Generalized geometry is a unifying approach to geometric structures where, for
example, complex and symplectic structures become particular instances of a more
general structure: a generalized complex structure. After a self-contained
introduction to generalized complex geometry (which is only possible for
even-dimensional manifolds), I will explain how generalized geometry can be upgraded
to Bn-generalized geometry, in which the generalized-complex approach applies as well
to odd dimensions. Finally, I will comment on some ongoing joint work with J. Porti in
which we look at the case of three-manifolds.
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