Extending results of Van Der Vlugt,
I shall derive a new non-trivial upper bound for the dimension of trace
codes connected to algebraic-geometric codes. Furthermore, I will deduce
their true dimension if certain conditions are satisfied. Finally,
potential areas of improvement and other related results will be outlined.

We consider a polymer measure based on random walks which are based on sums of iid stable random variables.
A Gibbs measure is defined which models an attraction to the origin for these walks. A phase transition occurs as the the strength of the attraction to the origin occurs.
We examine various "thermodynamic" quantities and show they are all related to each other in a simple way and exhibit universality.

Lagrangian coherent structures (LCS) are best described as moving curves in a fluid that separate particles that have qualitatively different trajectories. For instance, particles that circulate in an ocean bay have a separate behavior from particles that go on by the bay and don't get caught up in the circulation. Interestingly, these two classes of particles are separated by a sharp, but moving curve. Similar structures are found in Hurricanes: which particle are going to get swept up in the Hurricane and which don't? Likewise in Jellyfish, some particles enter the underbelly of the jellyfish and bring nutrients, while others are swept downstream to help propel the jellyfish. The way blood flows over a clot, as revealed by LCS can indicate whether or not the clot is dangerous. This lecture will give examples of this sort, explain how the LCS are computed and are connected with other mathematical constructions, such as Smale horseshoes in dynamical systems.

Computation on large combinatorial structures -- graphs, strings,
partial orders, etc. -- has become fundamental in many areas of data
analysis and scientific modeling. The field of high-performance
combinatorial computing, however, is in its infancy. By way of
contrast, in numerical supercomputing we possess standard algorithmic
primitives, high-performance software libraries, powerful
rapid-prototyping tools, and a deep understanding of effective
mappings of problems to high-performance computer architectures.

This talk will describe several challenges for the field of
combinatorial scientific computing in algorithms, tools,
architectures, and mathematics. I will draw examples from several
applications, and I will highlight our group's work on
high-performance implementation of algebraic primitives for
computation on large graphs.