This series of lectures will be concerned with the asymptotic behavior of some random dynamical system. The push-forward and pull-back approaches will be discussed. Some applications to stochastic reaction-diffusion equations and stochastic Navier-Stokes equations will be given.

The classical isoperimetric inequality states that a curve in the plane of length L bounds a disk whose area is at most L^2/4\pi. This inequality was generaized to curves in R^3 in the early 1900's. Such a curve bounds an immersed disk whose area is at most L^2/4\pi. It also bounds an embedded surface satisfying the same area bound.

An unknotted curve bounds an embedded disk in R^3. We show, in contrast to the above, that given any positive constant A, there are unknotted smooth curves of length 1 that do not bound embedded disks of area less than A. If we control the size of a tubular neighborhood of a curve then we do get explicit isoperimetric bounds.
(joint work with J. Lagarias and W. Thurston)

Accurate and reliable computational predictions for biological systems must be based on a) accurate experimental measurements; b) reliable modeling; c) well-chosen and sufficiently validated risk indicators (biological and mechanical markers). For arterial diseases, experimental measurements include vessel morphology, material properties, and flow information. Models for blood flow in diseased arteries have evolved from 1-D model, 2D and 3D models, to our recently introduced 3D multi-component models with fluid-structure interactions based on in vivo patient-specific geometries. Results from different models can differ considerably (difference can be from 50% to more than 800%) so proper bench mark should be set so that accurate and reliable predictions can be made using computational models.

It is well-accepted that atherosclerosis initiation and progression correlate positively with low and oscillating flow wall shear stresses. However, this low and oscillating shear stress hypothesis cannot explain why intermediate and advanced plaques continue to grow under elevated high flow shear stress conditions. It is also natural that people think that plaque rupture may be related to maximum stress conditions. We will challenge those popular views and present evidence which support new hypotheses for plaque progression and rupture conditions. Patient-specific multi-year serial MRI were acquired to provide plaque morphology and progression data. A 3D multi-component model with fluid-structure interactions (FSI) was introduced to obtain the flow and stress/strain distributions in the plaque to better understand mechanisms governing plaque progression and rupture process. Our results indicate that plaque thickness and plaque progression correlate positively with low structure wall stress for intermediate and advanced plaques which supports a possible new hypothesis: Low structure stress in the plaque has positive correlation with plaque growth, and may create favorable mechanical conditions for further plaque progression. For plaque vulnerability assessment, our results also indicate that maximum stress conditions are often found at healthy site of the vessel and are not good indicators of rupture risk. A computational plaque vulnerability index (CPVI) based on local stress conditions at critical sites was proposed. Plaque assessments (34 plaque MRI samples) using CPVI method had 90% agreement rate with histopathological analysis. With more patient study validations, our research may serve as the starting points for further plaque progression and rupture investigations. The work has been supported by the National Sciences Foundation (DMS and BIO), National Institutes of Health (NIBIB and NIGMS), and the Whitaker Foundation.

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.