L4

We would like to share two challenges that, if solved, could improve our domestic lives.  

 Firstly, having appliances that are as unobtrusive as possible is a strong desire, unwanted noise can cause a negative impact on relaxation.  A key target for refrigerators is low sound level, a key noise source is the capillary tube.  The capillary tube effects the phase change that is required for the refrigerant to be in the gaseous state in the evaporator for cooling.  Noise is generated during this process due to two phases being present within the flow through the tube.  The challenge is to create a numerical model and analysis of refrigerant flow properties in order to estimate the acoustic behaviour.

 Secondly, we would like to maximise the information that can be gathered from our new range of connected devices.  By analysing the data generated during usage we would like to be able to predict faults and understand user behaviour in more detail.  The challenge regarding fault prediction is the scarcity of the failure data and the impact of false positives.  Due to the number of units in the field, a relatively small fraction of false positives can remove the ROI from such an initiative.  We would like to understand if advanced machine learning methods can be used to reduce this risk.

Global singularity theory is a classical subject which classifies singularities of maps between manifolds, and describes topological reasons for their appearance. I will start with explaining a central problem of the subject regarding multipoint and multisingularity loci, then give an introduction into some recent major developments by Kazarian, Rimanyi, Szenes and myself.

We will review some open questions about automorphisms of free groups, give some partial answers, and explain the deformation spaces of trees that they act on, as well as the geometry of these spaces arising from the Lipschitz metric. This will be a gentle introduction to the topic, focused on introducing the concepts.

The first edition of Thompson’s famous book On Growth and Form appeared in 1917. It has subsequently been regarded as a foundational work in mathematical biology and a revolutionary contribution to the field of morphology. Most existing literature credits Thompson as a lone genius who produced the 793 pages of the 1917 edition and 1116 pages of the 1942 edition. Thompson’s correspondence presents a very different picture of this tome as one arising from extensive and ongoing – perhaps sometimes unwitting? – collaboration.

In this talk, I will introduce the BPS cohomology of the moduli space of Higgs bundles on a smooth projective curve of rank r and degree d using cohomological Donaldson-Thomas theory. The BPS cohomology and the intersection cohomology coincide when r and d are coprime, but they are different in general. We will see that the BPS cohomology does not depend on d. This is a generalization of the Hausel-Thaddeus conjecture to non-coprime case. I will also explain that Toda's χ-independence conjecture (and hence the strong rationality conjecture) for local curves can be proved in the same manner. This talk is based on a joint work with Naoki Koseki and another joint work with Naruki Masuda.

Let X be an algebraic variety over an infinite field k. In arithmetic geometry we are interested in the set X(k) of k-rational points on X. For example, is X(k) empty or not? And if it is not empty, is X(k) dense in X with respect to the Zariski topology?

Del Pezzo surfaces are surfaces classified by their degree d, which is an integer between 1 and 9 (for d >= 3, these are the smooth surfaces of degree d in P^d). For del Pezzo surfaces of degree at least 2 over a field k, we know that the set of k-rational points is Zariski dense provided that the surface has one k-rational point to start with (that lies outside a specific subset of the surface for degree 2). However, for del Pezzo surfaces of degree 1 over a field k, even though we know that they always contain at least one k-rational point, we do not know if the set of k-rational points is Zariski dense in general.

I will talk about density of rational points on del Pezzo surfaces, state what is known so far, and show a result that is joint work with Julie Desjardins, in which we give sufficient and necessary conditions for the set of k-rational points on a specific family of del Pezzo surfaces of degree 1 to be Zariski dense, where k is finitely generated over Q.

Independent sets in bipartite regular graphs have been studied extensively in combinatorics, probability, computer science and more. The problem of counting independent sets is particularly interesting in the d-dimensional hypercube $\{0,1\}^d$, motivated by the lattice gas hardcore model from statistical physics. Independent sets also turn out to be very interesting in the context of random graphs.

The number of independent sets in the hypercube $\{0,1\}^d$ was estimated precisely by Korshunov and Sapozhenko in the 1980s and recently refined by Jenssen and Perkins.

In this talk we will discuss new results on the number of independent sets in a random subgraph of the hypercube. The results extend to the hardcore model and rely on an analysis of the antiferromagnetic Ising model on the hypercube.

This talk is based on joint work with Yinon Spinka.

1pm Jonathan Tam: Markov decision processes with observation costs

We present a framework for a controlled Markov chain where the state of the chain is only given at chosen observation times and of a cost. Optimal strategies therefore involve the choice of observation times as well as the subsequent control values. We show that the corresponding value function satisfies a dynamic programming principle, which leads to a system of quasi-variational inequalities (QVIs). Next, we give an extension where the model parameters are not known a priori but are inferred from the costly observations by Bayesian updates. We then prove a comparison principle for a larger class of QVIs, which implies uniqueness of solutions to our proposed problem. We utilise penalty methods to obtain arbitrarily accurate solutions. Finally, we perform numerical experiments on three applications which illustrate our framework.

Preprint at https://arxiv.org/abs/2201.07908

1.45pm Remy Messadene: signature asymptotics, empirical processes, and optimal transport

Rough path theory provides one with the notion of signature, a graded family of tensors which characterise, up to a negligible equivalence class, and ordered stream of vector-valued data. In the last few years, use of the signature has gained traction in time-series analysis, machine learning, deep learning and more recently in kernel methods. In this work, we lay down the theoretical foundations for a connection between signature asymptotics, the theory of empirical processes, and Wasserstein distances, opening up the landscape and toolkit of the second and third in the study of the first. Our main contribution is to show that the Hambly-Lyons limit can be reinterpreted as a statement about the asymptotic behaviour of Wasserstein distances between two independent empirical measures of samples from the same underlying distribution. In the setting studied here, these measures are derived from samples from a probability distribution which is determined by geometrical properties of the underlying path.

2.30-3.00 Tea & coffee in the mezzananie

3-4pm Julien Berestycki: Extremal point process of the branching Brownian motion