We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of their results in terms of mirror symmetry and symplectic geometry. As an interesting application, we construct an explicit global deformation quantization of an affine del Pezzo surface as a noncommutative mirror to an elliptic orbifold.

The diffusive transport of passive tracers or particles can be enhanced by incompressible, turbulent flow fields. Analyzing the effective behavior is a challenging problem with theoretical and practical importance in many areas of science and engineering, ranging from the transport of mass, heat, and pollutants in geophysical flows to sea ice dynamics and turbulent combustion. The long time, large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity tensor D*. Two different formulations of integral representations for D* were developed for the case of time-independent fluid velocity fields, involving spectral measures of bounded self-adjoint operators acting on vector fields and scalar fields, respectively. Here, we extend both of these approaches to the case of space-time periodic velocity fields, allowing for chaotic dynamics, providing rigorous integral representations for D* involving spectral measures of unbounded self-adjoint operators.We prove the different formulations are equivalent. Their correspondence follows from a one-to-one isometry between the underlying Hilbert spaces. We also develop a Fourier method for computing D*, which captures the phenomenon of residual diffusion related to Lagrangian chaos of a model flow. This is reflected in the spectral measure by a concentration of mass near the spectral origin.

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We prove the spacetime positive mass theorem in dimensions less than eight. This theorem asserts that for any asymptotically flat initial data set that satisfies the dominant energy condition, the inequality E>=|P| holds, where (E, P) is the ADM energy-momentum vector. Previously, this theorem was only known for spin manifolds [38]. Our approach is a modification of the minimal hypersurface technique that was used by the last named author and S.-T. Yau to establish the time-symmetric case of this theorem [30, 27]. Instead of minimal hypersurfaces, we use marginally outer trapped hypersurfaces (MOTS) whose existence is guaranteed by earlier work of the first named author [14]. An important part of our proof is to introduce an appropriate substitute for the area functional that is used in the time-symmetric case to single out certain minimal hypersurfaces. We also establish a density theorem of independent interest and use it to reduce the general case of the spacetime positive mass theorem to the special case of initial data that has harmonic asymptotics and satisfies the strict dominant energy condition.

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