12/03/18 09:43:20.21
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ヤン＝ミルズ理論補足
URLﾘﾝｸ(en.wikipedia.org)
Biography
Donaldson's father was an electrical engineer in the physiology department at the University of Cambridge[citation needed].
Donaldson gained a BA degree in mathematics from Pembroke College, Cambridge in 1979, and in 1980 began postgraduate work at Worcester College, Oxford, at first under Nigel Hitchin and later under Michael Atiyah's supervision.
Still a graduate student, Donaldson proved in 1982 a result that would establish his fame.
He published the result in a paper Self-dual connections and the topology of smooth 4-manifolds which appeared in 1983.
In the words of Atiyah, the paper "stunned the mathematical world" (Atiyah 1986).
Whereas Michael Freedman classified topological four-manifolds,
Donaldson's work focused on four-manifolds admitting a differentiable structure, using instantons, a particular solution to the equations of Yang-Mills gauge theory which has its origin in quantum field theory.
One of Donaldson's first results gave severe restrictions on the intersection form of a smooth four-manifold.
As a consequence, a large class of the topological four-manifolds do not admit any smooth structure at all.
Donaldson also derived polynomial invariants from gauge theory.
These were new topological invariants sensitive to the underlying smooth structure of the four-manifold.
They made it possible to deduce the existence of "exotic" smooth structures?certain topological four-manifolds could carry an infinite family of different smooth structures.
（つづく）