Degenerate neckpinches in Ricci flow

In earlier work, we derived formal matched asymptotic profiles for families of Ricci flow solutions developing Type-II degenerate neckpinches. In the present work, we prove that there do exist Ricci flow solutions that develop singularities modeled on each such profile. In particular, we show that for each positive integer $k\geq3$, there exist compact solutions in all dimensions $m\geq3$ that become singular at the rate (T-t)^{-2+2/k}$

Uniqueness of two-convex closed ancient solutions to the mean curvature flow

In this paper we consider closed non-collapsed ancient solutions to the mean curvature flow ($n \ge 2$) which are uniformly two-convex. We prove that any two such ancient solutions are the same up to translations and scaling. In particular, they must coincide up to translations and scaling with the rotationally symmetric closed ancient non-collapsed solution constructed by Brian White in (2000), and by Robert Haslhofer and Or Hershkovits in (2016).Comment: 74 pages, 5 figure

Formal matched asymptotics for degenerate Ricci flow neckpinches

Gu and Zhu have shown that Type-II Ricci flow singularities develop from nongeneric rotationally symmetric Riemannian metrics on $S^m$, for all $m\geq 3$. In this paper, we describe and provide plausibility arguments for a detailed asymptotic profile and rate of curvature blow-up that we predict such solutions exhibit

A Density-Dependent Switch Drives Stochastic Clustering and Polarization of Signaling Molecules

Positive feedback plays a key role in the ability of signaling molecules to form highly localized clusters in the membrane or cytosol of cells. Such clustering can occur in the absence of localizing mechanisms such as pre-existing spatial cues, diffusional barriers, or molecular cross-linking. What prevents positive feedback from amplifying inevitable biological noise when an un-clustered “off” state is desired? And, what limits the spread of clusters when an “on” state is desired? Here, we show that a minimal positive feedback circuit provides the general principle for both suppressing and amplifying noise: below a critical density of signaling molecules, clustering switches off; above this threshold, highly localized clusters are recurrently generated. Clustering occurs only in the stochastic regime, suggesting that finite sizes of molecular populations cannot be ignored in signal transduction networks. The emergence of a dominant cluster for finite numbers of molecules is partly a phenomenon of random sampling, analogous to the fixation or loss of neutral mutations in finite populations. We refer to our model as the “neutral drift polarity model.” Regulating the density of signaling molecules provides a simple mechanism for a positive feedback circuit to robustly switch between clustered and un-clustered states. The intrinsic ability of positive feedback both to create and suppress clustering is a general mechanism that could operate within diverse biological networks to create dynamic spatial organization

Curve shortening and the topology of closed geodesics on surfaces

We study ¿flat knot types¿ of geodesics on compact surfaces M2. For every flat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening flow on the space of immersed curves on M2. We conclude existence of closed geodesics with prescribed flat knot types, provided the associated Conley index is nontrivial

The dynamics of a degenerate reaction diffusion equation

We consider the initial-boundary value problem for a degenerate reaction diffusion equation consisting of the porous medium operator plus a nonlinear reaction term. The structure of the set of equilibria depends on the length of the spatial domain. There are two critical lengths $\scriptstyle 0L_1$. Using a topological argument we show existence of connecting orbits joining the unstable equilibrium with the two stable equilibria for $\scriptstyle L\in(L_0, L_1]$, when there are three equilibria. By showing that the principle of linearized stability can sometimes be applied with succes to degenerate parabolic equations, these connections are found to be unique for $\scriptstyle L_