This talk will give a basic introduction to ocean-ice interactions, which is a major research topic for those concerned with predicting sea level rise. The talk will review the mechanisms involved in studying this problem, and will present some experimental (laboratory), model and observational results from David Holland's students working on the 9th floor. Some photos and scientific findings from the recent field season to the McMurdo Ice Shelf, Antarctica, will be shown.

We shall give a very elementary description, understandable to all, no matter what their mathematical training, of several problems in Discrete Geometry, also called Combinatorial Geometry. Then we shall do the same for the sub area which has been my special interest, Geometric Transversal Theory.

Bates, Johnson, Lindenstrauss, Preiss, and Schechtman proved that Lipschitz maps from the unit ball of a finite dimensional space into a superreflexive Banach space must be approximately affine on some smaller ball of a controlled radius r. However, one cannot read any kind of estimate of r from their proof. We present a new proof that gives a concrete lower bound for r. We also apply the affine approximation estimate to reprove Bourgain's discretization theorem for a restricted case and give a background to the related Ribe program.

Nonlinear interactions between waves and vortices arise in many areas of fluid dynamics, with the most prominent examples probably occurring in the geophysical fluid dynamics of the atmosphere and the oceans. Indeed, our ability to predict the long-term dynamics of these systems hinges delicately on our ability to model small-scale waves and their unresolvable interactions with the large-scale vortical flow. This has led to a substantial body of wave-mean interaction theory that has been developed for these GFD problems, but which is also very useful for other applications. In this talk, I will present some of the relevant theory and various applications of it to diverse problems in fluid dynamics. Specifically, examples will include the driving of longshore currents by breaking ocean waves, the creation of deep ocean vortex motion by dissipating internal tides, the micro-mixing of small droplets using acoustic waves, and some interactions between quantized vortices and phonon waves in superfluids.

Stochastic models are often formed by adding noise to existing deterministic models of physical systems. These stochastic models are most interesting when they display qualitatively different behavior from their deterministic counterparts. In this talk, I will discuss examples from biology and physics, highlighting cases when it is important to include noise within the model. I will also mention a few methods used to simulate and analyze these stochastic models.

A convex set is said to have constant width 1 if the projection of the body onto every line has length 1. It is well known that, in any dimension, the ball of diameter 1 encloses the most volume of any constant width set. A famous theorem of Lesbesgue and Blaschke asserts that the Reuleaux triangle has least area amongst any two dimensional set of constant width. In this talk, we present a few ideas that might be used to approach the minimum problem in dimension greater than two.

In the last five years, several difficult combinatorial problems have been solved by an unexpected argument using polynomials. The combinatorial problems involved have to do with the way that lines intersect in Euclidean space. We will discuss the example of the joints problem - a problem about the intersections of lines in 3-dimensional space. This problem was posed in the early 90's and was open for close to twenty years. We now have a one page proof, which I want to explain in detail.

Characteristic classes in their various guises (Chern classes in complex geometry, Stiefel-Whitney and Pontrjagen classes in real geometry, etc.) underlay major 20th century mathematical developments in a vast range of areas, including topology, global analysis, differential geometry, complex variables, algebraic geometry, number theory. PDE's, etc. Their roots go back to the Euler characteristic and they began as invariants of manifolds and vector bundles. But in recent decades have very fruitfully been extended and applied in ever more general settings. This talk will give an introductory survey which will not assume any prior familiarity with them.