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I will explain that a geometric theory built upon the theory of complex surfaces can be used to understand wide variety of phenomena in five-dimensional supersymmetric theories, which includes the following:
Classification of 5d superconformal field theories (SCFTs).
Enhanced flavor symmetries of 5d SCFTs.
5d N=1 gauge theory descriptions of 5d and 6d SCFTs.
Dualities between 5d N=1 gauge theories.
T-dualities between 6d N=(1,0) little string theories.
This relationship between geometry and 5d theories is based on M-theory and F-theory compactifications.