Abstract
The foundations of matrix geometry are discussed, which provides the basis for recent progress on the effective geometry and gravity in Yang-Mills matrix models. Basic examples lead to a notion of embedded noncommutative spaces (branes) with emergent Riemannian geometry. This class of configurations turns out to be preserved under small deformations, and is therefore appropriate for matrix models. The relation with spectral geometry is discussed. A possible realization of sufficiently generic 4-dimensional geometries as noncommutative branes in D= 10 matrix models is sketched.
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