Speaker: Xiaofeng Ren (George Washington University)
 Title: Analysis and computation of some pattern formation problems

 Abstract:  The Ohta-Kawasaki density functional theory of block
 copolymers is a variational problem with a nonlocal term in the
 integrand. It models the lamellar, cylindrical, and spherical
 morphological phases observed in diblock copolymers.

 The Gierer-Meinhardt problem is a reaction-diffusion system of partial
 differential equations for pattern formation and morphogenesis observed
 in the development of a higher organism out of a single
 fertilized egg. It is a minimal model that provides a theoretical bridge
 between observations on the one hand and the deduction of the underlying
 molecular-genetic mechanisms on the other hand.

 Interestingly in some important parameter ranges, these two different
 problems converge to the same singular limit: a geometric problem of
 finding a subset in plane or space that satisfies an equation involving
 the curvature of the boundary of the set.

 In this talk, I will present some direct computations that numerically
 simulate the Ohta-Kawasaki theory. Then I will discuss some analytic
 result that further reduce the geometric singular limit to a finite
 dimensional problem. And finally the finite dimensional problem is
 solved numerically by Matlab.