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We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. These provide a solution to the long-standing problem of finding a good notion of Ricci flow through singularities, in the 3-dimensional case. We prove that Ricci flow with surgery, starting from a fixed initial condition, subconverges to a singular Ricci flow as the surgery parameter tends to zero. We establish a number of geometric and analytical properties of singular Ricci flows.

Let (W, I) be a finite Coxeter group. In the case where W is a Weyl group, Berenstein and Kazhdan in [A. Berenstein, D. Kazhdan, Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases, in: Quantum Groups, in: Contemp. Math., vol. 433, Amer. Math. Soc., Providence, RI, 2007, pp. 1388] constructed a monoid structure on the set of all subsets of I using unipotent -linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type.

The propagation of wave envelopes in two-dimensional (2-D) simple periodic lattices is studied. A discrete approximation, known as the tight-binding (TB) approximation, is employed to find the equations governing a class of nonlinear discrete envelopes in simple 2-D periodic lattices. Instead of using Wannier function analysis, the orbital approximation of Bloch modes that has been widely used in the physical literature, is employed. With this approximation the Bloch envelope dynamics associated with both simple and degenerate bands are readily studied. The governing equations are found to be discrete nonlinear Schrödinger (NLS)-type equations or coupled NLS-type systems. The coefficients of the linear part of the equations are related to the linear dispersion relation. When the envelopes vary slowly, the continuous limit of the general discrete NLS equations are effective NLS equations in moving frames. These continuous NLS equations (from discrete to continuous) also agree with those derived via a direct multiscale expansion. Rectangular and triangular lattices are examples.

We study analytic properties of harmonic maps from Riemannian polyhedra into CAT(κ) spaces for κ∈{0,1}. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into CAT(κ) spaces. We compute a target variation formula that captures the curvature bound in the target, and use it to prove an Lp Liouville-type theorem for harmonic maps from admissible polyhedra into convex CAT(κ) spaces. Another consequence we derive from the target variation formula is the Eells–Sampson Bochner formula for CAT(1) targets.