C3

Wreath-like product groups were introduced recently and used to construct the first positive examples of rigidity conjectures of Connes and Jones. In this talk, I will review those examples, as well as discuss some ideas to construct examples with other rigidity phenomena by modifying the wreath-like product construction.

This is a continuation of https://www.maths.ox.ac.uk/node/61240

In the operator algebraic approach to quantum field theory, the DHR category is a braided tensor category describing topological point defects of a theory with at least 1 (+1) dimensions. A single von Neumann algebra with no extra structure can be thought of as a 0 (+1) dimensional quantum field theory. In this case, we would not expect a braided tensor category of point defects since there are not enough dimensions to implement a braiding. We show, however, that one can think of central sequence algebras as operators localized ``at infinity", and apply the DHR recipe to obtain a braided tensor category of bimodules of a von Neumann algebra M, which is a Morita invariant. When M is a II_1 factor, the braided subcategory of automorphic objects recovers Connes' chi(M) and Jones' kappa(M). We compute this for II_1 factors arising naturally from subfactor theory and show that any Drinfeld center of a fusion category can be realized. Based on joint work with Quan Chen and Dave Penneys.

In view of Takesaki-Takai duality, we can go back and forth between C*-dynamical systems of an abelian group and ones of its Pontryagin dual by taking crossed products. In this talk, I present a similar duality between actions on C*-algebras of two constructions of locally compact quantum groups: one is the bicrossed product due to Vaes-Vainerman, and the other is the double crossed product due to Baaj-Vaes. I will explain the situation by illustrating the example coming from groups. If time permits, I will also discuss its consequences in the case of quantum doubles.

Baseline T cell infiltration and the spatial distribution of T cells within a tumour has been found to be a significant indicator of patient outcomes. This observation, coupled with the increasing availability of spatially-resolved imaging data of individual cells within the tumour tissue, motivates the development of mathematical models which capture the spatial dynamics of T cells. Agent-based models allow the simulation of complex biological systems with detailed spatial resolution, and generate rich spatio-temporal datasets. In order to fully leverage the information contained within these simulated datasets, spatial statistics provide methods of analysis and insight into the biological system modelled, by quantifying inherent spatial heterogeneity within the system. We present a cellular automaton model of interactions between tumour cells and cytotoxic T cells, and an analysis of the model dynamics, considering both the temporal and spatial evolution of the system. We use the model to investigate some of the standard assumptions made in these models, to assess the suitability of the models to accurately describe tumour-immune dynamics.

A Hele-Shaw Newton's cradle: Circular bubbles in a Hele-Shaw channel. (Daniel Booth)

We present a model for the motion of approximately circular bubbles in a Hele-Shaw cell. The bubble velocity is determined by a balance between the hydrodynamic pressures from the external flow and the drag due to the thin films above and below the bubble. We find that the qualitative behaviour depends on a dimensionless parameter and is found to agree well with experimental observations.  Furthermore, we show how the effects of interaction with cell boundaries and/or other bubbles also depend on the value of this dimensionless parameter For example, in a train of three identical bubbles travelling along the centre line, the middle bubble either catches up with the one in front or is caught by the one behind, forming what we term a Hele-Shaw Newton's cradle.
 

Reciprocity in Fluids (Matthew Cotton)

Reciprocity is a useful, and often underused, way to calculate integrated quantities when a to solution to a related problem is known. In the remaining time, I will overview these ideas and give some example use cases

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation (SDE) or neither of these. Furthermore, we are able to formally prove the linear convergence of gradient descent to a global optimum for the training of deep residual networks with constant layer width and smooth activation function. We further prove that if the trained weights, as a function of the layer index, admit a scaling limit as the depth increases, then the limit has finite 2-variation.

Molecules can be broken apart with a high-powered laser or an electron beam. The position of charged fragments can then be detected on a screen. From the mass to charge ratio, the identity of the fragments can be determined. The covariance of two fragments then gives us the projection of a distribution related to the initial scattering distribution. We formulate the mathematical transformation from the scattering distribution to the covariance distribution obtained from experiments. We expand the scattering distribution in terms of basis functions to obtain a linear system for the coefficients, which we use to solve the inverse problem. Finally, we show the result of our method on three examples of test data, and also with experimental data.