Anna Skorobogatova | Limiting Sobolev-type inequalities for vector fields

Abstract: The classical Gagliardo-Nirenberg-Sobolev inequality on $n$ dimensions guarantees that for any $1 \leq p < n$, we can estimate the $L^{\frac{np}{n-p}}$ norm of any smooth compactly supported map $u$ by the $L^p$ norm of $Du$. Now consider a more general first order differential operator $\mathbb{B}$ instead of the full gradient. Can we get an analogous inequality when we replace $D$ by $\mathbb{B}$? Or do we really need all of the information that the full gradient provides? Which are the ‘correct’ conditions that need be imposed on the operator $\mathbb{B}$? In particular, we will see that the limiting case $p=1$ is special and needs to be handled separately.

Peter Mühlbacher | Random walks on the symmetric group and induced integer partitions

Abstract: Random walks on the symmetric group and induced integer partitions Abstract: How often do you have to swap (iid) elements to observe large permutation cycles? Does this mean we're close to a uniform permutation? Can we directly describe how these permutation cycles evolve in time? Heuristics and partial results suggesting some kind of universality will be presented.

Jaro Sant | Ergodicity & Speed of Convergence for 1D Diffusions

Abstract: Ergodicity is a central theme in the study of diffusion processes, and ensures that the long time behaviour of a process can be studied meaningfully. I'll introduce the notions of ergodicity and Harris recurrence for general state space stochastic processes, show how the two notions are linked in a general setting, and illustrate how this allows one to relate time averages to state space averages via Birkhoff's ergodic theorem. I will then focus on obtaining rates of convergence in this theorem, specifically for the case of 1D diffusions where the scenario is much easier to deal with. Through the use of speed and scale one can control the rate of convergence in the ergodic theorem by means of moments of hitting times of singletons. I will briefly comment on the extra problems that need to be tackled in higher dimensions time permitting.

Cathie Wells, University of Reading
Reducing Aviation Emissions and Fuel Burn by Re-routing Transatlantic Flights

Introduction: Commercial long haul flight is at the brink of major change. After decades of limited communications across the North Atlantic, full satellite coverage will soon be available. At the same time, airlines are facing increasing pressure to ensure their flights are more fuel efficient and thus less polluting. Here we show that changes to current fixed track trajectories could significantly reduce fuel use for aircraft making the journey between London and New York.

Methods: Each simulation is completed for a constant air speed and at FL 350, an altitude of 35 000 feet. Optimal Control Theory is used to minimise time of flight in routing aircraft through actual wind fields obtained from a global atmospheric re-analysis dataset. New formulae are then applied to calculate fuel burn rate at the given air speed and the fuel used for the trajectory is found. Applying this method to routes obtained at a variety of air speeds, the most fuel efficient simulation can be chosen.

Results:Total fuel burned in simulations was compared with actual flight data. Three different models of fixed wing aircraft were considered, the Boeing 747-436, 777-236 and 787-9. Each has a unique physical profile leading to different fuel burn rates. This is reflected in the varied levels of savings possible. The importance of the change to air speed, compared with the re-routing of the flight, was also considered.

Conclusions: By making use of the available wind fields, this low cost method not only allows punctual arrivals, but also reduces fuel use and emissions, ensuring a solution that is for once both good for business and for the environment.

Matthew King | Sound and critical layers in sheared flow over acoustic liners

Abstract: Acoustics within mean flows, governed by the linearised Euler equations, has applications within various real world settings such as aeroacoustics. One area of study is the modelling of acoustic liners within turbo-fan jet engines. For this, a uniform flow profile is often considered within a cylindrical duct. More recently however, the need to consider a sheared flow profile has become apparent. We make use of the Pridmore-Brown equation in order to study the acoustics of such a system. This talk will look at solving the Pridmore-Brown equation near a regular singularity known as the critical layer, the behaviour of these solutions, and the difficulties induced by the presence of the critical layer.

Marco Palma | Quantifying uncertainty in brain-predicted age using scalar-on-image quantile regression

Abstract: Prediction of subject age from brain anatomical MRI has the potential to provide a sensitive summary of brain changes, indicative of different neurodegenerative diseases. However, existing studies typically neglect the uncertainty of these predictions.

In this work we take into account this uncertainty by applying methods of functional data analysis. In particular, we propose a penalised functional quantile regression model of age on brain structure with cognitively normal (CN) subjects in the Alzheimer’s Disease Neuroimaging Initiative (ADNI), and use it to predict brain age in Mild Cognitive Impairment (MCI) and Alzheimer’s Disease (AD) subjects. Unlike the machine learning approaches available in the literature of brain age prediction, which provide only point predictions, the outcome of our model is a prediction interval for each subject.

In this talk an overview of all the theoretical aspects (functional data analysis, multidimensional smoothing, quantile regression) will be provided, followed by a discussion of the main results and of their meaning in the field of neuroscience of ageing.

Carla Groenland, University of Oxford |
Cyclically Covering Subspaces

Abstract: How 'small' can you take a linear subspace of F_2^n such that it contains at least one cyclic shift of all vectors of F_2^n, where the cyclic shift rotates the coordinates? I will explain why Artin's conjecture about primes has implications for this question, and how this led us to conjecture a generalisation of the so-called handshake lemma to Cayley graphs. All the definitions in this abstract will be explained within the talk.

Suzie Brown | Resampling in Sequential Monte Carlo

Abstract: Sequential Monte Carlo refers to a wide class of stochastic algorithms that generate samples from a sequence of posterior distributions by simulating a population of particles evolving through time. A key step in the algorithm is resampling, where a new set of particles is generated from the weighted particles of the previous generation. There are many ways to do resampling, leading to different properties that can affect the behaviour of the resulting Monte Carlo estimates. I will introduce and discuss a range of commonly used resampling schemes, and compare their properties. Time permitting, I will explore the link between resampling and particle genealogies.

Ian Hamilton | Predicting the past: A retrodictive model for modern rugby union

Abstract: It is not uncommon for competitors in sports tournaments to have results related to schedules of varying strength, consisting of stronger (weaker) opposition, more (fewer) matches, and a higher (lower) proportion of matches at home. One popular means of producing a rating model in such circumstances is the Bradley-Terry model. This talk introduces that model, discusses the extensions in the literature, and proposes a further one to handle modern rugby union, where teams gain points for wins and draws, but also bonus points for losing within a certain score, and for the number of tries scored. Additionally it will touch on the differences between predictive and retrodictive models, present a principled entropy-based motivation for the model, and argue for a pragmatic understanding of the use of a prior in this setting. The proposed model is applied to a national schools rugby tournament in order to investigate their current ranking method and to suggest improvements.

Trishen Gunaratnam, University of Bath | Phase transition for $\phi^4$

Abstract: $\phi^4$ is a continuum statistical mechanics model with unbounded spin. It arises in the study of (Euclidean) quantum field theory, as the continuum limit of Ising-type models near criticality, as the invariant measure of singular stochastic PDEs, and as the invariant measure of nonlinear Schrodinger equations. At large scales this model has a remarkable resemblance to the famous Ising model: It undergoes phase transition in 2 and 3 dimensions as the temperature is varied.

In this talk we describe how contour arguments can be used to prove phase transition for $\phi^4$. Precisely, we look at the sophisticated modification of the classical Peierls' argument (used to prove phase transition for the Ising model) adapted to the case of $\phi^4$ in 2 dimensions by Glimm, Jaffe and Spencer (CMP 1975). Time permitting, we discuss the extension to 3 dimensions and phase separation results that have recently been obtained in joint work with Ajay Chandra and Hendrik Weber.

Eleanor Archer | Edge-reinforced random walk and the localisation phenomenon

Abstract: Edge-reinforced random walk (ERRW) is a classical example of a self-interacting process. Focusing mainly on the case of ERRW on trees, we will introduce the ERRW and discuss some fundamental connections with random urn models which enable us to describe it as a random "mixture" of non-interacting random walks. This representation allows us to construct the scaling limit of ERRW on trees. Time permitting we will also discuss some properties of the scaling limit, including a quenched localisation result showing that the walk moves logarithmically slowly.

Dr. Alice Corbella | Towards automatic Zig Zag sapling and its use for epidemic inference

Abstract: Zig-Zag sampling, introduced by Bierkens et al. 2019, is based on the simulation of a piecewise deterministic Markov process (PDMP) whose switching rate $\lambda(t)$ is governed by the derivative of the log-target density. To our knowledge, Zig-Zag sampling has been used mainly on simple targets for which derivatives can be computed manually in a reasonable time.

To expand the applicability of this method, we incorporate Automatic Differentiation (AD) tools in the Zig-Zag algorithm, computing $ \lambda(t) $ automatically from the functional form of the log-target density. Moreover, to allow the simulation of the PDMP via thinning, we use standard optimization routines to find a local upper bound for the rate.

We present several implementations of our automatic Zig-Zag sampling and we measure the potential loss in computational time caused by AD and optimization routines. Among the examples, we consider the case of data arising from an epidemic which can be approximated by a deterministic system of equations; here manual derivation of the posterior density is practically infeasible due to the recursive relationships contained the likelihood function. Automatic Zig-Zag sampling successfully explores the parameter space and samples efficiently from the posterior distribution.

(Joint work with Gareth O. Roberts and Simon E. F. Spencer)