We give an introduction to the principle ISP and its relatives as well as their connections to supercompact cardinals and the proper forcing axiom.

We discuss self-organized dynamics of agent-based models
with focus on a prototype model driven by non-symmetric self-alignment
introduced in [1].
Unconditional consensus and flocking emerge when the self-alignment is
driven by global interactions with a sufficiently slow decay rate.  In
more realistic models, however, the interaction of self-alignment is
compactly supported, and open questions arise regarding the emergence
of clusters/flocks/consensus, which are related to the propagation of
connectivity of the underlying graph.
In particular, we discuss heterophilious self-alignment: here, the
pairwise interaction between agents increases with the diversity of
their positions and we assert that this diversity enhances
flocking/consensus. The methodology carries over from agent-based to
kinetic and hydrodynamic descriptions.

[1] A new model for self-organized dynamics and its flocking behavior, J.
Stat. Physics 144(5) (2011) 923-947.
 

I will present a new PDE approach to obtain large time behavior
of Hamilton-Jacobi equations. This applies to usual Hamilton-Jacobi
equations, as well as the degenerate viscous cases, and weakly coupled
systems. The degenerate viscous case was an open problem in last 15 years.
This is the joint work with Cagnetti, Gomes, and Mitake.

Advancement to Candidacy Committee:
Daqing Wan (Chair)
Karl Rubin
Zhiqin Lu
Chuu-Lian Terng
Andromache Karanika (outside)

 
Experience suggests that uncertainties often play an important role in quantifying the performance of complex systems. Therefore, uncertainty needs to be treated as a core element in modeling, simulation and optimization of complex systems. In this talk, a new formulation for analyzing uncertainty sensitivity, quantifying uncertainty and visualizing uncertainty will be discussed. An integrated simulation framework will be presented that quantifies both numerical and modeling errors in an effort to establish “error bars” in CFD. In particular, stochastic formulations based on Galerkin and collacation versions of the generalized Polynomial Chaos (gPC) will be discussed. Additionally, we will present some effective new ways of dealing with this “curse of dimensionality”. Particularly, adaptive ANOVA decomposition, and some stochastic sensitivity analysis techniques will be discussed in some detail. Several specific examples on sensitivity analysis and predictive modeling of thrombin production in blood coagulation chemical reaction network, flow and transport in randomly heterogeneous porous media, random roughness problem, and uncertainty quantification in carbon sequestration will be presented to illustrate the main idea of our approach.
 
A mesoscale particle based numerical method, Dissipative Particle Dynamics (DPD) is employed to model the red blood cell (RBC) deformation. RBC’s have highly deformable viscoelastic membranes exhibiting complex rheological response and rich hydrodynamic behavior governed by special elastic and bending properties and by the external/internal fluid and membrane viscosities. We present a multiscale RBC model that is able to predict RBC mechanics, rheology, and dynamics in agreement with experiments. The dynamics of RBC’s in shear and Poiseuille flows is tested against experiments.