University of Cambridge

Do classical solutions of the compressible Navier-Stokes equations converge to an entropy solution of their inviscid counterparts, the Euler equations? In this talk we present a result which answers this question affirmatively, in the one-dimensional case, for a particular class of fluids. Specifically, we consider gases that exhibit approximately polytropic behaviour in the vicinity of the vacuum, and that are isothermal for larger values of the density (which we call approximately isothermal gases). Our approach makes use of methods from the theory of compensated compactness of Tartar and Murat, and is inspired by the earlier works of Chen and Perepelitsa, Lions, Perthame and Tadmor, and Lions, Perthame and Souganidis. This is joint work with Matthew Schrecker.

The finite element method (FEM) is one of the great triumphs of applied mathematics, numerical analysis and software development. Recent developments in sensor and signalling technologies enable the phenomenological study of systems. The connection between sensor data and FEM is restricted to solving inverse problems placing unwarranted faith in the fidelity of the mathematical description of the system. If one concedes mis-specification between generative reality and the FEM then a framework to systematically characterise this uncertainty is required. This talk will present a statistical construction of the FEM which systematically blends mathematical description with observations.

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World wide, unconstrained lava flows kill people almost each year and cause extensive damage, costing millions of pounds. Defending against lava flows is possible by using topographic variations sensibly, placing buildings considerately, constructing defending walls of appropriate size and the like. Hinton, Hogg and Huppert have recently published three rather mathematical papers outlining how viscous flows down slopes interact with a variety of geometrical shapes; evaluating, in particular, the conditions under which “dry zones” form – safe places for people and belongings – and the size of a protective wall required to defend a given size building.

Following a desktop experimental demonstration, we will discuss these analyses and their consequences.

I will talk about recent work with Jack Thorne in which we find the average size of the Selmer group for a family of genus-2 curves by analyzing a graded Lie algebra of type E_8. I will focus on the role representation theory plays in our proofs.

When S is a finite set of natural numbers, a GCD-sum is a particular kind of double-sum over the elements of S, and they arise naturally in several settings. In particular, these sums play a role when one studies the local statistics of point sequences on the unit circle. There are known upper bounds for the size of a GCD-sum in terms of the size of the set S, most recently due to de la Bretèche and Tenenbaum, and these bounds are sharp. Yet the known examples of sets S for which the GCD-sum over S provides a matching lower bound all possess strong multiplicative structure, whereas in applications the set S often comes with additive structure. In this talk I will describe recent joint work with Thomas Bloom in which we apply an estimate from sum-product theory to prove a much stronger upper bound on a GCD-sum over an additively structured set. I will also describe an application of this improvement to the study of the distribution of points on the unit circle, with a further application to arbitrary infinite subsets of squares. 

Seminal work of Hida tells us that if a modular eigenform is ordinary at p then we can always find other eigenforms, of different weights, that are congruent to our given form. Even better, it says that we can find q-expansions with coefficients in p-adic analytic function of the weight variable k that when evaluated at positive integers give the q-expansion of classical eigenforms. His construction of these families uses mainly the geometry of the modular curve and its ordinary locus.
In a joint work with Marc-Hubert Nicole, we obtained similar results for Drinfeld modular forms over function fields. After an extensive introduction to Drinfeld modules, their moduli spaces, and Drinfeld modular forms, we shall explain how to construct Hida families for ordinary Drinfeld modular forms.

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.

Many fluid flows in natural systems are highly complex, with an often beguilingly intricate and confusing detailed structure. Yet, as with many systems, a good deal of insight can be gained by testing the consequences of simple mathematical models that capture the essential physics.  We’ll tour two such problems.  In the summer melt seasons in Greenland, lakes form on the surface of the ice which have been observed to rapidly drain.  The propagation of the meltwater in the subsurface couples the elastic deformation of the ice and, crucially, the flow of water within the deformable subglacial till.  In this case the poroelastic deformation of the till plays a subtle, but crucial, role in routing the surface meltwater which spreads indefinitely, and has implications for how we think about large-scale motion in groundwater aquifers or geological carbon storage.  In contrast, when magma erupts onto the Earth’s surface it flows before rapidly cooling and crystallising.  Using analogies from the kitchen we construct, and experimentally test, a simple model of what sets the ultimate extent of magmatic intrusions on Earth and, as it turns out, on Venus.  The results are delicious!  In both these cases, we see how a simplified mathematical analysis provides insight into large scale phenomena.