Using the moduli space of instantons (anti-self-dual connections), Donaldson de-fined diffeomorphism invariants of four-manifolds [5]. Donaldson invariants were studied in detail from the late 1980s to the 1990s and many excellent results were proved. After that period, Seiberg-Witten invariants were then defined [41] and many researchers switched to Seiberg-Witten invariants. Concerning the geome-try of the moduli spaces themselves, however, many interesting problems remained unsolved. Also there were fundamental open problems on Donaldson invariants. Seiberg-Witten's physical analyses were introduced based on N = 2 supersymmetric gauge theory [34]. But their results were not formulated as statements which mathematicians can understand.
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