Georgia Institute of Technology College of Sciences

We will discuss properties of manifolds obtained by deleting a totally geodesic ``divisor'' from hyperbolic manifold. Fundamental groups of these manifolds do not generally fit into any class of groups studied by the geometric group theory, yet the groups turn out to be relatively hyperbolic when the divisor is ``sparse'' and has ``normal crossings''.

We examine a variety of problems in delay-differential equations. Among the new results we discuss are existence and asymptotics for multiple-delay problems, global bifurcation of periodic solutions, and analyticity (or lack thereof) in variable-delay problems. We also plan to discuss some interesting open questions in the field.

In this lecture the analog of Riemannian manifold will be introduced through the notion of spectral triple. The recent work on the case of a metric Cantor set, endowed with an ultrametric, will be described in detail during this lecture. An analog of the Laplace Beltrami operator for a metric Cantor set will be defined and studied

For general graphs, approximating the maximum clique is a notoriously hard problem even to approximate to a factor of nearly n, the number of vertices. Does the situation get better with random graphs? A random graph on n vertices where each edge is chosen with probability 1/2 has a clique of size nearly 2\log n with high probability. However, it is not know how to find one of size 1.01\log n in polynomial time. Does the problem become easier if a larger clique were planted in a random graph? The current best algorithm can find a planted clique of size roughly n^{1/2}. Given that any planted clique of size greater than 2\log n is unique with high probability, there is a large gap here. In an intriguing paper, Frieze and Kannan introduced a tensor-based method that could reduce the size of the planted clique to as small as roughly n^{1/3}. Their method relies on finding the spectral norm of a 3-dimensional tensor, a problem whose complexity is open. Moreover, their combinatorial proof does not seem to extend beyond this threshold. We show how to recover the Frieze-Kannan result using a purely probabilistic argument that generalizes naturally to r-dimensional tensors and allows us recover cliques of size as small as poly(r).n^{1/r} provided we can find the spectral norm of r-dimensional tensors. We highlight the algorithmic question that remains open. This is joint work with Charlie Brubaker.