L3

Networks of interacting oscillators give rise to collective dynamics such as localized frequency synchrony. In networks of neuronal oscillators, for example, the location of frequency synchrony could encode information. We discuss some recent persistence results for certain dynamically invariant sets called weak chimeras, which show localized frequency synchrony of oscillators. We then explore how the network structure and interaction allows for dynamic switching of the spatial location of frequency synchrony: these dynamics are induced by stable heteroclinic connections between weak chimeras. Part of this work is joined with Peter Ashwin (Exeter).

Building on advancements in computer vision we now have an array of visual tracking methods that allow the reliable estimation of cellular motion in high-throughput settings as well as more complex biological specimens. In many cases the underlying assumptions of these methods are still not well defined and result in failures when analysing large scale experiments.

Using organotypic co-culture systems we can now mimic more physiologically relevant microenvironments in vitro.  The robust analysis of cellular dynamics in such complex biological systems remains an open challenge. I will attempt to outline some of these challenges and provide some very preliminary results on analysing more complex cellular behaviours.

In the first part of my presentation, I will briefly summarize a dynamic view of the cell cycle created in collaboration with Prof John Tyson over the past 25 years. 
In our view, the decisions a cell must make during DNA synthesis and mitosis are controlled by bistable switches, which provide abrupt and irreversible transition 
between successive cell cycle phases. In addition, bistability provides the foundation for 'checkpoints' that can stop cell proliferation if problems arise 
(e.g., DNA damage by UV irradiation). In the second part of my talk, I will highlight a few representative examples from our ongoing BBSRC Strategic LoLa grant 
(http://cellcycle.org.uk/) in which we are testing the predictions of our theoretical ideas in human cells in collaboration with four experimental groups.

All data intelligence processes are designed for processing a finite amount of data within a time period. In practice, they all encounter
some difficulties, such as the lack of adequate techniques for extracting meaningful information from raw data; incomplete, incorrect 
or noisy data; biases encoded in computer algorithms or biases of human analysts; lack of computational resources or human resources; urgency in 
making a decision; and so on. While there is a great enthusiasm to develop automated data intelligence processes, it is also known that
many of such processes may suffer from the phenomenon of data processing inequality, which places a fundamental doubt on the credibility of these 
processes. In this talk, the speaker will discuss the recent development of an information-theoretic measure (by Chen and Golan) for optimizing 
the cost-benefit ratio of a data intelligence process, and will illustrate its applicability using examples of data analysis and 
visualization processes including some in bioinformatics.

The past few years have seen a surge of interest in nonconvex reformulations of convex optimizations using nonlinear reparameterizations of the optimization variables. Compared with the convex formulations, the nonconvex ones typically involve many fewer variables, allowing them to scale to scenarios with millions of variables. However, one pays the price of solving nonconvex optimizations to global optimality, which is generally believed to be impossible. In this talk, I will characterize the nonconvex geometries of several low-rank matrix optimizations. In particular, I will argue that under reasonable assumptions, each critical point of the nonconvex problems either corresponds to the global optimum of the original convex optimizations, or is a strict saddle point where the Hessian matrix has a negative eigenvalue. Such a geometric structure ensures that many local search algorithms can converge to the global optimum with random initializations. Our analysis is based on studying how the convex geometries are transformed under nonlinear parameterizations.

Many systems in nature consist of stochastically interacting agents or particles. Stochastic processes have been widely used to model such systems, yet they are notoriously difficult to analyse. In this talk I will show how ideas from statistics can be used to tackle some challenging problems in the field of stochastic processes.

In the first part, I will consider the problem of inference from experimental data for stochastic reaction-diffusion processes. I will show that multi-time distributions of such processes can be approximated by spatio-temporal Cox processes, a well-studied class of models from computational statistics. The resulting approximation allows us to naturally define an approximate likelihood, which can be efficiently optimised with respect to the kinetic parameters of the model. 

In the second part, we consider more general path properties of a certain class of stochastic processes. Specifically, we consider the problem of computing first-passage times for Markov jump processes, which are used to describe systems where the spatial locations of particles can be ignored.  I will show that this important class of generally intractable problems can be exactly recast in terms of a Bayesian inference problem by introducing auxiliary observations. This leads us to derive an efficient approximation scheme to compute first-passage time distributions by solving a small, closed set of ordinary differential equations.

Part of the series "What do historians of mathematics do?"

The talk will set out the key debate in England at the Restoration, the need for a new orientation in mathematics towards algebra and the new "analysis". It will focus on efforts by three central players in England's mathematical community, John Pell, John Collins, and the Oxford mathematician John Wallis to produce an English language algebra text which would play a pioneering role in promoting this change. What was the background to the work we now call Pell's Algebra and why was it so significant?