Admissible Lagrangians, Fukaya categories, and homological mirror symmetry.

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The idea of noncommutative geometry, where one tries to think of spaces in nonlocal terms and concepts such as position no longer make sense, has emerged as a promising language to unify diverse areas of mathematics. In symplectic geometry -- the geometry of phase spaces of classical mechanics -- points are "too small" to be relevant, and it is more natural to consider a class of half-dimensional subspaces called Lagrangian submanifolds. Considering the geometry of these submanifolds, rather than points, naturally gives rise to a noncommutative space: the Fukaya category. A remarkable mathematical conjecture inspired by ideas from theoretical physics, "homological mirror symmetry," asserts that many Fukaya categories are in fact equivalent to conventional commutative spaces such as those studied in algebraic geometry. The main goal of this research project is to expand the range of settings to which mirror symmetry is applicable. Specifically, the goal is to establish homological mirror symmetry in a broad enough setting to exhibit all (commutative) algebraic spaces defined by systems of polynomial equations as instances of Fukaya categories. A key step in this program is to study the non-commutative geometry that arises from a symplectic manifold equipped with one or more (commuting) functions. More specifically, the main goal of this project is to prove Kontsevich's homological mirror symmetry conjecture for all complete intersections in (possibly noncompact) toric varieties. The mirror spaces in this setting are so-called toric Landau-Ginzburg models (i.e., noncompact toric Calabi-Yau varieties equipped with regular functions). A key ingredient in the study of their Fukaya categories is the concept of simultaneous admissibility with respect to a collection of toric monomials. This gives a new approach to the Floer-theoretic calculations needed to prove homological mirror symmetry. The expected outcome will be a framework for understanding mirror symmetry that places varieties of general type (including noncompact ones) on the same footing as the more classical Calabi-Yau and Fano cases. This project will also investigate some apparently new structural features of Fukaya categories for Landau-Ginzburg models and for affine varieties, which suggest previously unnoticed functoriality properties of homological mirror symmetry, as well as a new approach to computations for affine varieties.