Many scientists, and in particular mathematicians, report difficulty in understanding thermodynamics. So why is thermodynamics so difficult? To attempt an answer, we begin by looking at the components in an exposition of a scientific theory. These include a mathematical core, a motivation for the choice of variables and equations, some historical remarks, some examples and a discussion of how variables, parameters, and functions (such as equations of state) can be inferred from experiments. There are other components too, such as an account of how a theory relates to other theories in the subject.

It will be suggested that theories of thermodynamics are hard to understand because (i) many expositions appear to argue from the particular to the general (ii) there are several different thermodynamic theories that have no obvious logical or mathematical equivalence (iii) each theory really is subtle and requires intense study (iv) in most expositions different theories are mixed up, and the different components of a scientific exposition are also mixed up. So, by presenting one theory at a time, and by making clear which component is being discussed, we might reduce the difficulty in understanding any individual thermodynamic theory. The key is perhaps separation of the mathematical core from the physical motivation. It is also useful to realise that a motivation is not generally the same as a proof, and that no theory is actually true.

By way of illustration we will attempt expositions of two of the simplest thermodynamic theories – reversible and then irreversible thermodynamics of homogeneous materials – where the mathematical core and the motivation are discussed separately. In conclusion we’ll relate these two simple theories to other, foundational and generalised, thermodynamic theories.

The accurate modelling of geophysical flows often requires information which is difficult to measure and therefore poorly quantified. Such information may relate to the fluid properties or an unknown boundary condition, for example. The premise of this talk is that when the flow is bounded by a free surface, the deformation of this free surface contains useful information which can be used to infer such unknown quantities. The increasing availability of free surface data through remote sensing using drones and satellites provides the impetus to develop new mathematical methods and numerical tools to interpret the signature embedded in the free surface deformation. This talk will explore two recent examples drawn from glaciology and inspired from volcanology for which free surface data was successfully used to reconstruct an unknown field.

The magma chamber - an underground vat of fluid magma that is tapped during volcanic eruptions - has been the foundation of models of volcanic eruptions for many decades and successfully explains many geological observations.  However, geophysics has failed to image the postulated large magma chambers, and the chemistry and ages of crystals in erupted magmas indicate a more complicated history.  New conceptual models depict subsurface magmatic systems as dominantly uneruptible crystalline networks with interstitial melt (mushes) extending deep into the Earth's crust to the mantle, containing lenses of potentially eruptible (low-crystallinity) magma.  These lenses would commonly be less dense than the overlying mush and so Rayleigh Taylor instabilities should develop leading to ascent of blobs of magma unless the growth rate is sufficiently slow that other processes (e.g. solidification) dominate.  The viscosity contrast between a buoyant layer and mush is typically extremely large; a consequence is that the horizontal dimension of a magma reservoir is commonly much less than the theoretical fastest growing wavelength assuming an infinite horizontal layer.  

I will present laboratory experiments and linear stability analysis for low Reynolds number, laterally confined Rayleigh Taylor instabilities involving one layer that is much thinner and much less viscous than the other.  I will then apply the results to magmatic systems, comparing timescales for development of the instability and the volumes of packets of rising melt generated, with the frequencies and sizes of volcanic eruptions.  I will then discuss limitations of this work and outstanding fluid dynamical problems in this new paradigm of trans-crustal magma mush systems.

Glacier flow is an example of a gravity driven non-linear viscous flow at low Reynolds numbers. As a glacier flows over an undulating bed, the surface topography is modified in response. Some information about bed conditions is therefore contained in the shape of the surface and the surface velocity field. I will present theoretical and numerical work on how basal conditions on glaciers affect ice flow, and how one can obtain information about basal conditions through surface-to-bed inversion. I’ll give an overview over inverse methodology currently used in glaciology, and how satellite data is now routinely used to invert for bed properties of the Greenland and the Antarctic Ice Sheets.

Plumes are a characteristic feature of convective flow through porous media. Their dynamics are an important part of numerous geological processes, ranging from mixing in magma chambers to the convective dissolution of sequestered carbon dioxide. In this talk, I will discuss models for the spread of convective plumes in a heterogeneous porous environment. I will focus particularly on the effect of thin, roughly horizontal, low-permeability barriers to flow, which provide a generic form of heterogeneity in geological settings, and are a particularly widespread feature of sedimentary formations. With the aid of high-resolution numerical simulations, I will explore how a plume spreads and flows in the presence of one or more of these layers, and will briefly consider the implications of these findings in physical settings.

 
We study continuous theories of classes of finite dimensional Hilbert spaces expanded by 
a finite family (of a fixed size) of unitary operators. 
Infinite dimensional models of these theories are called pseudo finite dimensional dynamical Hilbert spaces. 
Our main results connect decidability questions of these theories with the topic of approximations of groups by metric groups. 

The (total) Steenrod square is a ring homomorphism from the cohomology of a topological space to the Z/2-equivariant cohomology of this space, with the trivial Z/2-action. Given a closed monotone symplectic manifold, one can define a deformed notion of the Steenrod square for quantum cohomology, which will not in general be a ring homomorphism, and prove some properties of this operation that are analogous to properties of the classical Steenrod square. We will then link this, in a more general setting, to a definition by Seidel of a similar operation on Floer cohomology.
 

Let F be a fixed infinite, vertex-transitive graph. We say a graph G is `r-locally F' if for every vertex v of G, the ball of radius r and centre v in G is isometric to the ball of radius r in F. For each positive integer n, let G_n = G_n(F,r) be a graph chosen uniformly at random from the set of all unlabelled, n-vertex graphs that are r-locally F. We investigate the properties that the random graph G_n has with high probability --- i.e., how these properties depend upon the fixed graph F. 
We show that if F is a Cayley graph of a torsion-free group of polynomial growth, then there exists a positive integer r_0 such that for every integer r at least r_0, with high probability the random graph G_n = G_n(F,r) defined above has largest component of size between n^{c_1} and n^{c_2}, where 0 < c_1 < c_2  < 1 are constants depending upon F alone, and moreover that G_n has at least exp(poly(n)) automorphisms. This contrasts sharply with the random d-regular graph G_n(d) (which corresponds to the case where F is replaced by the infinite d-regular tree).
Our proofs use a mixture of results and techniques from group theory, geometry and combinatorics, including a recent and beautiful `rigidity' result of De La Salle and Tessera.
We obtain somewhat more precise results in the case where F is L^d (the standard Cayley graph of Z^d): for example, we obtain quite precise estimates on the number of n-vertex graphs that are r-locally L^d, for r at least linear in d, using classical results of Bieberbach on crystallographic groups.
Many intriguing open problems remain: concerning groups with torsion, groups with faster than polynomial growth, and what happens for more general structures than graphs.
This is joint work with Itai Benjamini (Weizmann Institute).
 

 I will introduce two obstructions for a rational homology 3-sphere to smoothly bound a rational homology 4-ball- one coming from Donaldson's theorem on intersection forms of definite 4-manifolds, and the other coming from correction terms in Heegaard Floer homology. If L is a nonunimodular definite lattice, then using a theorem of Elkies we will show that whether L embeds in the standard definite lattice of the same rank is completely determined by a collection of lattice correction terms, one for each metabolizing subgroup of the discriminant group. As a topological application this gives a rephrasing of the obstruction coming from Donaldson's theorem. Furthermore, from this perspective it is easy to see that if the obstruction to bounding a rational homology ball coming from Heegaard Floer correction terms vanishes, then (under some mild hypotheses) the obstruction from Donaldson's theorem vanishes too.