The well known (and notoriously hard) P vs NP problem asks whether every Boolean function with polynomial-size proofs is also computable in
polynomial time.

The standard approach to the P vs NP problem is via circuit complexity. For progressively richer classes of Boolean circuits (networks of AND, OR and NOT
logic gates), one wishes to show super-polynomial lower bounds on the sizes of circuits (as a function of the size of the input) computing some Boolean
function known to be in NP, such as the Satisfiability problem.

However, there is a more logic-oriented approach initiated by Cook and Reckhow, going through proof complexity rather than circuit complexity. For
progressively richer proof systems, one wishes to show super-polynomial lower bounds on the sizes of proofs (as a function of the size of the tautology) of
some sequence of propositional tautologies.

I will give a brief overview on known results along these two directions, and on their limitations. Somewhat surprisingly, similar techniques have been found
to be useful for these seemingly different approaches. I will say something about known connections between the approaches, and pose the question of
whether there are deeper connections.

Finally, I will discuss how the perspective of proof complexity can be used to formalize the difficulty of proving lower bounds on the sizes of computations
(or of proofs).

 

Energy transfers from small-scale turbulence and waves to large-scale flows are ubiquituous in oceans, atmospheres, planetary cores and stars.

Therefore, turbulence and waves have a direct effect on the large-scale organization of geophysical and astrophysical fluids and can affect their long-term dynamics.

In this talk I will discuss recent direct numerical simulation (DNS) results of two upscale energy transfer mechanisms that emerge from the dynamics of a fluid that is self-organized in a turbulent layer next to a stably-stratified one. This self-organization in an adjacent "two-layer" turbulent-stratified system is ubiquituous in nature and is representative of e.g. Earth's troposphere-stratosphere system, the oceans' surface mixed layer-thermocline system, and stars' convective-radiative interiors. The first set of DNS results will demonstrate how turbulent motions can generate internal waves, which then force a slowly-reversing large-scale flow, akin to Earth's Quasi-Biennial Oscillation (QBO). The second set of DNS results will show how the stratified layer regulates the emergence of large-scale vortices (LSV) in the turbulent layer under rapid rotation in the regime known as geostrophic turbulence. I will demonstrate why it is important to resolve both the turbulence and the waves, as otherwise the natural variability of the QBO is lost and LSV cannot form. I will discuss future works and highlight how the results may guide the implementation of upscale energy transfers in global earth system models.

Partially molten materials resist shearing and compaction. This resistance

is described by a fourth-rank effective viscosity tensor. When the tensor

is isotropic, two scalars determine the resistance: an effective shear and

an effective bulk viscosity. In this seminar, calculations are presented of

the effective viscosity tensor during diffusion creep for a 3D tessellation of

tetrakaidecahedrons (truncated octahedrons). The geometry of the melt is

determined by assuming textural equilibrium.  Two parameters

control the effect of melt on the viscosity tensor: the porosity and the

dihedral angle. Calculations for both Nabarro-Herring (volume diffusion)

and Coble (surface diffusion) creep are presented. For Nabarro-Herring

creep the bulk viscosity becomes singular as the porosity vanishes. This

singularity is logarithmic, a weaker singularity than typically assumed in

geodynamic models. The presence of a small amount of melt (0.1% porosity)

causes the effective shear viscosity to approximately halve. For Coble creep,

previous modelling work has argued that a very small amount of melt may

lead to a substantial, factor of 5, drop in the shear viscosity. Here, a

much smaller, factor of 1.4, drop is obtained.

Ice streams are fast flowing regions of ice that generally slide over a layer of unconsolidated, water-saturated subglacial sediment known as till.  A striking feature that has been observed geophysically is that subglacial till has been found to accumulate distinctively into sedimentary wedges at the grounding zones (regions where ice sheets begin to detach from the bedrock to form freely floating ice shelves) of both past and present-day ice sheets. These grounding-zone wedges have important implications for ice-sheet stability against grounding zone retreat in response to rising sea levels, and their origins have remained a long-standing question. Using a combination of mathematical modelling, a series of laboratory experiments, field data and numerical simulations, we develop a fluid-mechanical model that explains the mechanism of the formation of these sedimentary wedges in terms of the loading and unloading of deformable till in three dynamical regimes. We also undertake a series of analogue laboratory experiments, which reveal that a similar wedge of underlying fluid accumulates spontaneously in experimental grounding zones, we formulate local conditions relating wedge slopes in each of the scenarios and compare them to available geophysical radargram data at the well lubricated, fast-flowing Whillans Ice Stream.

We consider a well-studied optimal execution problem under little assumptions on the underlying midprice process. We do so by using signatures from rough path theory, that allows converting the original problem into a more computationally tractable problem. We include a few numerical experiments where we show that our methodology is able to retrieve the theoretical optimal execution speed for several problems studied in the literature, as well as some cases not included in the literatture. We also study some estensions of our framework to other settings.