These Lecture Notes are an expanded version of the Fermi Lectures I
gave at Scuola Normale Superiore in Pisa, the Loeb Lectures at
Harvard and the Whittemore Lectures at Yale, in 1978. In all cases
I was addressing a mixed audience of mathematicians and physicists
and the presentation had to be tailored accordingly. Throughout, I
presented the mathematical material in a somewhat unorthodox order,
following a pattern which I felt would relate the new techniques to
familiar ground for physicists. The main new results presented in
the lectures, namely the construction of all multi-istanton
solutions of Yang-Mills fields, is the culmination of several years
of fruitful interaction between many physicists and mathematicians.
The major breakthrough came with the observation by Ward that the
complex methods developed by Penrose in his a oetwistor programmea
were ideally suited to the study of the Yang-Mills equations. The
instanton problem was then seen to be equivalent to a problem in
complex analysis and to one in algebraic geometry. Using the
powerful methods of modern algebraic geometry it was not long
before the problem was finally solved.
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