Georgia Institute of Technology College of Sciences

1-bit compressed sensing combines the dimension reduction of compressed sensing with extreme quantization -- only the sign of each linear measurement is retained. We discuss recent convex-programming approaches with strong theoretical guarantees. We also discuss connections to related statistical models such as sparse logistic regression. Behind these problems lies a geometric question about random hyperplane tessellations. Picture a subset K of the unit sphere, as in the continents on the planet earth. Now slice the sphere in half with a hyperplane, and then slice it several times more, thus cutting the set K into a number of sections. How many random hyperplanes are needed to ensure that all sections have small diameter? How is the geodesic distance between two points in K related to the number of hyperplanes separating them? We show that a single geometric parameter, the mean width of K, governs the answers to these questions.

Front propagation in fluid flows arise in power generation of automobile engines, forest fire spreading, and material interfaces of solidification to name a few. In this talk, we introduce the level set formulation and the resulting Hamilton-Jacobi equation, known as G-equation in turbulent combustion. When the fluid flow has enough intensity, G-equation becomes non-coercive and non-linearity no longer dominates. When front curvature and flow stretching effects are included, the extended G-equation is also non-convex. We discuss recent progress in analysis and computation of homogenization and large time front speeds in cellular flows (two dimensional Hamiltonian flows) from both Lagrangian and Eulerian perspectives, and the recovery of experimental observations from the G-equations.

For dimensions n greater than or equal to 3, and integers N greater than 1, there is a distribution of points P in a unit cube [0,1]^{n}, of cardinality N, for which the discrepancy function D_N associated with P has an optimal Exponential Orlicz norm. In particular the same distribution will have optimal L^p norms, for 1 < p < \infty. The collection P is a random digit shift of the examples of W.L. Chen and M. Skriganov.