Myxobacteria are social bacteria, that can glide in two dimensions and form counterpropagating, interacting waves. Here, we present a novel age-structured, continuous macroscopic model for the movement of myxobacteria. The derivation is based on microscopic interaction rules that can be formulated as a particle-based model and set within the Self-Organized Hydrodynamics (SOH) framework. The strength of this combined approach is that microscopic knowledge or data can be incorporated easily into the particle model, whilst the continuous model allows for easy numerical analysis of the different effects. However, we found that the derived macroscopic model lacks a diffusion term in the density equations, which is necessary to control the number of waves, indicating that a higher order approximation during the derivation is crucial. Upon ad hoc addition of the diffusion term, we found very good agreement between the age-structured model and the biology. In particular, we analyzed the influence of a refractory (insensitivity) period following a reversal of movement. Our analysis reveals that the refractory period is not necessary for wave formation, but essential to wave synchronization, indicating separate molecular mechanisms.

We classify three fold isolated quotient Gorenstein singularity C3=G. These singularities are rigid, i.e. there is no non-trivial deformation, and we conjecture that they dene 4d N = 2 SCFTs which do not have a Coulomb branch.

We introduce a new concept of strong vanishing property and investigate this property for minimally elliptic singularities. Acomparison theorem about strong vanishing property is also obtained. We use Zariski decomposition to give a formula of plurigenera for Gorenstein singularities and use it to test whether a divisor is f-nef.

One of the fundamental questions in CR geometry is: Given two strongly pseudoconvex CR manifolds X1 and X2 of dimension 2n􀀀 1, is there a non-constant CR morphism between them? In this paper, we use Kohn{Rossi cohomology to show the non-existence of non-constant CR morphism between such two CR manifolds. Specically, if dimHp;q KR(X1) < dimHp;q KR(X2) for any (p; q) with 1  q  n 􀀀 2, then there is no non-constant CR morphism from X1 to X2.

Let R = C[x1, x2, . . . , xn]/(f1, . . . , fm) be a positively graded Artinian algebra. A long-standing conjecture in algebraic geometry, differential geometry, and rational homotopy theory is the non-existence of negative weight derivations on R. Alexsandrov conjectured that there are no negative weight derivations when R is a complete intersection algebra, and Yau conjectured there are no negative weight derivations on R when R is the moduli algebra of a weighted homogeneous hypersurface singularity. This problem is also important in rational homotopy theory and differential geometry. In this paper we prove the non-existence of negative weight derivations on R when the degrees of f1, . . . ,fm are bounded below by a constant C depending only on the weights of x1, . . . , xn. Moreover this bound C is improved in several special cases.