Starting from Hilbert's 10th problem, I will explain how to characterise the set of integers by non-solubility of a set of polynomial equations and discuss related challenges. The methods needed are almost entirely elementary; ingredients from algebraic number theory will be explained as we go along. No knowledge of first-order logic is necessary.

Parallelizing the time domain in numerical simulations is non-intuitive, but has been proven to be possible using various algorithms like parareal, PFASST and RIDC. Temporal parallelizations adds an entire new dimension to parallelize and significantly enhances use of super computing resources. Exploiting this technique serves as a big step towards exascale computation.

Starting with relatively simple problems, the parareal algorithm (Lions et al, A ''parareal'' in time discretization of PDE's, 2001) has been successfully applied to various complex simulations in the last few years (Samaddar et al, Parallelization in time of numerical simulations of fully-developed plasma turbulence using the parareal algorithm, 2010). The algorithm involves a predictor-corrector technique.

Numerical studies of the edge of magnetically confined, fusion plasma are an extremely challenging task. The complexity of the physics in this regime is particularly increased due to the presence of neutrals as well as the interaction of the plasma with the wall. These simulations are extremely computationally intensive but are key to rapidly achieving thermonuclear breakeven on ITER-like machines.

The SOLPS code package (Schneider et al, Plasma Edge Physics with B2‐Eirene, 2006) is widely used in the fusion community and has been used to design the ITER divertor. A reduction of the wallclock time for this code has been a long standing goal and recent studies have shown that a computational speed-up greater than 10 is possible for SOLPS (Samaddar et al, Greater than 10x Acceleration of fusion plasma edge simulations using the Parareal algorithm, 2014), which is highly significant for a code with this level of complexity.

In this project, the aim is to explore a variety of cases of relevance to ITER and thus involving more complex physics to study the feasibility of the algorithm. Since the success of the parareal algorithm heavily relies on choosing the optimum coarse solver as a predictor, the project will involve studying various options for this purpose. The tasks will also include performing scaling studies to optimize the use of computing resources yielding maximum possible computational gain.

Firstly, we will discuss how the category of strict polynomial functors can be endowed with a monoidal structure, including concrete calculations. It is well-known that the above category is equivalent to the category of modules over the Schur algebra. The so-called Schur functor in turn relates the category of modules over the Schur algebra to the category of representations of the symmetric group which posseses a monoidal structure given by the Kronecker product. We show that the Schur functor is monoidal with respect to these structures.
Finally, we consider the right and left adjoints of the Schur functor. We explain how these can be expressed in terms of one another using Kuhn duality and the central role the monoidal structure on strict polynomial functors plays in this context.