For a one-parameter family (V, \Omega_{i}) with I=1,...,p_g of general type hypersurfaces with bases of holomorphic n-forms, we construct open covers V = Spg i=1 Ui using tropical geometry. We show that after normalization, each i is approximately supported on a unique Ui and such a pair approximates a Calabi-Yau hypersurface together with its holomorphic n-form as the parameter becomes large. We also show that the Lagrangian fibers in the fibration constructed by Mikhalkin [9] are asymptotically special Lagrangian. As the holomorphic n-form plays an important role in mirror symmetry for Calabi-Yau manifolds, our results is a step toward understanding mirror symmetry for general type manifolds.

No abstract uploaded!

No abstract uploaded!

Given a loop on a surface, its homotopy class can be specified as a word consisting of letters representing the homotopy group generators. One of the interesting problems is how to compute the shortest word for a given loop. This is an NP-hard problem in general. However, for a closed surface that allows a hyperbolic metric and is equipped with a canonical set of fundamental group generators, the shortest word problem can be reduced to finding the shortest loop that is homotopic to the given loop, which can be solved efficiently. In this paper, we propose an efficient algorithm to compute the shortest words for loops given on triangulated surface meshes. The design of this algorithm is inspired and guided by the work of Dehn and BirmanSeries. In support of the shortest word algorithm, we also propose efficient algorithms to compute shortest paths and shortest loops under hyperbolic metrics using a novel

We discuss the connections between the complex SYK model at the conformal limit and warped conformal field theories. Both theories have an SL(2 ,R)×U(1) global symmetry. We present comparisons on symmetries, correlation functions, the effective action and the entropy formula. We also use modular covariance to reinterpret results in the complex SYK model.