In most practical applications of fluid mechanics, it is the interaction of the fluid with the boundary that is most critical to understanding the behavior of the fluid. Physically important parameters, such as the lift and drag of a wing, are determined by the sharp transition the air makes from being at rest on the wing to flowing freely around the airplane near the wing. Mathematically, the behavior of such flows are modeled by the Navier-Stokes equations. In this talk, I will discuss the asymptotic behavior of solutions to the Navier-Stokes equations at small viscosity under various boundary conditions.

In this talk, which will be will be aimed at a general math audience, we shall overview an approach to analyze coupled slow-fast dynamics via Young measures, namely, via probability measure valued maps. The framework is ordinary differential equations, possibly controlled.

We shall address the modeling issue, motivating examples, the mathematical analysis and, in brief, numerical prospects.

Composite materials can have properties unlike any found in nature, and in this case they are known as metamaterials. Recent attention has been focused on obtaining metamaterials which have an interesting dynamic behavior. Their effective mass density can be anisotropic, negative, or even complex. Even the eigenvectors of the effective mass density tensor can vary with frequency. Within the framework of linear elasticity, internal masses can cause the effective elasticity tensor to be frequency dependent, yet not contribute at all to the effective mass density at any frequency. One may use coordinate transformations of the elastodynamic equations to get novel unexpected behavior. A classical propagating wave can have a strange behavior in the new abstract coordinate system. However the problem becomes to find metamaterials which realize the behavior in the new coordinate system. This can be solved at a discrete level, by replacing the original elastic material with a network of masses and springs and then applying transformations to this network. The realization of the transformed network requires a new type of spring, which we call a torque spring. The forces at the end of the torque spring are equal and opposite but not aligned with the line joining the spring ends. We show how torque springs can theoretically be realized. This is joint work with Lindsay Botton, Mark Briane, Andrej Cherkaev, Fernando Guevara Vasquez, Ross McPhedran, Nicolae Nicorovici, Daniel Onofrei, Pierre Seppecher, and John Willis.

A typical step in matrix algebra is elimination, and its description as a triangular factorization. For a doubly infinite banded Toeplitz matrix A, that step is made easy by factoring the polynomial a(z) whose coefficients come from the diagonals of A. What to do if A is not Toeplitz?

A nice case is a permutation matrix (on Z). Which is the main diagonal? For the (Toeplitz) example of a shift matrix, the main diagonal contains the 1's. We identify the correct diagonal for every banded permutation. Then we consider banded matrices (not Toeplitz!) as operators on L2(Z) and ask about their factorization.

A special case is when the inverse of A is also banded -- these matrices factor into block-diagonal matrices. The help coming from analysis is the theory of Fredholm operators.

In this talk we will consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on N vertices. The processes are allowed to spread with different rates, start from vertex subsets of different sizes or at different times. We obtain tight results regarding the sizes of the vertex sets occupied by each process, showing that in the generic situation one process will occupy roughly N^alpha vertices, for some 0 < alpha < 1. The value of alpha is calculated in terms of the relative rates of the processes, as well as the sizes of the initial vertex sets and the possible time advantage of one process. These results are in sharp contrast with the picture in the lattice case.
This is a joint work with Yael Dekel, Elchanan Mossel and Yuval Peres.