Isostasy is one of the earliest quantitative geophysical theories still in current use. It explains why observed gravity anomalies are generally much weaker than what is inferred from visible topography, and why planetary crusts can support large mountains without breaking up. At large scale, most topography (including bathymetry) is in isostatic equilibrium, meaning that surface loads are buoyantly supported by crustal thickness variations or density variations within the crust and lithosphere, in such a way that deeper layers are hydrostatic. On Earth, examples of isostasy are the average depth of the oceans, the elevation of the Himalayas, and the subsidence of ocean floor away from mid-ocean ridges, which are respectively attributed to the crust-ocean thickness difference, to crustal thickening under mountain belts, and to the density increase due to plate cooling. Outside Earth, isostasy is useful to constrain the crustal thickness of terrestrial planets and the shell thickness of icy moons with subsurface oceans.

Given the apparent simplicity of the isostatic concept – buoyant support of mountains by iceberg-like roots – it is surprising that a debate has been going on for over a century about its various implementations. Classical isostasy is indeed not self-consistent, neglects internal stresses and geoid contributions to topographical support, and yields ambiguous predictions of geoid anomalies at the planetary scale. In the last few years, these problems have attracted renewed attention when applying isostasy to planetary bodies with an unbroken crustal shell. In this talk I will discuss isostatic models based on the minimization of stress, on time-dependent viscous evolution, and on stationary viscous flow. I will show that these new isostatic approaches are mostly equivalent and discuss their implications for the structure of icy moons.

Rivers choose their size and shape, and spontaneously organize into ramified networks. Yet, they are essentially a channelized flow of water that carries sediment. Based on laboratory experiments, field measurements and simple theory, we will investigate the basic mechanisms by which rivers form themselves, and carve the landscapes that surround us.

We outline Dirac cohomology for Lie algebras and Vogan’s conjecture. We then cover some basic material on Schur-Weyl duality and Arakawa-Suzuki functors. Finishing with current efforts and results on generalising Vogan’s conjecture to a Schur-Weyl duality setting. This would relate the centre of a Lie algebra with the centre of the relevant tantaliser algebra. We finish by considering a unitary module X and giving a bound on the action of the tantalizer algebra.

I will discuss algebraic conditions that for a given group guarantee or characterize the vanishing of cohomology in a given degree with coefficients in any unitary representation. These conditions will be expressed in terms positivity of certain elements over group algebras, where positivity is meant as being a sum of hermitian squares. I will explain how conditions like this can be used to give computer-assisted proofs of vanishing of cohomology.