One can rarely identify extremizers of nontrivial inequalities, yet one can not infrequently show that extremizers exist. Thus one asks what qualitative and quantitative properties can be established. One way to attack such questions is to exploit Euler-Lagrange equations which extremizers must satisfy. Inequalities involving L^p norms, with p not equal to 2, lead to nonlinear equations. In this talk we discuss the nonlinear, nonlocal Euler-Lagrange equation which arises in connection with such an inequality for the Radon transform. We show that all solutions are infinitely differentiable, and have a certain rate of decay at spatial infinity. (joint work with Qingying Xue)

TBD

This is joint work with Louis-Pierre Arguin, Michael Damron and Dan Stein (arXiv:0911.4201). It is an open problem to determine the number of infinite-volume ground states in the Edwards-Anderson (nearest neighbor) spin glass modelon Z^d for d \geq 2 (with, say, mean zero Gaussian couplings). This is a limiting case of the problem of determining the number of extremal Gibbs states at low temperature. In both cases, there are competing conjectures for d \geq 3, but no complete results even for d=2. I report on new results which go some way toward proving that (with zero external field, so that ground states come in pairs, related by a global spin flip) there is only a single ground state pair (GSP). Our result is weaker in two ways: First, it applies not to the full plane Z^2, but to a half-plane. Second, rather than showing that a.s. (with respect to the quenched random coupling realization J) there is a single GSP, we show that there is a natural joint distribution on J and GSP's such that for a.e. J, the conditional distribution on GSP's given J is supported on only a single GSP. The methods used are a combination of percolation-like geometric arguments with translation invariance (in one of the two coordinate directions of the half-plane) and uses as a main tool the "excitation metastate" which is a probability measure on GSP's and on how they change as one or more individual couplings vary.

Bregman iteration has been around since 1967. It turns out to be unreasonably effective for optimization problems involving L1, BV and related penalty terms. This is partly because of a miraculous cancellation of error. We will discuss this and give biomedical imaging applications, related to compressive sensing and Total Variation based restoration.

I will discuss the reliability of large networks of coupled oscillators in response to fluctuating inputs. The networks considered are quite generic. In this talk, I view them as idealized models from neuroscience and borrow some of the associated language. Reliability is the opposite of trial-to-trial variability; a system is reliable if a signal elicits identical responses upon repeated presentations. I will address the problem on two levels: neuronal reliability, which concerns the behavior of individual neurons (or oscillators) embedded in the network, and pooled-response reliability, which measures total outputs from subpopulations. The effects of network structure, cell heterogeneity and noise on reliability will be discussed. Our findings are based largely on dynamical systems ideas (with a slight statistical mechanics flavor) and are supported by simulations. This is joint work with Kevin Lin and Eric Shea-Brown.