In 1953 Roth proved that any positive density subset of the integers contains a non-trivial three term arithmetic progression. I will present a recent quantitative improvement for this theorem, give an overview of the main ideas of the proof, and discuss its relation to other recent work in the area. I will also discuss some closely related problems. 

We discuss a new method to bound the number of primes in certain very thin sets. The sets $S$ under consideration have the property that if $p\in S$ and $q$ is prime with $q|(p-1)$, then $q\in S$. For each prime $p$, only 1 or 2 residue classes modulo $p$ are omitted, and thus the traditional small sieve furnishes only the bound $O(x/\log^2 x)$ (at best) for the counting function of $S$. Using a different strategy, one related to the theory of prime chains and Pratt trees, we prove that either $S$
contains all primes or $\# \{p\in S : p\le x \} = O(x^{1-c})$ for some positive $c$. Such sets arise, for example, in work on Carmichael's conjecture for Euler's function.

In this talk, we will discuss the dimension of $J_0(N)[m]$ at an Eisenstein prime m for
square-free level N. We will also study the structure of $J_0(N)[m]$ as a Galois module.
This work generalizes Mazur’s work on Eisenstein ideals of prime level to the case of
arbitrary square-free level up to small exceptional cases.

Let f be an elliptic modular newform of weight at least 2. The 
problem of the automorphy of the symmetric power L-functions of f is a 
key example of Langlands' functoriality conjectures. Recently, the 
potential automorphy of these L-functions has been established, using 
automorphy lifting techniques, and leading to a proof of the Sato-Tate 
conjecture. I will discuss a new approach to the automorphy of these 
L-functions that shows the existence of Sym^m f for m = 1,...,8.

Marine-ice formation occurs on a vast range of length scales: from millimetre scale frazil crystals, to consolidated sea ice a metre thick, to deposits of marine ice under ice shelves that are hundreds of kilometres long. Scaling analyses is therefore an attractive and powerful technique to understand and predict phenomena associated with marine-ice formation, for example frazil crystal growth and the convective desalination of consolidated sea ice. However, there are a number of potential pitfalls arising from the assumptions implicit in the scaling analyses. In this talk, I tease out the assumptions relevant to these examples and test them, allowing me to derive simple conceptual models that capture the important geophysical mechanisms affecting marine-ice formation. 

Explosive volcanic eruptions often produce large amounts of ash that is transported high into the atmosphere in a turbulent buoyant plume.  The ash can be spread widely and is hazardous to aircraft causing major disruption to air traffic.  Recent events, such as the eruption of Eyjafjallajokull, Iceland, in 2010 have demonstrated the need for forecasts of ash transport to manage airspace.  However, the ash dispersion forecasts require boundary conditions to specify the rate at which ash is delivered into the atmosphere.

Models of volcanic plumes can be used to describe the transport of ash from the vent into the atmosphere.  I will show how models of volcanic plumes can be developed, building on classical fluid mechanical descriptions of turbulent plumes developed by Morton, Taylor and Turner (1956), and how these are used to determine the volcanic source conditions.  I will demonstrate the strong atmospheric controls on the buoyant plume rise.  Typically steady models are used as solutions can be obtained rapidly, but unsteadiness in the volcanic source can be important.  I'll discuss very recent work that has developed unsteady models of volcanic plumes, highlighting the mathematical analysis required to produce a well-posed mathematical description.

One of the main obstacles to forecasting sea level rise over the coming centuries is the problem of predicting changes in the flow of ice sheets, and in particular their fast-flowing outlet glaciers. While numerical models of ice sheet flow exist, they are often hampered by a lack of input data, particularly concerning the bedrock topography beneath the ice. Measurements of this topography are relatively scarce, expensive to obtain, and often error-prone. In contrast, observations of surface elevations and velocities are widespread and accurate.

In an ideal world, we could combine surface observations with our understanding of ice flow to invert for the bed topography. However, this problem is ill-posed, and solutions are both unstable and non-unique. Conventionally, this problem is circumvented by the use of regularization terms in the inversion, but these are often arbitrary and the numerical methods are still somewhat unstable.

One philosophically appealing option is to apply a fully Bayesian framework to the problem. Although some success has been had in this area, the resulting distributions are extremely difficult to work with, both from an interpretive standpoint and a numerical one. In particular, certain forms of prior information, such as constraints on the bedrock slope and roughness, are extremely difficult to represent in this framework.

A more profitable avenue for exploration is a semi-Bayesian approach, whereby a classical inverse method is regularized using terms derived from a Bayesian model of the problem. This allows for the inclusion of quite sophisticated forms of prior information, while retaining the tractability of the classical inverse problem. In particular, we can account for the severely non-Gaussian error distribution of many of our measurements, which was previously impossible.

We will present results from studies of the impact of the non-slow (typically fast) components of a rotating, stratified flow on its slow dynamics. We work in the framework of fast singular limits that derives from the work of Bogoliubov and Mitropolsky [1961], Klainerman and Majda [1981], Shochet [1994], Embid and Ma- jda [1996] and others.

In order to understand how the flow approaches and interacts with the slow dynamics we decompose the full solution, where u is a vector of all the unknowns, as

u = u α + u ′α where α represents the Ro → 0, F r → 0 or the simultaneous limit of both (QG for

quasi-geostrophy), with

P α u α = u α    P α u ′α = 0 ,

and where Pαu represents the projection of the full solution onto the null space of the fast operator. We use this decomposition to find evolution equations for the components of the flow (and the corresponding energy) on and off the slow manifold.

Numerical simulations indicate that for the geometry considered (triply periodic) and the type of forcing applied, the fast waves act as a conduit, moving energy onto the slow manifold. This decomposition clarifies how the energy is exchanged when either the stratification or the rotation is weak. In the quasi-geostrophic limit the energetics are less clear, however it is observed that the energy off the slow manifold equilibrates to a quasi-steady value.

We will also discuss generalizations of the method of cancellations of oscillations of Schochet for two distinct fast time scales, i.e. which fast time scale is fastest? We will give an example for the quasi-geostrophic limit of the Boussinesq equations.

At the end we will briefly discuss how understanding the role of oscillations has allowed us to develop convergent algorithms for parallel-in-time methods.

Beth A. Wingate - University of Exeter

Jared Whitehead - Brigham Young University

Terry Haut - Los Alamos National Laboratory