Optical super-resolution microscopy enables the observations of individual bio-molecules. The arrangement and dynamic behaviour of such molecules is studied to get insights into cellular processes which in turn lead to various application such as treatments for cancer diseases. STFC's Central Laser Facility provides (among other) public access to super-resolution microscope techniques via research grants. The access includes sample preparation, imaging facilities and data analysis support. Data analysis includes single molecule tracking algorithms that produce molecule traces or tracks from time series of molecule observations. While current algorithms are gradually getting away from "connecting the dots" and using probabilistic methods, they often fail to quantify the uncertainties in the results. We have developed a method that samples a probability distribution of tracking solutions using the Metropolis-Hastings algorithm. Such a method can produce likely alternative solutions together with uncertainties in the results. While the method works well for smaller data sets, it is still inefficient for the amount of data that is commonly collected with microscopes. Given the observations of the molecules, tracking solutions are discrete, which gives the proposal distribution of the sampler a peculiar form. In order for the sampler to work efficiently, the proposal density needs to be well designed. We will discuss the properties of tracking solutions and the problems of the proposal function design from the point of view of discrete mathematics, specifically in terms of graphs. Can mathematical theory help to design a efficient proposal function?

In 1985, McDowell introduced a family of parabolically induced Whittaker modules over a complex semisimple Lie algebra, which includes both Verma modules and the nondegenerate Whittaker modules studied by Kostant. Many classical results for Verma modules and the Bernstein--Gelfand--Gelfand category O have been generalized to the category of Whittaker modules introduced by Milicic--Soergel, including the classification of irreducible objects and the Kazhdan--Lusztig conjectures. Contravariant forms on Verma modules are unique up to scaling and play a key role in the definition of the Jantzen filtration. In this talk I will discuss a classification of contravariant forms on parabolically induced Whittaker modules. In a recent result, joint with Anna Romanov, we show that the dimension of the space of contravariant forms on a parabolically induced Whittaker module is given by the cardinality of a Weyl group. This result illustrates a divergence from classical results for Verma modules, and gives insight to two significant open problems in the theory of Whittaker modules: the Jantzen conjecture and the absence of an algebraic definition of duality.

Charles Babbage (1791–1871) was Lucasian Professor of mathematics in Cambridge from 1828–1839. He displayed a fertile curiosity that led him to study many contemporary processes and problems in a way which emphasised an analytic, data driven view of life.

In popular culture Babbage has been celebrated as an anachronistic Victorian engineer. In reality, Babbage is best understood as a figure rooted in the enlightenment, who had substantially completed his core investigations into 'mechanisation of thought' by the mid 1830s: he is thus an anachronistic Georgian: the construction of his first difference engine design is contemporary with the earliest public railways in Britain.

A fundamental question that must strike anybody who examines Babbage's precocious designs is: how could one individual working alone have synthesised a workable computer design, designing an object whose complexity of behaviour so far exceeded that of contemporary machines that it would not be matched for over a hundred years?

We shall explore the extent to which the answer lies in the techniques Babbage developed to reason about complex systems. His Notation which shows the geometry, timing, causal chains and the abstract components of his machines, has a direct parallel in the Hardware Description Languages developed since 1975 to aid the design of large scale electronics. In this presentation, we shall provide a basic tutorial on Babbage's notation showing how his concepts of 'pieces' and 'working points' effectively build a graph in which both parts and their interactions are represented by nodes, with edges between part-nodes and interaction-nodes denoting ownership, and edges between interaction-nodes denoting the transmission of forces between individual assemblies within a machine. We shall give examples from Babbage's Difference Engine 2 for which a complete set of notations was drawn in 1849, and compare them to a design of similar complexity specified in 1987 using the Inmos HDL.