In 1960-70's, Erickson and Leslie proposed the so-called
Ericksen-Leslie evolution equation
that models the hydrodynamic flow of liquid crystals. The underlying
equation is a coupled, dissipative system
between the optical axis $n$ of the nematic liquid cyrstal and its
macroscopic motion, represented by the velocity
field $u$. Roughly speaking, it is a strong coupling between
Navier-Stokes equation and "heat flow of harmonic maps".
In this talk, we will discuss some recent results on the global
existence of almost regular solutions in dimension two.
This is a joint work with F.H.
We study a model, due to J.M. Lasry and P.L. Lions, describing
the evolution of a scalar price which is realized as a free boundary in
a 1D diffusion equation with dynamically evolving, non-standard sources.
We establish global existence and uniqueness. This is joint work with L.
Chayes, M. Gualdani and I. Kim.
EXPLICIT CONSTRUCTION OF MODULI SPACE OF BOUNDED COMPLETE REINHARDT DOMAINS IN C^n AND HILBERT 14th PROBLEMS
One of the most fundamental problems in complex
geometry is to determine when two bounded domains
in C^n are biholomorphically equivalent. Even for complete
Reinhardt domains, this fundamental problem remains unsolved
for many years. Using the Bergmann function theory,
we construct an infinite family of numerical invariants from
the Bergman functions for complete Reinhardt domains in
C^n. These infinite family of numerical invariants are actually
a complete set of invariants if the domains are pseudoconvex
with C^1 boundaries.
The study of injective envelopes of metric spaces, also known as metric trees (R-trees or T-theory), has its motivation in many sub-disciplines of mathematics as well as biology/medicine and computer science. Its relationship with biology and medicine stems from the construction of phylogenetic trees .Concepts of string matching in computer science is closely related with the structure of metric trees . A metric tree is a metric space such that for every in M there is a unique arc between x and y and this arc is isometric to an interval in R. ,.
Starting from a nondecreasing function $K:[0,\infty)\to [0,\infty)$,
we consider a M\"obius-invariant Banach space $Q_K$ of functions
analytic in the unit disk. For $0
In this talk, I'll discuss some recent developments
in Nevanlinna theoy, as well as its applications in the study of the
Gauss map of minimal surfaces, and in the study of Diophantine approximations.