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Wedge product on deRham complex of a Riemannian manifold M can be pulled back to H^∗(M) via explicit homotopy, constructed using Green’s operator, to give higher product structures. We prove Fukaya’s conjecture which suggests that Witten deformation of these higher product structures have semiclassical limits as operators defined by counting gradient flow trees with respect to Morse functions, which generalizes the remarkable Witten deformation of deRham differential from a statement concerning homology to one concerning real homotopy type of M. Various applications of this conjecture to mirror symmetry are also suggested by Fukaya.

In this note, we prove the sharp Davies-Gaffney-Grigor’yan lemma for minimal heat kernels on graphs.

No abstract uploaded!

Recovering low-rank structures via eigenvector perturbation analysis is a common problem in statistical machine learning, such as in factor analysis, community detection, ranking, matrix completion, among others. While a large variety of results provide tight bounds on the average errors between empirical and population statistics of eigenvectors, fewer results are tight for entrywise analyses, which are critical for a number of problems such as community detection and ranking.