Hadwiger's conjecture states that if a graph is not t-colorable then it contains the complete graph on t+1 vertices as a minor. The case t=4 is equivalent to the four color theorem and the case t=5 was proved by Robertson, Seymour, and Thomas with the use of the four color theorem. For t>5, the conjecture remains open. Reed and Seymour have also proved that Hadwiger's conjecture holds for line graphs and in a recent work with Maria Chudnovsky we proved that Hadwiger's conjecture holds for a proper superset of the class of line graphs, known as the class of quasi-line graphs (graphs where the neighbor set of every vertex is the union of two cliques).

A graph is claw-free if it does not have a vertex with three pairwise nonadjacent neighbors. The class of claw-free graphs is a proper superset of the class of quasi-line graphs. In this talk I will outline the proof of a weakened version of Hadwiger's conjecture for claw-free graphs. The proof uses a structure theorem from a recent work of Chudnovsky and Seymour. At the end of the talk I will discuss the progress that has been made on the actual Hadwiger's conjecture for claw-free graphs. This is a joint work with Maria Chudnovsky.

In many applications, the main goal is to obtain a global low dimensional representation of the data, given some local noisy geometric constraints. In this talk we will show how the problems listed below can be efficiently solved by constructing suitable operators on their data and computing a few eigenvectors of sparse matrices corresponding to the data operators.

* Cryo Electron Microscopy for protein structuring: reconstructing the three-dimensional structure of a molecule from projection images taken at random unknown orientations (unlike classical tomography, where orientations are known).
* NMR spectroscopy for protein structuring: finding the global positioning of all hydrogen atoms in a molecule from their local distances. Distances between neighboring hydrogen atoms are estimated from the spectral lines corresponding to the short ranged spin-spin interaction.
* Sensor networks: finding the global positioning from noisy local distances.

Joint work with Ronald Coifman, Yoel Shkolnisky (Yale Applied Math) and Fred Sigworth (Yale School of Medicine).

Wilf asked in the 1980s about f(n,m,l), the maximum number of l-colorings that a graph with n vertices and m edges can have. We essentially solve this problem for l=3, in particular proving, for all large n, the conjecture of Lazebnik (1989) that if m\le n^2/4 then the maximum number of 3-colorings is achieved by a semi-complete biparite graph. This is joint work with Po-Shen Loh and Benny Sudakov.

The hydrodynamics of complex fluids, such as polymer solutions and colloidal suspensions, has attracted great interest due to recent advances in fabrication of micro- and nano-fluidic devices. I will first review recent advances in mesoscopic numerical methods for simulating the interaction between complex fluid flow and suspended macro molecules and structures. Computational issues at play include coarse-graining to bridge the large gap in timescales and length scales, coupling between disparate methods such as molecular dynamics and Navier-Stokes solvers, the inclusion of thermal fluctuations.

I will then present my recent work at LLNL to develop novel particle methods for modeling polymer chains in flow. Typically, Molecular Dynamics (MD) is used for the polymer chains, and the solvent is modeled with a mesoscopic method. In our algorithm, termed Stochastic Event-Driven Molecular Dynamics (SEDMD) [A. Donev and A. L. Garcia and B. J. Alder, J. Comp. Phys., 227(4), 2644-2665, 2008], polymers are modeled as chains of hard spheres and the solvent is modeled using a dense-fluid generalization of the Direct Simulation Monte Carlo (DSMC) method [Phys. Rev. Lett., 101, 075902, 2008]. Even with all of the speedup compared to brute-force MD the algorithm is still time-consuming due to the large number of solvent particles necessary to fill the computational domain. It is natural to restrict the particle model only to regions close to a polymer chain and use a lower-resolution continuum model elsewhere. I will present a hybrid method that couples an explicit fluctuating compressible Navier-Stokes solver with the particle method. The coupling is flux-based and generalizes previous work [J. B. Bell and A. L. Garcia and S. A. Williams, SIAM Multiscale Modeling and Simulation, 6, 1256-1280, 2008] to dense fluids as appropriate for polymer problems.

I will conclude with a look into the challenges of developing a simulation methodology capable of simulating macroscopic flows of complex fluids with atomistic fidelity.

The main goal of the talk is to introduce a new cap-and-trade scheme design for the control and the reduction of atmospheric pollution. The tools developed for the purpose of the study are intended to help policy makers and regulators understand the pros and cons of the emissions markets at a quantitative level.

We propose a model for an economy where risk neutral firms produce goods to satisfy an inelastic demand and are endowed with permits by the regulator in order to offset their pollution at compliance time and avoid having to pay a penalty. Firms that can easily reduce emissions do so, while those for which it is harder buy permits from those firms anticipating that they will not need them, creating a financial market for pollution credits.

Our model captures most of the features of the European Union Emissions Trading Scheme. We show existence of an equilibrium and uniqueness of emissions credit prices. We also characterize the equilibrium prices of goods and the optimal production and trading strategies of the firms. We choose the electricity market in Texas to illustrate numerically the qualitative properties observed during the implementation of the first phase of the European Union cap-and-trade CO2 emissions scheme, comparing the results of cap-and-trade schemes to the Business As Usual benchmark. In particular, we confirm the presence of windfall profits criticized by the opponents of these markets. We also demonstrate the shortcomings of tax and subsidy alternatives. Finally we introduce a relative allocation scheme which, despite its ease of implementation, leads to smaller windfall profits than the standard scheme.

Recent work in the emerging field of compressed sensing indicates that, when feasible, judicious selection of the type of image transformation induced by imaging systems may dramatically improve our ability to perform reconstruction, even when the number of measurements is small relative to the size and resolution of the final image. The basic idea of compressed sensing is that when an image is very sparse (i.e. zero-valued at most locations) or highly compressible in some basis, relatively few incoherent observations suffice to reconstruct the most significant non-zero basis coefficients. These theoretical results have profound implications for the design of new imaging systems, particularly when physical and economical limitations require that these systems be as small, mechanically robust, and inexpensive as possible.

In this talk I will describe the primary theory underlying compressed sensing and discuss some of the key mathematical challenges associated with its application to practical imaging systems. In particular, I will explore several novel imaging system designs based on compressed sensing, including compressive coded aperture and hyperspectral imagers, and examine the interplay between compressed sensing theory and the practical physical constraints which must be considered to effectively exploit this theory.

The microscopic structure of a macroscopic system in a steady state is described locally, i.e. at a suitably scaled macroscopic point $x$, by a time invariant measure of the corresponding infinite system with translation invariant dynamics. This measure may be extremal, with good decay of correlations, or a superposition of extremal measures, with weights depending on $x$ (and possibly even on the way one scales).

I will illustrate the above by some exact results for 1D lattice systems with two types of particles (plus holes) evolving according to variants of the simple asymmetric exclusion process, in open or closed systems. Somewhat surprisingly, the spatially asymmetric local dynamics satisfy (in some cases) detailed balance with respect to a global Gibbs measure with long range pair interactions.