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Mirror symmetry is supposed to involve an identification between the Fukaya category $A = D^\pi Fuk(X)$ of a symplectic manifold $X$ and the derived category of coherent sheaves $B = D^bCoh(X)$ of a complex manifold $X^\vee$. It has been productive to think of this an identification of \emph{noncommutative spaces} $A = B$, and to attempt to understand the intrinsic invariants of this 'noncommutative space'. The basic invariant of a noncommutative space is its Hochschild homology $HH(A) = HH( B )$ which is supposed to be the complex of differential forms on this space. On the $A$ side the Hochschild homology maps to a symplectic invariant, symplectic cohomology, via the \emph{open-closed map}; this map is a basic computational tool in homological mirror symmetry calculations, and the map intertwines purely noncommutative and operadic structures associated to $HH(A)$ with symplectic enumerative data on $X$. On the $B$ side the corresponding map maps to $2$-periodic differential forms on $X^\vee$ and partially recovers Hodge theoretic data; thus, homological mirror symmetry is closely tied to classical, enumerative mirror symmetry. I will explain this package of ideas, with the aim of conveying how ideas from symplectic geometry have been entwined with advances in `noncommutative geometry' and continue to drive an active research program.

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Miniseries Symplectic topology 3.pdf | 9.32 MB |

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