University of British Columbia

Of the canonical stratified shear flow instabilities (Kelvin–Helmholtz, Holmboe-wave and Taylor–Caulfield), the Taylor–Caulfield instability (TCI) has received relatively little attention, and forms the focus of the presentation. A diagnostic of the linear instability dynamics is developed that exploits the net pseudomomentum to distinguish TCI from the other two instabilities for any given flow profile. Next, the nonlinear dynamics of TCI is shown across its range of unstable horizontal wavenumbers and bulk Richardson numbers. At small bulk Richardson numbers, a cascade of billow structures of sequentially smaller size may form. For large bulk Richardson numbers, the primary nonlinear travelling waves formed by the linear instability break down via a small-scale, Kelvin– Helmholtz-like roll-up mechanism with an associated large amount of mixing. In all cases, secondary parasitic nonlinear Holmboe waves appear at late times for high Prandtl number. Finally, a nonlinear diagnostic is proposed to distinguish between the saturated states of the three canonical instabilities based on their distinctive density–streamfunction and generalised vorticity–streamfunction relations.

Ice streams are narrow bands of rapidly sliding ice within an otherwise

slowly flowing continental ice sheet. Unlike the rest of the ice sheet,

which flows as a typical viscous gravity current, ice streams experience

weak friction at their base and behave more like viscous 'free films' or

membranes. The reason for the weak friction is the presence of liquid

water at high pressure at the base of the ice; the water is in turn

generated as a result of dissipation of heat by the flow of the ice

stream. I will explain briefly how this positive feedback can explain the

observed (or inferred, as the time scales are rather long) oscillatory

behaviour of ice streams as a relaxation oscillation. A key parameter in

simple models for such ice stream 'surges' is the width of an ice stream.

Relatively little is understood about what controls how the width of an

ice stream evolves in time. I will focus on this problem for most of the

talk, showing how intense heat dissipation in the margins of an ice stream

combined with large heat fluxes associated with a switch in thermal

boundary conditions may control the rate at which the margin of an ice

stream migrates. The relevant mathematics involves a somewhat non-standard

contact problem, in which a scalar parameter must be chosen to control the

location of the contact region. I will demonstrate how the problem can be

solved using the Wiener-Hopf method, and show recent extensions of this

work to more realistic physics using a finite element discretization.

Donaldson-Thomas invariants are fundamental deformation invariants of Calabi-Yau threefolds. We describe a recent conjecture of Oberdieck and Pandharipande which predicts that the (three variable) generating function for the Donaldson-Thomas invariants of K3xE is given by the reciprocal of the Igusa cusp form of weight 10. For each fixed K3 surface of genus g, the conjecture predicts that the corresponding (two variable) generating function is given by a particular meromorphic Jacobi form. We prove the conjecture for K3 surfaces of genus 0 and genus 1. Our computation uses a new technique which mixes motivic and toric methods.

It is well known that electrical resistance arguments provide (usually) the best method for determining whether a graph is transient or recurrent. In this talk I will discuss a similar characterization of 'sub-diffusive behaviour' -- this occurs in spaces with many obstacles or traps.

The characterization is in terms of the energy of functions in annuli.

Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically by introducing an artificial time variable. The actual computational model involves a discretization of the now time-dependent differential system, usually employing forward Euler. The resulting dynamics of such an algorithm is then a discrete dynamics, and it is expected to be ''close enough'' to the dynamics of the continuous system (which is typically easier to analyze) provided that small -- hence many -- time steps, or iterations, are taken. Indeed, recent papers in inverse problems and image processing routinely report results requiring thousands of iterations to converge. This makes one wonder if and how the computational modeling process can be improved to better reflect the actual properties sought.

In this talk we elaborate on several problem instances that illustrate the above observations. Algorithms may often lend themselves to a dual interpretation, in terms of a simply discretized differential equation with artificial time and in terms of a simple optimization algorithm; such a dual interpretation can be advantageous. We show how a broader computational modeling approach may possibly lead to algorithms with improved efficiency.

It is now known that the overall behaviour of a simple random walk (SRW) on

supercritical (p>p_c) percolation cluster in Z^d is similiar to that of the SRW

in Z^d. The critical case (p=p_c) is much harder, and one needs to define the

'incipient infinite cluster' (IIC). Alexander and Orbach conjectured in 1982

that the return probability for the SRW on the IIC after n steps decays like

n^{2/3} in any dimension. The easiest case is that of trees; this was studied by

Kesten in 1986, but we can now revisit this problem with new techniques.