In this talk, I will be discussing various funding and other opportunities for mathematical and statistical sciences at NSF. New programs in computational and data-enabled sciences will be discussed. Also,this is an opportunity to hear feedback from the community regarding future needs.

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.

Several recent results on the regularity of streamlines beneath a rotational travelling wave, along with the wave profile itself, will be discussed.

The talk includes the classical water wave problem in both finite and infinite depth, capillary waves, and solitary waves as well. A common assumption in all models to be discussed is the absence of stagnation points.

In statistical physics, systems like percolation and Ising models are of particular interest at their critical points. Critical systems have long-range correlations that typically decay like inverse powers. Their continuum scaling limits, in which the lattice spacing shrinks to zero, are believed to have universal dimension-dependent properties. In recent years critical two-dimensional scaling limits have been studied by Schramm, Lawler, Werner, Smirnov and others with a focus on the boundaries of large clusters. In the scaling limit these can be described by Schramm-Loewner Evolution (SLE) curves.

In this talk, I'll discuss a different but related approach, which focuses on cluster area measures. In the case of the two-dimensional Ising model, this leads to a representation of the continuum Ising magnetization field in terms of sums of certain measure ensembles with random signs. This is based on joint work with F. Camia (PNAS 106 (2009) 5457-5463) and on work in progress with F. Camia and C. Garban.

Alexandrov geometry reflects the geometry of Riemannian manifolds when stripped from everything but their structure as metric spaces with a (local) lower curvature bound. In this talk I will define Alexandrov spaces and discuss basic properties, constructions and examples. By now there are numerous applications of Alexandrov geometry, including Perelman's spectacular solution of the geometrization conjecture for 3-manifolds.

The utility of Alexandrov geometry to Riemannian geometry is due to a large extend by the fact that there are several geometrically natural constructions that are closed in Alexandrov geometry but not in Riemannian geometry. These include, but are not limited to (1) Taking Gromov-Hausdorff limits, (2) Taking quotients, and (3) forming cones, jones etc of positively curved spaces. In the talk I will give examples of applications of each of these and one additional new construction.