This paper gives a new way of constructing Landau-Ginzburg mirrors using deformation theory of Lagrangian immersions motivated by the works of Seidel, Strominger-Yau-Zaslow and Fukaya-Oh-Ohta-Ono. Moreover we construct a canonical functor from the Fukaya category to the mirror category of matrix factorizations. This functor derives homological mirror symmetry under some explicit assumptions. As an application, the construction is applied to spheres with three orbifold points to produce their quantum-corrected mirrors and derive homological mirror symmetry. Furthermore we discover an enumerative meaning of the (inverse) mirror map for elliptic curve quotients.

Recently Freed and Hopkins [11] proved that there is no parity anomaly in M-theory on pin+ manifolds in the low-energy field theory approximation, and they also developed an algebraic theory of cubic forms. Earlier Witten [33] proved the anomaly cancellation for spin manifolds by introducing the E8-bundle technique. Motivated by the cubic forms and the anomaly cancellation formulas of Witten-Freed-Hopkins, we give some new cubic forms on spin, spinc, spinω2 and orientable 12- manifolds respectively. We relate them to η-invariants when the manifolds are with boundary, and mod 2 indices on 10 dimensional characteristic submanifolds when the manifolds are spinc or spinω2 . Our method of producing these cubic forms is a combination of (generalized) Witten classes and the character of the basic representation of affine E8.

In 3D printing, it is critical to use as few as possible supporting materials for efficiency and material saving. Multiple model decomposition methods and multi-DOF (degrees of freedom) 3D printers have been developed to address this issue. However, most systems utilize model decomposition and multi-DOF independently. Only a few existing approaches combine the two, i.e. partitioning the models for multi-DOF printing. In this paper, we present a novel model decomposition method for multidirectional 3D printing, allowing consistent printing with the least cost of supporting materials. Our method is based on a global optimization that minimizes the surface area to be supported for a 3D model. The printing sequence is determined inherently by minimizing a single global objective function. Experiments on various complex 3D models using a five-DOF 3D printer have demonstrated the effectiveness of our approach.

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We study the heat kernel of a regular symmetric Dirichlet form on a metric space with doubling measure, in particular, a connection between the properties of the jump measure and the long time behaviour of the heat kernel. Under appropriate optimal hypotheses, we obtain the Hölder regularity and lower estimates of the heat kernel.