The questions we would like to answer are as follows:

1. Given three distributions pdf1, pdf2 and pdf-so, is it always possible to find a joint-distribution consistent with those 3 one-dimensional distributions?
2. Assuming that we are in a situation where (1) holds, can we find a nonparametric joint-distribution consistent with the 3 given one-dimensional distributions?
3. If (2) leads to an under-determined problem, can we find a joint-distribution that is “as close as possible” to the historical joint distribution?
4. Can we achieve (3) with a neural network?
5. If we observe the marginal and spread distributions for multiple maturities T, can we specify the evolution of pdf(T), possibly using neural differential equations?

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.

The string 2-group is a fundamental object in string geometry, which is a refinement of spin geometry required to describe the spinning string. While many models for the string 2-group exist, the construction of a representation for it is new. In this talk, I will recall the notion of strict 2-group, and then give two examples: the automorphism 2-group of a von Neumann algebra, and the string 2-group. I will then describe the representation of the string 2-group on the hyperfinite III_1 factor, which is a functor from the string 2-group to the automorphism 2-group of the hyperfinite III_1 factor.

Let G(R) be the real points of a complex reductive algebraic group G. There are many difficult questions about admissible representations of real reductive groups which have (relatively) easy answers in the case of complex groups. Thus, it is natural to look for a relationship between representations of G and representations of G(R). In this talk, I will introduce a functor from admissible representations of G to admissible representations of G(R). This functor interacts nicely with many natural invariants, including infinitesimal character, associated variety, and restriction to a maximal compact subgroup, and it takes unipotent representations of G to unipotent representations of G(R).

By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group admits a derangement (an element with no fixed points). More recently, the existence of derangements with additional properties has attracted much attention, especially for primitive actions of almost simple groups. Surprisingly, there exist almost simple groups with elements that are derangements in every faithful primitive action; we say that these elements are totally deranged. I'll talk about ongoing work to classify the totally deranged elements of almost simple groups, and I'll mention how this solves a question of Garzoni about invariable generating sets for simple groups.

We prove a positive characteristic analogue of the classical result that the centralizer of a nonconstant differential operator in one variable is commutative. This leads to a new, short proof of that classical characteristic zero result, by reduction modulo p. This is joint work with Justin Desrochers available at https://arxiv.org/abs/2201.04606.