Many problems that involve the propagation of time-harmonic waves are naturally posed in unbounded domains. For instance, a common problem in the are a of acoustic scattering is the determination of the sound field that is generated when an incoming time-harmonic wave (which is assumed to arrive ``from infinity'') impinges onto a solid body (the scatterer). The boundary
conditions to be applied on the surface of the scatterer (most often of Dirichlet, Neumann or Robin type) tend to be easy to enforce in most numerical solution schemes. Conversely, the imposition of a suitable decay condition (typically a variant of the Sommerfeld radiation condition), which is required to ensure the well-posedness of the solution, is considerably more involved. As a result, many numerical schemes generate spurious reflections from the outer boundary of the finite computational domain.

Perfectly matched layers (PMLs) are in this context a versatile alternative to the usage of classical approaches such as employing absorbing boundary conditions or Dirichlet-to-Neumann mappings, but unfortunately most PML formulations contain adjustable parameters which have to be optimised to give the best possible performance for a particular problem. In this talk I will present a parameter-free PML formulation for the case of the two-dimensional Helmholtz equation. The performance of the proposed method is demonstrated via extensive numerical experiments, involving domains with smooth and polygonal boundaries, different solution types (smooth and singular, planar and non-planar waves), and a wide range of wavenumbers (R. Cimpeanu, A. Martinsson and M.Heil, J. Comp. Phys., 296, 329-347 (2015)). Possible extensions and generalisations will also be touched upon.

This talk will review the main Tikhonov and Bayesian smoothing formulations of inverse problems for dynamical systems with partially observed variables and parameters. The main contenders: strong-constraint, weak-constraint and penalty function formulations will be described. The relationship between these formulations and associated optimisation problems will be revealed.  To close we will indicate techniques for maintaining sparsity and for quantifying uncertainty.

TBA

Motivic homotopy theory gives a way of viewing algebraic varieties and topological spaces as objects in the same category, where homotopies are parametrised  by the affine line.  In particular, there is a notion of $\mathbb A^1$ contractible varieties.  Affine spaces are $\mathbb A^1$ contractible by definition.  The Koras-Russell threefold KR defined by the equation $x + x^2y + z^2 + t^3 = 0$ in $\mathbb A^4$ is the first nontrivial example of an $\mathbb A^1$ contractible smooth affine variety.  We will discuss this example in some detail, and speculate on whether one can use motivic homotopy theory to distinguish between KR and $\mathbb A^3$.

Let $P$ be a random polynomial of degree $d$ such that the leading and constant coefficients are 1 and the rest of the coefficients are independent random variables taking the value 0 or 1 with equal probability. Odlyzko and Poonen conjectured that $P$ is irreducible with probability tending to 1 as $d$ grows.  I will talk about an on-going joint work with Emmanuel Breuillard, in which we prove that GRH implies this conjecture. The proof is based on estimates for the mixing time of random walks on $\mathbb{F}_p$, where the steps are given by the maps $x \rightarrow ax$ and $x \rightarrow ax+1$ with equal probability.

Claudia Scheimbauer

Title: Quantum field theory meets higher categories

Abstract: Studying physics has always been a driving force in the development of many beautiful pieces of mathematics in many different areas. In the last century, quantum field theory has been a central such force and there have been several fundamentally different approaches using and developing vastly different mathematical tools. One of them, Atiyah and Segal's axiomatic approach to topological and conformal quantum field theories, provides a beautiful link between the geometry of "spacetimes” (mathematically described as cobordisms) and algebraic structures. Combining this approach with the physical notion of "locality" led to the introduction of the language of higher categories into the topic. The Cobordism Hypothesis classifies "fully local" topological field theories and gives us a recipe to construct examples thereof by checking certain algebraic conditions generalizing the existence of the dual of a vector space. I will give an introduction to the topic and very briefly mention on my own work on these "extended" topological field theories.

Alberto Paganini

Title: Shape Optimization with Finite Elements

Abstract: Shape optimization means looking for a domain that minimizes a target cost functional. Such problems are commonly solved iteratively by constructing a minimizing sequence of domains. Often, the target cost functional depends on the solution to a boundary value problem stated on the domain to be optimized. This introduces the difficulty of solving a boundary value problem on a domain that changes at each iteration. I will suggest how to address this issue using finite elements and conclude with an application from optics.

Research suggests that students with a 'growth mindset' may do better than those with a 'fixed mindset'.

• What does that mean for our teaching?
• How can we support students to develop a growth mindset?
• What sorts of mindsets do we ourselves have?
• And how does that affect our teaching and indeed the rest of our work?