We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions codimensions, we classify ${\bf S}^n$ with low area: they are steady or shrinking equatorial spheres. the case S}^3$, more relaxed area bounds: equators Clifford tori. embedded curve sh...

The Smyth line (SL) chicken, a model for autoimmune human vitiligo, is characterized by a spontaneous posthatch epidermal pigment loss (vitiligo). Even though the immunological and morphological changes accompanying the vitiligo process have been well studied, the genetics of this phenomenon remains elusive. The SL lines have been maintained by nonpedigreed matings since their inception, and th...

Complexity of geometric symmetry for differential operators with mixed homogeniety is examined here. Sharp Sobolev estimates are calculated for the Grushin operator in low dimensions using hyperbolic symmetry and conformal geometry. Considerable interest exists in understanding differential operators with mixed homogeneity. A simple example is the Grushin operator on R ∆G = ∂ ∂t2 + 4t ∂ ∂x2 . T...