentaries to the earlier volumes, there is an extensive
account of my main mathematical collaborations, More recently, I was asked to provide an integrated analysis of these collaborations, both from the mathematical and personal points of view. This appeared as
[72]
and is appropriately the first article in this volume
[6 of the *Collected works*].

My monograph
[8]
written jointly with
Nigel Hitchin,
and based on the Porter lectures I gave at Rice University in 1987, is now out of print and so is included in Volume 6
[of the *Collected works*].
It provides a full account of the results announced briefly in papers
[1]
and
[2]
which appeared in Volume 5. The main result is the explicit determination of the natural metric on the moduli space of centered __\( \mathit{SU}(2) \)__-monopoles of charge 2. The key property which is exploited is that the metric is hyperkähler, a property arising from super-symmetry, and of increasing interest to geometers and physicists. An expository account of hyperkähler manifolds is contained in
[17].
The next paper
[4]
is a lengthy one centering round cohomological and arithmetical aspects of the Dedekind __\( \eta \)__-function, interpreted in terms of the index theorem in various forms. It was the subject of the Rademacher Lectures that I gave at the University of Pennsylvania in 1987.

The spectacular results of Donaldson on 4-dimensional manifolds which led to his Fields Medal are briefly described in the citation [3] I presented at the ICM in Berkeley in 1986. Four years later, there was a significant follow up when I wrote the citation [20] for Witten’s Fields Medal at the Kyoto ICM (1990). Witten is the only physicist so far to have received the Fields Medal and, although some eyebrows were raised at the time, there are few who would now dispute the enormous influence of his work on mathematics.

The new vistas opened up in geometry by Donaldson and Witten transformed the subject in the last part of the 20th century, and this is reflected in my own papers, many of which were expository or speculative. Donaldson theory and the new polynomial knot invariants of Vaughan Jones (also a Fields Medalist) were discussed by me in [6], at the Hermann Weyl Symposium, where I speculated on the possible physical significance of the geometric results. This challenge was successfully taken up by Witten in the next few years and led to the emergence of topological quantum field theories (TQFT). I summarized this story in an axiomatic form that I thought would be palatable to mathematicians in [7], and I elaborated on the 2-dimensional case later in [38]. I am glad to say that mathematicians did in fact take on board the concept of a TQFT and built substantially on it, at least in the 3-dimensional context.

The Jones theory, transformed by Witten into a TQFT, was the subject of my Lincei Lectures in [15], delivered in Florence in 1988, and was given a popular exposition at the Royal Institution [67] shortly afterwards. A slightly different presentation was the subject of my Milne Iecture in Oxford [9]. A topological subtlety in the theory, the “gravitational anomaly”, led to the note [14].

Witten’s interpretation of Donaldson theory as a TQFT involved an elaborate super-symmetric version of Yang–Mills theory, whose
mathematical significance was far from clear. My paper
[16]
with my last student,
Lisa Jeffrey,
gave a formal mathematical explanation of it in terms of an equivariant (infinite-dimensional) Euler class. A very different object, with a similar nature, was introduced in my joint paper
[10]
with
Graeme Segal
as a __\( K \)__-theory interpretation of what physicists called the “string theory Euler number of an orbifold”. This was perhaps the first indication that __\( K \)__-theory might have relevance for string theory (beyond its role in index theory).

Magnetic monopoles are, like instantons, solitons and their theory is in some sense “integrable”. In particular, they are described by Hitchin’s “spectral curve”. My paper [18], based on work with my former student Michael Murray, establishes a connection between these spectral curves and those arising from the Yang–Baxter equations. The result is intriguing but still mysterious and it has not yet been taken any further.

My work on monopoles in [8] had been much influenced by Nick Manton whom I met as a graduate student in Cambridge. In fact it was my old friend and contemporary John Polkinghorne, just on the verge of forsaking physics for the Church, who suggested to me that I might find Nick interesting to talk to. It was Nick who told me that the metric on the monopole moduli space should determine the low-energy dynamics, and this eventually led to [8]. Subsequently he got me interested in skyrmions, a classical non-linear soliton model of the nucleus that was introduced much earlier by Tony Skyrme and which has now reacquired some popularity. Nick was persuaded that skyrmions had something in common with monopoles. After much struggle we finally found a link (actually via instantons) and this led to our two papers [11] and [22].

The next set of four short articles are concerned with the general relation between geometry and physics. In [26] I responded to a deliberately provocative article by Jaffe and Quinn, and the lecture [31] follows up the same theme. [5] was my lecture at the first ICIAM conference and aimed at a broad audience of applied mathematicians, while [42] was delivered on the occasion of my honorary degree at Brown University.

Between 1990 and 1995, much of my time was taken up by my duties as President of the Royal Society and, from the large number of speeches I had to deliver in that capacity. I have selected a small sample of three. The first two [21], [43] and [23] were delivered in Philadelphia at a joint meeting with the American Philosophical Society. [24] was only one of my five Presidential Addresses to the Royal Society where I concentrated on mathematics. I tried to describe its fundamental nature, particularly in relation to science. I elaborated the view put forward more briefly in [21] of mathematics as a language. A slightly different perspective on mathematics, related more to philosophical questions, is contained in my book review [29].

The next eight papers relate to individual mathematicians and their work. [39] was a lecture delivered at the Royal Society in connection with the laying of a commemorative stone to Dirac in Westminster Abbey. [34] and [53] were lectures in honour of my friend and collaborator Fritz Hirzebruch, and [27] was my contribution to the collected works of another close colleague, Raoul Bott. [40] was given at a conference honouring Roger Penrose and it gave me an opportunity of acknowledging the influence of my Oxford professorial colleague and erstwhile fellow student. [41] and [44] were obituary articles for my first teacher J. A. Todd and the great Japanese mathematician K. Kodaira, at whose feet I had studied in Princeton in 1955.

The obituary article [58] on Hermann Weyl was a very unusual one and calls for some comment. Weyl died in 1955, but for some reason the National Academy of Sciences had not published an obituary in the years after his death. I was extremely surprised to be asked to undertake this so many years later. Given the wide range and importance of Weyl’s work, it would have been a herculean task to have produced a really comprehensive obituary, and perhaps this is why the project lapsed. However, Weyl has always been one of my heroes, and the present interaction between geometry and physics is so much in the spirit of Weyl’s own work that I felt I could not turn it down. I used the occasion to assess Weyl’s contributions in the retrospective light of the progress of the last fifty years, with emphasis on the relations to physics.

The year 1900 saw the great lecture by Hilbert at the Paris ICM, when he produced his famous list of problems. It was widely recognised that this feat could not be successfully repeated but various attempts were made to recognise the end of the century (and of the millennium). The three next articles reflect this. [45] was given at the opening of the new mathematical centre at the American University of Beirut in Lebanon, my father’s home country. [55] was an extremely broad-brush survey lecture I gave, looking back at the mathematics of the 20th Century. This lecture was given in many places and was reproduced in many journals and in different languages. The IMU commissioned a special volume for 2000 for which I was one of the four editors. I failed to contribute an article but my co-editors kindly allowed me to write a lengthy preface which is reproduced as [46]. My most recent survey on geometry and physics is [70], given on the occasion of the Gelfand 90th birthday conference in Harvard. It concludes with some speculation about the future significance of all these developments for both mathematics and physics.

The next six papers all had their origin in a simple question about
geometry that was put to me by
Michael Berry.
It emerged from a study
(jointly with
Jonathan Robbins)
of the spin statistics theorem of quantum
mechanics and it concerned configurations of __\( n \)__ distinct points in __\( \mathbb{R}^3 \)__.
It turned out to be remarkably fruitful and it enabled me to re-enter mathematical research after my
long period of involvement with Trinity and the Royal Society. In
[32]
I described the problem and gave a first solution, but I also made a conjecture which would lead to a much more elegant solution. In
[52]
I deduced some cohomological consequences, while in
[51]
I elaborated on my conjecture, defining a certain __\( n\times n \)__ determinant associated to __\( n \)__ points. The conjecture asserts that this determinant never vanishes. In
[56]
I collaborated with
Paul Sutcliffe
whose computational skills enabled us to provide strong numerical evidence in favour of the conjecture. We also reversed the problem by studying those configuration of points which maximized the norm of the determinant. We found these were remarkably symmetric configurations, similar to those arising in a variety of physical problems. In my “Leonardo” lecture at Milan I gave a survey of such problems and this was then written up by Paul as our second joint paper
[63].
Finally, in a slightly different direction, my paper
[60]
with
Roger Bielawski
gave yet another solution to the Berry–Robbins problem. This came from a study of Nahm’s equation and was related to other questions in physics.

It also generalized the problem for Lie groups — the original problem being the case of __\( U(n) \)__.

The final five papers are on physics or on closely related geometry. My joint paper
[50]
with
Maldacena
and
Vafa
originated from an earlier discussion I had had with Vafa about deriving an open string — closed
string duality in 6 dimensions by going up to 7 dimensions. This fitted in with current ideas on M-theory and was my introduction to the subject. The much larger joint paper with
Witten
[59]
started off as a follow-up trying to exploit a hidden triality in
[50].
However, as a result of a two-month stay at Cal Tech it grew into a much more ambitious project.
I learnt a great deal more physics and contributed a modest amount of topology. I reported on some of this at a conference in Durham at which
Jürgen Berndt
was present. He suggested that my results on the quaternion projective plane should extend to the Cayley plane and even to a certain homogeneous space of the exceptional group __\( E_6 \)__. This led in due course to our joint paper
[62]
and to my education on the exceptional Lie groups.

The last two papers are both concerned with generalizations of __\( K \)__-theory motivated by physics. My joint paper with
Michael Hopkins
[66]
arose from my trying to understand “orientifold string theory” while I was at Cal Tech, and following up a suggestion of Hopkins. The paper
[69]
with
Graeme Segal
deals with a more substantial extension or “twisting” of __\( K \)__-theory which is needed for M-theory and has recently become popular.
[69]
is a full account of this theory and will be followed later by a second paper dealing with its relations with cohomology. These two papers amplify (and correct) the sketch outlined in
[53].