The tree amplituhedron A(n, k, m) is a geometric object generalizing the positive Grassmannian, which was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. I will give a gentle introduction to the amplituhedron, and then describe what it looks like in various special cases. For example, one can use the theory of sign variation and matroids to show that the amplituhedron A(n, k, 1) can be identified with the complex of bounded faces of a cyclic hyperplane arrangement. I will also present some conjectures relating the amplituhedron A(n, k, m) to combinatorial objects such as non-intersecting lattice paths and plane partitions. This is joint work with Steven Karp, and part of it is additionally joint work with Yan Zhang.

The vast majority of the world's electricity is generated by converting thermal energy into electric energy by use of a steam turbine. Siemens are one of the worlds leading manufacturers of such
turbines, and aim to design theirs to be as efficient as possible. Using an internally built software, Siemens can estimate the efficiency which would result from a turbine design. In this presentation, we present the approaches that have been taken to improve turbine design using mathematical optimisation software. In particular, we focus on the failings of the approach currently taken, the obstacles in place which make solving this problem difficult, and the approach we intend to take to find a locally optimal solution.

In the 80s, Hatcher and Thurston introduced the pants graph as a tool to prove that the mapping class group of a closed, orientable surface is finitely presented. The pants graph remains relevant for the study of the mapping class group, sitting between the marking graph and the curve graph. More precisely, there is a sequence of natural coarse lipschitz maps taking the marking graph via the pants graph to the curve graph.

A second motivation for studying the pants graph comes from Teichmüller theory. Brock showed that the pants graph can be interpreted as a combinatorial model for Teichmüller space with the Weil-Petersson metric.

In this talk I will introduce the pants graph, discuss some of its properties and state a few open questions.

Calcination describes the heat treatment of anthracite particles in a furnace to produce a partially-graphitised material which is suitable for use in electrodes and for other met- allurgical applications. Electric current is passed through a bed of anthracite particles, here referred to as a coke bed, causing Ohmic heating and high temperatures which result in the chemical and structural transformation of the material.

Understanding the behaviour of such mechanisms on the scale of a single particle is often dealt with through the use of computational models such as DEM (Discrete Element Methods). However, because of the great discrepancy between the length scale of the particles and the length scale of the furnace, we can exploit asymptotic homogenisation theory to simplify the problem.  

In this talk, we will present some results relating to the electrical and thermal conduction through granular material which define effective quantities for the conductivities by considering a microscopic representative volume within the material. The effective quantities are then used as parameters in the homogenised macroscopic model to describe calcination of anthracite. 

Joint work with Zsolt Vizi (Bolyai Institute, University of Szeged, Hungary), Istvan Kiss (Department
of Mathematics, University of Sussex, United Kingdom)

Pairwise models have been proven to be a flexible framework for analytical approximations
of stochastic epidemic processes on networks that are in many situations much more accurate
than mean field compartmental models. The non-Markovian aspects of disease transmission
are undoubtedly important, but very challenging to incorporate them into both numerical
stochastic simulations and analytical investigations. Here we present a generalization of
pairwise models to non-Markovian epidemics on networks. For the case of infectious periods
of fixed length, the resulting pairwise model is a system of delay differential equations, which
shows excellent agreement with results based on the explicit stochastic simulations. For more
general distribution classes (uniform, gamma, lognormal etc.) the resulting models are PDEs
that can be transformed into systems of integro-differential equations. We derive pairwise
reproduction numbers and relations for the final epidemic size, and initiate a systematic
study of the impact of the shape of the particular distributions of recovery times on how
the time evolution of the disease dynamics play out.

(Joint work with Coralia Cartis) The problem of finding the most extreme value of a function, also known as global optimization, is a challenging task. The difficulty is associated with the exponential increase in the computational time for a linear increase in the dimension. This is known as the ``curse of dimensionality''. In this talk, we demonstrate that such challenges can be overcome for functions with low effective dimensionality --- functions which are constant along certain linear subspaces. Such functions can often be found in applications, for example, in hyper-parameter optimization for neural networks, heuristic algorithms for combinatorial optimization problems and complex engineering simulations.
We propose the use of random subspace embeddings within a(ny) global minimisation algorithm, extending the approach in Wang et al. (2013). We introduce a new framework, called REGO (Random Embeddings for GO), which transforms the high-dimensional optimization problem into a low-dimensional one. In REGO, a new low-dimensional problem is formulated with bound constraints in the reduced space and solved with any GO solver. Using random matrix theory, we provide probabilistic bounds for the success of REGO, which indicate that this is dependent upon the dimension of the embedded subspace and the intrinsic dimension of the function, but independent of the ambient dimension. Numerical results demonstrate that high success rates can be achieved with only one embedding and that rates are for the most part invariant with respect to the ambient dimension of the problem.