Let's take a step back to understand what it means to use maths in society: Which maths, and whose society? I'll talk about some of the options I've come across, including time I spent at the US Census Bureau, and we will hear your ideas too. We might even crowdsource a document of maths in society opportunities together...

RF-engineering defines the “perfect” parabolic shape a foldable reflector antenna (e.g. the membrane) should have. In practice it is virtually impossible to design a deployable backing structure that can meet all RF-imposed requirements. Inevitably the shape of the membrane will deviate from its ideal parabolic shape when material properties and pragmatic mechanical design are considered. There is therefore a challenge to model such membranes in order to find the form they take and then use the model as a design tool and perhaps in an optimisation objective function, if tractable. 

The variables we deal with are:
Elasticity of the membrane (anisotropic or orthotropic typ)
Boundary forces (by virtue of the interaction between the membrane and it’s attachment)
Elasticity of the backing structure (e.g. the elasticity properties of the attachment)
Number, location and elasticity of the membrane fixing points

There are also in-orbit environmental effects on such structures for which modelling could also be of value. For example, the structure can undergo thermal shocks and oscillations can occur that are un-dampened by the usual atmospheric interactions at ground level etc. There are many other such points to be considered and allowed for.

The $k$-th complete homogeneous symmetric polynomial in $m$ variables $h_{k,m}$ is the sum of all the monomials of degree $k$ in $m$ variables. They are related to the Symmetric powers of vector spaces. In this talk we will present some of their standard properties, some classic combinatorial results using the "stars and bars" argument, as well as an interesting result: the complete homogeneous symmetric polynomial applied to $(1+X_i)$ can be written as a linear combination of complete homogeneous symmetric poynomials in the $X_i$. To compute the coefficients of this linear combination, we extend the classic "stars and bars" argument.

Do classical solutions of the compressible Navier-Stokes equations converge to an entropy solution of their inviscid counterparts, the Euler equations? In this talk we present a result which answers this question affirmatively, in the one-dimensional case, for a particular class of fluids. Specifically, we consider gases that exhibit approximately polytropic behaviour in the vicinity of the vacuum, and that are isothermal for larger values of the density (which we call approximately isothermal gases). Our approach makes use of methods from the theory of compensated compactness of Tartar and Murat, and is inspired by the earlier works of Chen and Perepelitsa, Lions, Perthame and Tadmor, and Lions, Perthame and Souganidis. This is joint work with Matthew Schrecker.

In this talk, we consider the random version of some classical extremal problems in the context of long cycles. This type of problems can also be seen as random analogues of the Turán number of long cycles, established by Woodall in 1972.

For a graph $G$ on $n$ vertices and a graph $H$, denote by $\text{ex}(G,H)$ the maximal number of edges in an $H$-free subgraph of $G$. We consider a random graph $G\sim G(n,p)$ where $p>C/n$, and determine the asymptotic value of $\text{ex}(G,C_t)$, for every $A\log(n)< t< (1- \varepsilon)n$. The behaviour of $\text{ex}(G,C_t)$ can depend substantially on the parity of $t$. In particular, our results match the classical result of Woodall, and demonstrate the transference principle in the context of long cycles.

Using similar techniques, we also prove a robustness-type result, showing the likely existence of cycles of prescribed lengths in a random subgraph of a graph with a nearly optimal density (a nearly ''Woodall graph"). If time permits, we will present some connections to size-Ramsey numbers of long cycles.

Based on joint works with Michael Krivelevich and Adva Mond.

A broad challenge in the theory of finitely generated groups is to understand their subgroups. A group is commensurably coHopfian if its finite index subgroups are distinct from its infinite index subgroups (that is to say not abstractly isomorphic). We will focus primarily on hyperbolic groups, and give the first examples of one-ended hyperbolic groups that are not commensurably coHopfian.
This is joint work with Emily Stark.
 

In this talk we will survey a novel domain of computational group theory: computing with linear groups over infinite fields.  We will provide an introduction to the area, and will discuss available methods and algorithms. Special consideration is given to algorithms for Zariski dense subgroups. This includes a computer realization of the strong approximation theorem, and algorithms for arithmetic groups. We illustrate applications of our methods to the solution of problems further afield by computer experimentation.