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.

Let E be an elliptic curve over an algebraically closed field k of
characteristic p>0. Then the physical p-torsion E[p](k) is either trivial,
and E is called supersingular, or E[p](k) is a group of order p. More
generally, if X/k is an abelian variety of dimension g, then X[p](k)
is isomorphic to (Z/p)^f for some number f, called the p-rank of X.
The p-rank induces a stratification of the moduli space of abelian
varieties; via the Torelli functor, it induces a stratification of the
moduli space of (hyperelliptic) curves.
I'll discuss recent results on the geometry of these strata, with
special attention to their structure at the boundary of the moduli
space. This information yields new applications about the prime-to-p
part of the class group of a quadratic function field with specified geometric
p-rank; the existence of absolutely simple hyperelliptic Jacobians of
every p-rank; and the stratification of the moduli space of curves by
Newton polygon.

Generalized ideal class groups can be described adelically in terms of a coset space for the group GL1, and this in turn leads to a notion of "class number" (as the size of a certain set, if finite) for an arbitrary affine algebraic group over a global field. Related to this is the notion of the "Tate-Shafarevich set" of an algebraic group, which is tied up with questions relating global and local information. Finiteness of class numbers and Tate-Shafarevich sets for affine algebraic groups was proved by Borel and his coworkers over number fields, andif one grants the finiteness of Tate-Shafarevich groups for abelian varieties then Mazur showed how to get such finiteness for all algebraic group varieties over number fields (which has applications to the local-to-global principle for projective varieties over number fields).

The above methods do not apply over global function fields. After reviewing some history, I will explain the content of a recent classification theorem of "pseudo-reductive groups" proved jointly with Gabber and G. Prasad that makes it possible to prove the analogous finiteness theorems in the function field case away from characteristic 2. If time permits I will say something about how this classification theorem is used to get such results.