The general average distance problem, introduced by Buttazzo, Oudet, and Stepanov,  asks to find a good way to approximate a high-dimensional object, represented as a measure, by a one-dimensional object. We will discuss two variants of the problem: one where the one-dimensional object is a measure with connected one-dimensional support and one where it is an embedded curve. We will present examples that show that even if the data measure is smooth the nonlocality of the functional can cause the minimizers to have corners. Nevertheless the curvature of the minimizer can be considered as a measure. We will discuss a priori estimates on the total curvature and ways to obtain information on topological complexity of the minimizers. We will furthermore discuss functionals that take the transport along the network into account and model best ways to design transportation networks. (Based on joint works with Xin Yang Lu and Slav Kirov.)

High-dimensional data can be difficult to analyze. Assume data are distributed on a low-dimensional manifold. The Vector Diffusion Mapping (VDM), introduced by Singer-Wu, is a non-linear dimension reduction technique and is shown robust to noise. It has applications in cryo-electron microscopy and image denoising and has potential application in time-frequency analysis.

 
In this talk, I will present a theoretical analysis of the effectiveness of the VDM. Specifically, I will discuss parametrisation of the manifold and an embedding which is equivalent to the truncated VDM. In the differential geometry language, I use eigen-vector fields of the connection Laplacian operator to construct local coordinate charts that depend only on geometric properties of the manifold. Next, I use the coordinate charts to embed the entire manifold into a finite-dimensional Euclidean space. The proof of the results relies on solving the elliptic system and provide estimates for eigenvector fields and the heat kernel and their gradients.

 In the 1960s, Benjamin and Feir, and Whitham, discovered that a Stokes wave would be unstable to long wavelength perturbations, provided that (the carrier wave number) x (the undisturbed water depth) > 1.363.... In the 1990s, Bridges and Mielke studied the corresponding spectral instability in a rigorous manner. But it leaves some important issues open, such as the spectrum away from the origin. The governing equations of the water wave problem are complicated. One may resort to simpler approximate models to gain insights.

I will begin by Whitham's shallow water equation and the modulational instability index for small amplitude and periodic traveling waves, the effects of surface tension and vorticity. I will then discuss higher order corrections, extension to bidirectional propagation and two-dimensional surfaces. This is partly based on joint works with Jared Bronski (Illinois), Mat Johnson (Kansas), and Ashish Pandey (Illinois).

In the first part of this talk I will do a brief introduction to the recent theory of Mean Field Games (MFG) initiated by J.-M. Lasry and P.-L. Lions. The main objective of the MFG theory is the study of the limit behavior of Nash equilibria for symmetric differential games with a very large number of “small” players. In its simplest form, as the number of players tends to infinity, limits of Nash equilibria can be characterized in terms of the solution of a coupled system of a Hamilton-Jacobi and Fokker-Planck (or continuity) equations. The first equation describes the evolution of the value function of a typical agent, while the second one characterizes the evolution of the agents’ density. In the second part, I will introduce a variational MFG model, where we impose a density constraint. From the modeling point of view, imposing this new constraint means that we are aiming to avoid congestion among the agents. We will see that a weak solution of the system contains an extra term, an additional price imposed on the saturated zones. I will show that this price corresponds to the pressure field from the models of incompressible Euler equations à la Brenier. If time permits, I will discuss the regularity properties of the pressure variable, which allows us to write optimality conditions at the level of single-agent trajectories and to define a weak notion of Nash equilibrium for our model. The talk is based on a joint work with P. Cardaliaguet (Paris Dauphine) and F. Santambrogio (Paris-Sud, Orsay).