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The mathematics of G_2 conifolds for M-theory

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G_2 manifolds play
the analogous role in M-theory that Calabi-Yau manifolds play in string
theory. There has been work in the physics community on conjectural
"mirror symmetry" in this context, and it has also been observed that
singularities are necessary for a satisfactory theory. After a very
brief review of these physical developments (by a mathematician who
doesn't necessarily understand the physics), I will give a mathematical
introduction to G_2 conifolds. I will then proceed to give a detailed
survey of recent mathematical developments on G_2 conifolds, including
desingularization, deformation theory, and possible constructions of G_2
conifolds. This includes separate joint works of myself with Jason
Lotay and with Dominic Joyce.