Outer Space is an important object in Geometric Group Theory and can be described from two viewpoints: as a space of marked graphs and a space of actions on trees. The latter viewpoint can be used to prove that Outer Space is contractible; and this fact together with some arguments using the first viewpoint enables us to say something about the Outer Automorphism group of a free group - I will sketch both these proofs.

# Past Forthcoming Seminars

The classical Ingham-Karamata Tauberian theorem has many applications in different fields of mathematics, varying from number theory to $C_0$-semigroup theory and is considered to be one of the most important Tauberian theorems. We will discuss how to obtain remainder estimates in the theorem if one strengthens the assumptions on the Laplace transform. Moreover, we will give new (remainder) versions of this theorem under the more general one-sided Tauberian condition of $\rho(x) \ge −f(x)$ where $f$ is an arbitrary function satisfying some regularity assumptions. The talk is based on collaborative work with Jasson Vindas.

The birth of fixed-wing, powered flight in the first decade of the twentieth century brought with it significant potential for pilots to return to Earth by unintended, often fatal, means. I will discuss the nature of the contemporary mathematical and engineering debates associated with these facets of flight, and the practical steps taken to facilitate safer aircraft and more robust operating procedures.

We discuss the connections and differences between the ZFC set theory and univalent foundations and answer the above question in the negative.

I will introduce a cohomology theory which combines topological and algebraic concepts. Interpretations of certain cohomology groups will be given. We also generalise the construction of the second Stiefel-Whitney class of a line bundle. As I will explain in my talk, the refined Stiefel-Whitney class of the canonical bundle on certain moduli stacks provides an obstruction for the construction of cohomological Hall algebras.

The design of shapes that are in some sense optimal is a task faced by engineers in a wide range of disciplines. In shape optimisation one aims to improve a given initial shape by iteratively deforming it - if the shape is represented by a mesh, then this means that the mesh has to deformed. This is a delicate problem as overlapping or highly stretched meshes lead to poor accuracy of numerical methods.

In the presented work we consider a novel mesh deformation method motivated by the Riemannian mapping theorem and based on conformal mappings.

The edge isoperimetric problem for a graph G is to find, for each n, the minimum number of edges leaving any set of n vertices. Exact solutions are known only in very special cases, for example when G is the usual cubic lattice on Z^d, with edges between pairs of vertices at l_1 distance 1. The most attractive open problem was to answer this question for the "strong lattice" on Z^d, with edges between pairs of vertices at l_infty distance 1. Whilst studying this question we in fact solved the edge isoperimetric problem asymptotically for every Cayley graph on Z^d. I'll talk about how to go from the specification of a lattice to a corresponding near-optimal shape, for both this and the related vertex isoperimetric problem, and sketch the key ideas of the proof. Joint work with Joshua Erde.

Pseudo-reductive groups are smooth connected linear algebraic groups over a field k whose k-defined unipotent radical is trivial. If k is perfect then all pseudo-reductive groups are reductive, but if k is imperfect (hence of characteristic p) then one gets a strictly larger collection of groups. They come up in a number of natural situations, not least when one wishes to say something about the simple representations of all smooth connected linear algebraic groups. Recent work by Conrad-Gabber-Prasad has made it possible to reduce the classification of the simple representations of pseudo-reductive groups to the split reductive case. I’ll explain how. This is joint work with Mike Bate.

Time parallelisation techniques provide an additional direction for the parallelisation of the solution of time-dependent PDEs or of systems of ODEs. In particular, the Parareal algorithm has imposed itself as the canonical choice to achieve parallelisation in time, also because of its simplicity and flexibility. The algorithm works by splitting the time domain in chunks, and iteratively alternating a prediction step (parallel), in which a "fine" solver is employed to achieve a high-accuracy solution within each chunk, to a correction step (serial) where a "coarse" solver is used to quickly propagate the update between the chunks. However, the stability of the method has proven to be highly sensitive to the choice of fine and coarse solver, even more so when applied to chaotic systems or advection-dominated problems.

In this presentation, an alternative formulation of Parareal is discussed. This aims to conduct the update by estimating directly the sensitivity of the solution of the integration with respect to the initial conditions, thus eliminating altogether the necessity of choosing the most apt coarse solver, and potentially boosting its convergence properties.

This will be a quick introduction to tropical algebra and the main results from the paper https://arxiv.org/pdf/1604.00113.pdf