A systematic understanding of traffic dynamics on road networks is crucial for many transportation studies and can help to develop more efficient ramp metering, evacuation, signal control, and other management and control strategies. In this study, we present a theory of multi-commodity network traffic flow based on the Lighthill-Whitham-Richards (LWR) model. In particular, we attempt to analyze kinematic waves of the Riemann problem for a general junction with multiple upstream and downstream links. In this theory, kinematic waves on a link can be determined by its initial condition and prevailing stationary state. In addition to stationary states, a flimsy interior state can develop next to the junction on a link. In order to pick out unique, physical solutions, we introduce two types of entropy conditions in supply-demand space such that (i) speeds of kinematic waves should be negative on upstream links and positive on downstream links, and (ii) fair merging and First-In-First-Out diverging rules are used to prescribe fluxes from interior states. We prove that, for given initial upstream demands, turning proportions, and downstream supplies, there exists a unique critical demand level satisfying the entropy conditions. It follows that stationary states and kinematic waves on all links exist and are unique, since they are uniquely determined by the critical demand level. For a simple model of urban or freeway intersections with four upstream and four downstream links, we demonstrate that theoretical solutions are consistent with numerical ones from a multicommodity Cell Transmission Model. In a sense, the proposed theory can be considered as the continuous version of the multi-commodity Cell Transmission Model with fair merging and First-In-First-Out diverging rules. Finally we discuss future research topics along this line.

Mathematical modeling, analysis and numerical simulation, combined with imaging and experimental validation, provide a powerful tool for studying various aspects of cardiovascular treatment and diagnosis. This talk will address two examples where such a synergy led to novel results. The first example concerns a mathematical study of fluid-structure interaction (FSI) in blood flow with clinical application to 2D and 3D Doppler assessment of mitral regurgitation (MR). Our computational studies, performed in collaboration with several experts in echocardiography, addressed current imaging challenges in Doppler assessment of MR, which led to refinement and reinforcement of the emerging 3D echocardiographic applications. The second example concerns a novel dimension reduction/multi-scale approach to modeling of endovascular stents as 3D meshes of 1D curved rods forming a 3D network of 1D hyperbolic conservation laws. Our computational studies, motivated by the questions posed to us by cardiologists at the Texas Heart Institute, provided novel insight into the mechanical properties of 4 currently available coronary stents on the US market, and suggested optimal stent design for a novel application of stents in transcatheter aortic valve replacement.
The applications discussed above gave rise to new mathematical problems whose solutions required a development of sophisticated mathematical ideas. They include a design of a novel unconditionally stable, loosely coupled partitioned scheme for numerical simulation of solutions to FSI in blood flow, and the development of the theory and numerics for nonlinear hyperbolic nets and networks arising in dimension reduction of the stent problem. An overview of the basic mathematical ideas associated with this research, and application to the two related problems in cardiovascular diagnosis and treatment, will be presented. This talk will be accessible to a wide scientific audience.

Abstract: I'll talk about the recently discovered strong negative dependence properties of the symmetric exclusion process, a model of non-intersecting random walkers. The negative dependence theory gives a simple way to show central limit theorems for the bulk motion of particles. Our results are general enough to deal with non-equilibrium systems of particles with inhomogeneous hopping rates.