Mitotic spindle is a complex molecular machine segregating
chromosomes
before cell division. While experiments have revealed the basic
mechanisms of spindle
dynamics, a complete picture of how molecular motor forces are
integrated in mitosis
is still lacking. In this study we undertook systemic analysis to
identify potential mechanisms
of force integration that will reproduce spindle development in
Drosophila embryo. First,
computer assembled millions of different models based on various
possible combinations
of molecular motors. Mathematically, each model is a system of ~ 20 ODEs
and algebraic
equations characterized by ~ 50 parameters. Then, searches in the 'model
space' based on repeated stochastic
optimization using genetic algorithms followed by cluster analysis
identified distinct strategies for
force integration. Our searches identified 1450 distinct groups of
models that can reproduce
experimentally observed pole separation in wild type embryo but only 21
distinct strategies
that can reproduce pole separation in both wild type and eight different
mutant or biochemically
inhibited embryos. Furthermore, out of these 21 plausible models only
one is supported by
additional data on chromosome motility. In addition, analysis of
different force balance strategies r
evealed general design principles that are common among all plausible
models.

The basic problem faced in geophysical fluid dynamics is that a mathematical description based only on fundamental physical principles, the so-called the ``Primitive Equations'', is often prohibitively expensive computationally, and hard to study analytically. In this talk I will survey the main obstacles in proving the global regularity for the three-dimensional Navier-Stokes equations and their geophysical counterparts. Even though the Primitive Equations look as if they are more difficult to
study analytically than the three-dimensional Navier-Stokes equations I will show in this talk that they have a unique global (in time) regular solution for all initial data.

Joint work with Chongsheng Cao.

We describe a method to improve both the accuracy and computational efficiency of a given finite difference scheme used to simulate a geophysical flow. The resulting modified scheme is at least as accurate as the original, has the same time step, and often uses the same spatial stencil. However, in certain parameter regimes it is higher order. As examples we apply the method to the shallow water equations, the Navier-Stokes equations, and to a sea breeze model.

Best constants are found for a class of multiplicative inequalities
that give an estimate of the C-norm of a function in terms of the product
of the L_2-norms of the powers of the Laplace operator.
Special attention is given to functions defined on the sphere S^n.

We show that the set of oscillatory motions in the Sitnikov restricted three body problem has full Hausdorff dimension. This is a joint work with V.Kaloshin.