High-dimensional data can be difficult to analyze. Assume data are distributed on a low-dimensional manifold. The Vector Diffusion Mapping (VDM), introduced by Singer-Wu, is a non-linear dimension reduction technique and is shown robust to noise. It has applications in cryo-electron microscopy and image denoising and has potential application in time-frequency analysis.
In this talk, I will present a theoretical analysis of the effectiveness of the VDM. Specifically, I will discuss parametrisation of the manifold and an embedding which is equivalent to the truncated VDM. In the differential geometry language, I use eigen-vector fields of the connection Laplacian operator to construct local coordinate charts that depend only on geometric properties of the manifold. Next, I use the coordinate charts to embed the entire manifold into a finite-dimensional Euclidean space. The proof of the results relies on solving the elliptic system and provide estimates for eigenvector fields and the heat kernel and their gradients.
We show that the total area of two distinct Gaussian curvature 1 surfaces with the same conformal factor on the boundary, which are also conformal to the Euclidean unit disk, must be at least 4π. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We refer to this lower bound of total areas as the Sphere Covering Inequality. This inequality and it's generalizations are applied to a number of open problems related to Moser-Trudinger type inequalities, mean field equations and Onsager vortices, etc, and yield optimal results. In particular we confirm the best constant of a Moser-Truidinger type inequality conjectured by A. Chang and P. Yang in 1987. This is a joint work Changfeng Gui.
For a bounded domain, we consider the L^\infty-functional involving a nonnegative Hamilton function. Under the continuous Dirichlet boundary condition and some assumptions of Hamiltonian H, the uniqueness of absolute minimizers for Hamiltonian H is established. This extendes the uniqueness theorem to a larger class of Hamiltonian $H(x,p)$ with $x$-dependence. As a corollary, we confirm an open question on the uniqueness of absolute minimizers posed by Jensen-Wang-Yu. Our proofs rely on geometric structure of the action function induced by Hamiltonian H(x,p), and the identification of the absolute subminimality with convexity of the associated Hamilton-Jacobi flow.
In the 1960s, Benjamin and Feir, and Whitham, discovered that a Stokes wave would be unstable to long wavelength perturbations, provided that (the carrier wave number) x (the undisturbed water depth) > 1.363.... In the 1990s, Bridges and Mielke studied the corresponding spectral instability in a rigorous manner. But it leaves some important issues open, such as the spectrum away from the origin. The governing equations of the water wave problem are complicated. One may resort to simpler approximate models to gain insights.
I will begin by Whitham's shallow water equation and the modulational instability index for small amplitude and periodic traveling waves, the effects of surface tension and vorticity. I will then discuss higher order corrections, extension to bidirectional propagation and two-dimensional surfaces. This is partly based on joint works with Jared Bronski (Illinois), Mat Johnson (Kansas), and Ashish Pandey (Illinois).