In this paper, we present a staggered discontinuous Galerkin immersed boundary method (SDGIBM) for the numerical approximation of fluid-structure interaction. The immersed boundary method is used to model the fluid-structure interaction, while the fluid flow is governed by incompressible Navier-Stokes equations. One advantage of using Galerkin method over the finite difference method with immersed boundary method is that we can avoid approximations of the Dirac Delta function. Another key ingredient of our method is that our solver for incompressible Navier-Stokes equations combines the advantages of discontinuous Galerkin methods and staggered meshes, and results in many good properties, namely local and global conservations and pointwise divergence-free velocity field by a local postprocessing technique. Furthermore, energy stability is improved by a skew-symmetric discretization of the convection term. We will present numerical results to show the performance of the method.

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We study Thurston’s Lipschitz and curve metrics, as well as the arc metric on the Teichmüller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston’s stretch maps and prove the following: (1) On the Teichmüller space of the torus with one boundary component, the Lipschitz and the curve metrics coincide and define a geodesic metric on this space. (2) On the same space, the arc and the curve metrics coincide when the length of the boundary component is ≤4arcsinh(1), but differ when the boundary length is large. We further apply our stretch map generalization to construct novel Thurston geodesics on the Teichmüller spaces of closed hyperbolic surfaces, and use these geodesics to show that the sum-symmetrization of the Thurston metric fails to exhibit Gromov hyperbolicity.

In its simplest form, convolution neural networks (CNNs) consist of a fully connected layer g composed with a sequence of convolution layers T . Although g is known to have the universal approximation property, it is not known if CNNs, which has the form g ◦ T inherits this property, especially when the kernel size in T is small. In this paper, we show that under suitable conditions, CNNs does inherit the universal approximation property and its sample complexity can be characterized. In addition, we discuss concretely how the nonlinearity of T can improve the approximation power. Finally, we show that when the target function class has a certain compositional form, convolutional networks are far more advantageous compared with fully connected networks, in terms of number of parameters needed to achieve a desired accuracy.

We introduce a quantum trace map for an ideally triangulated hyperbolic knot complement S^3∖K. The map assigns a quantum operator to each element of Kauffmann Skein module of the 3-manifold. The quantum operator lives in a module generated by products of quantized edge parameters of the ideal triangulation modulo some equivalence relations determined by gluing equations. Combining the quantum map with a state-integral model of SL(2,C) Chern-Simons theory, one can define perturbative invariants of knot K in the knot complement whose leading part is determined by its complex hyperbolic length. We then conjecture that the perturbative invariants determine an asymptotic expansion of the Jones polynomial for a link composed of K and K. We propose the explicit quantum trace map for figure-eight knot complement and confirm the length conjecture up to the second order in the asymptotic expansion both numerically and analytically.