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Goldman has investigated geometric structures, in various incarnations, on manifolds since his undergraduate thesis, "Affine manifolds and projective geometry on manifolds", supervised by William Thurston and Dennis Sullivan. This work led to work with Morris Hirsch and David Fried on affine structures on manifolds, and work in real projective structures on compact surfaces. In particular he proved that the space of convex real projective structures on a closed orientable surface of genus {\displaystyle g>1}g>1 is homeomorphic to an open cell of dimension {\displaystyle 16g-16}{\displaystyle 16g-16}. With Suhyoung Choi, he proved that this space is a connected component (the "Hitchin component") of the space of equivalence classes of representations of the fundamental group in {\displaystyle {\rm {SL}}(3,\mathbb {R} )}{\displaystyle {\rm {SL}}(3,\mathbb {R} )}. Combining this result with Suhyoung Choi's convex decomposition theorem, this led to a complete classification of convex real projective structures on compact surfaces.
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Geometry and Topologypp.129-145, (2020)
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