Brown University Graduate School
Dissertation Defense Information Seminar
Brown Analysis Seminar
Abstract: This talk will be devoted to a proof of the following theorem. For an "ideal" steady water wave, the pressure monotonically increases (i) horizontally away from the crest line and (ii) with depth. The proof is entirely based on repeated uses of the maximum principle.
Center for Fluid Mechanics Seminar
Abstract: Red blood cells (RBCs) exhibit rich behavior in viscous flows. For example, in steady shear flow, RBCs can tank-tread, tumble, or "swing" (tank-treading accompanied by oscillations in the inclination angle). I will present a model that quantitatively describes this behavior. The analysis accounts for the fact that the membrane is deformable, incompressible, and resistant to bending and shearing. Considering a nearly spherical shape, we obtain analytical solutions for the creeping-flow equations via a regular perturbation expansion in the excess area. In steady shear flows, the theory shows that a closed lipid membrane (vesicle or RBC) deforms into a prolate ellipsoid, which tumbles at low shear rates, and swings at higher shear rates. The amplitude of the oscillations decreases with shear rate. In quadratic flows, the theory predicts a peculiar coexistence of parachute- and bullet-like vesicle shapes at the flow centerline. Vesicles and RBCs always migrate towards the flow centerline unlike drops, whose direction of migration depends on the viscosity ratio. In time-dependent flows, vesicles can exhibit chaotic dynamics.
Stochastic Systems Seminar
Abstract:
We consider the problem of choosing an optimal portfolio to minimize
the probability that the growth rate of the wealth process falling below a
given level. In finance, such problems arise in risk management. The
mathematical problem involved is not a conventional optimization problem and a
solution is not readily found. We show a duality relation of this problem and
the portfolio optimization problem with risk sensitive criterion. The latter
has been studied in several recent papers. For some factor models, it can be
reformulated as a risk sensitive stochastic control problem (hence the name
for such portfolio optimization problem). The dynamic programming approach can
be used to derive the Bellman equation which is a nonlinear partial
differential equation. As in the theory of stochastic control, a candidate of
optimal portfolio can be derived from a solution of the Bellman equation.
Therefore, the duality relation mentioned above suggests a possibility to
indirectly use the theory of stochastic control in such an unconventional
optimization problem discussed here. Such duality relation has origin in the
theory of large deviations. A difficult part in the analysis involving the
idea of changing the probability measures that is used often in the theory of
large deviations. How to choose a suitable new probability measure seems to
require a nontrivial insight, since in our problem we need to deal with the
control process which is absent in the theory of large deviations. Stutzer
(2003) argued that such duality relation provides a foundation for using HARA
utilities.
Our study is closely related to the study of Pham (2003), who initiated the
mathematical study of the problem by considering the cases that down-side risk
probabilities are large, and hence the up-side chance probabilities need to be
considered instead.
CCMB Distinguished Lecture Series
Hosted by: Benjamin Raphael |
Abstract: New technology recently has facilitated the complete sequencing of individual human genomes. As the cost and efficiency of this approach continues to improve, we can envision a powerful new means for the study of genes and other genome elements and mechanisms that underlie cancer and other human diseases. I will discuss some of the discoveries made to date with emerging genome sequencing technologies, and how these methods will allow us to better understand both basic biology and human disease.
Scientific Computing Seminar
Abstract: Atherosclerosis (the vascular disease that causes heart attacks and strokes) is characterised by the formation of lipid rich lesions within the walls of large arteries. It has been observed that such lesions form preferentially at sites of arterial branching and in regions of high curvature. Since these particular regions are associated with complex blood flow patterns, it has been postulated that blood flow may regulate onset of the disease. Several mechanisms have been suggested that could cause flow dependent regulation of atherosclerosis. One particular mechanism relies on the formation of a lipid rich layer adjacent to the blood-side surface of the arterial wall (forming due to lipid concentration polarisation). If blood flow is able regulate formation of such a layer, then it could control the amount of lipid in contact with the arterial wall, and thus the local susceptibility of the arterial wall to atherosclerosis. In this seminar efforts to model blood flow and lipid concentration polarisation at both the vascular (~1cm) and cellular (~1cm) scale will be presented.
**Special Stochastic Systems Seminar**
Abstract: Controlled Stochastic PDE's with the cost functional of discounted and ergodic types will be considered and some recent results obtained jointly with Ben Goldys on the ergodic control will be outlined. The optimal cost and optimal control in these cases may be found in a feedback form by means of the solutions to appropriate stationary HJB equations in a Banach spaces, the existence and uniqueness of which has been proved. The results are applicable to controlled stochastic reaction-diffusion equations.
PDE Seminar
Abstract: The main goal of recent mathematical work on quasi-static fracture triggered by the Francfort-Marigo approach is to turn Griffith's criterion for crack growth into a model for predicting crack paths. The main drawback of the Francfort-Marigo model is its reliance on global minimization, which in particular results in violations of Griffith's criterion. I will discuss some efforts towards creating a variational model based on local minimization that also predicts crack paths, and why a fully satisfactory (variational) approach seems impossible. Time permitting, I will also discuss some recent work on dynamics.
PDE Seminar
Abstract: We discus existence and stability for Einstein-scalar field Lichnerowicz equations in the inhomogeneous context of compact Riemannian manifolds.