Giant lipid vesicles are closed compartments consisting of semi-permeable shells, which isolate femto- to pico-liter quantities of aqueous core from the bulk. Although water permeates readily across vesicular walls, passive permeation of solutes is hindered. In this study, we show that, when subject to a hypotonic bath, giant vesicles consisting of phase separating lipid mixtures undergo osmotic relaxation exhibiting damped oscillations in phase behavior, which is synchronized with swell–burst lytic cycles: in the swelled state, osmotic pressure and elevated membrane tension due to the influx of water promote domain formation. During bursting, solute leakage through transient pores relaxes the pressure and tension, replacing the domain texture by a uniform one. This isothermal phase transition—resulting from a well-coordinated sequence of mechanochemical events—suggests a complex emergent behavior allowing synthetic vesicles produced from simple components, namely, water, osmolytes, and lipids to sense and regulate their micro-environment. 

In this talk, we study continuous maximal regularity theory for a class of degenerate or singular differential operators on manifolds with singularities. Based on this theory, we show local existence and uniqueness of solutions for several nonlinear geometric flows and diffusion equations on non-compact, or even incomplete, manifolds, including the Yamabe flow and parabolic p-Laplacian equations. In addition, we also establish regularity properties of solutions by means of a technique consisting of continuous maximal regularity theory, a parameter-dependent diffeomorphism and the implicit function theorem.

 The probability mass function $1/j(j+1)$ for $j\geq 1$ belongs to the domain of attraction of a completely asymmetric Cauchy distribution.

The purpose of the talk is to review some of applications of this simple observation to limit theorems related to the destruction of random recursive trees.

Specifically, a random recursive tree of size $n+1$ is a tree chosen uniformly at random amongst the $n!$ trees on the set of vertices $\{0,1, 2, ...,  n\}$ such that the sequence of vertices along any segment starting from the root $0$ increases. One destroys this tree by removing its edges one after the other in a uniform random order. It was first observed by Iksanov and M\"ohle that the central limit theorems for the random walk with step distribution given above explains the fluctuations of the number of cuts needed to isolate the root. We shall discuss further results in the same vein.