The marriage of analytic power to geometric intuition drives many of today's mathematical
advances, yet books that build the connection from an elementary level remain scarce. This …

The multi-bubble isoperimetric conjecture in $ n $-dimensional Euclidean and spherical
spaces from the 1990's asserts that standard bubbles uniquely minimize total perimeter …

Abstract We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted
perimeter way to decompose R^n into q cells of prescribed (positive) Gaussian measure …

The free energy of a ternary system with a self-organization property includes an interface
energy and a longer ranging, inhibitory interaction energy. In a planar domain, if the two …

We investigate the minimal-perimeter configurations of two finite sets of points on the square
lattice. This corresponds to a lattice version of the classical double-bubble problem. We give …

A ternary inhibitory system is a three component system characterized by two properties:
growth and inhibition. A deviation from homogeneity has a strong positive feedback on its …

We consider a sharp-interface model of triblock copolymers, for which the surface tension
across the interface separating phase from phase may depend on the components. We …

We establish the Gaussian Double-Bubble Conjecture: the least Gaussian-weighted
perimeter way to decompose $\mathbb {R}^ n $ into three cells of prescribed (positive) …

We use a new approach that we call unification to prove that standard weighted double
bubbles in n-dimensional Euclidean space minimize immiscible fluid surface energy, that is …

Locally isoperimetric $ N $-partitions are partitions of the space $\mathbb R^ d $ into $ N $
regions with prescribed, finite or infinite measure, which have minimal perimeter (which is …