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In this work we investigate the decorated domain wall construction in bosonic group-cohomology symmetry-protected topological (SPT) phases and related quantum anomalies in bosonic topological phases. We first show that a general decorated domain wall construction can be described mathematically as an Atiyah-Hirzebruch spectral sequence, where the terms on the E2 page correspond to decorations by lower-dimensional SPT states at domain wall junctions. For bosonic group-cohomology SPT phases, the spectral sequence becomes the Lyndon-Hochschild-Serre (LHS) spectral sequence for ordinary group cohomology. We then discuss the physical interpretations of the differentials in the spectral sequence, particularly in the context of anomalous SPT phases and symmetry-enriched gauge theories. As the main technical result, we obtain a full description of the LHS spectral sequence concretely at the cochain level. The explicit formulae are then applied to explain Lieb-Schultz-Mattis theorems for SPT phases, and also derive a new LSM theorem for easy-plane spin model in a π flux lattice. We also revisit the classifications of symmetry-enriched 2D and 3D Abelian gauge theories using our results.

Let $F: \sum^n \times [0, T) \to \mathbb{R}^{n+m} be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps $\gamma : (\sum^n, g_t) \to G(n,m)$ form a harmonic heat flow with respect to the time-dependent induced metric $g_t$. This provides a more systematic approach to investigating higher codimension mean curvature flows. A direct consequence is any convex function on $G(n,m)$ produces a subsolution of the nonlinear heat equation on $(\sum, g_t)$. We also show the condition that the image of the Gauss map lies in a totally geodesic submanifold of $G(n,m)$ is preserved by the mean curvature flow. Since the space of Lagrangian subspaces is totally geodesic in $G(n, n)$, this gives an alternative proof that any Lagrangian submanifold remains Lagrangian along the mean curvature flow.

This paper presents an algorithm for extending B-spline curves and surfaces. Based on the unclamping algorithm for B-spline curves, we propose a new algorithm for extending B-spline curves that extrapolates using the recurrence property of the de Boor algorithm. This algorithm provides a nice extension, with maximum continuity, to the original curve segment. Moreover, it can be applied to the extension of B-spline surfaces. Extension to both single and multiple target points/curves are considered in this paper.

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