L3

Nekrasov, Rosly and Shatashvili observed that the generating function of a certain space of SL(2) opers has a physical interpretation as the effective twisted superpotential for a four-dimensional N=2 quantum field theory. In this talk we describe the ingredients needed to generalise this observation to higher rank. Important ingredients are spectral networks generated by Strebel differentials and the abelianization method. As an example we find the twisted superpotential for the E6 Minahan-Nemeschansky theory. 
 

A K3 surface is called attractive if and only if its Picard number is 20: The maximal possible. Attractive K3 surfaces possess complex multiplication. This property endows attractive K3 surfaces with rich and well understood arithmetic. For example, the associated Galois representation turns out to be a product of well known two dimensional representations and the  Hasse-Weil L-function turns out to be a product of well known L-functions. On the other hand, attractive K3 surfaces show up as solutions of the attractor equations in type IIB string theory compactified on the product of a K3 surface with an elliptic curve. As such, these surfaces dictate the near horizon geometry of a charged black hole in this theory. We will try to see which arithmetic properties of the attractive K3 surfaces lend a stringy interpretation and use them to shed light on physical properties of the charged black hole. 
 

In this talk we consider a system of interacting Brownian particles. When diffusing particles interact with each other their motions are correlated, and the configuration space is of very high dimension. Often an equation for the one-particle density function (the concentration) is sought by integrating out the positions of all the others. This leads to the classic problem of closure, since the equation for the concentration so derived depends on the two-particle correlation function. We discuss two  common closures, the mean-field (MFA) and the Kirkwood-superposition approximations, as well as an alternative approach, which is entirely systematic, using matched asymptotic expansions (MAE). We compare the resulting (nonlinear) diffusion models with Monte Carlo simulations of the stochastic particle system, and discuss for which types of interactions (short- or long-range) each model works best. 

Most motile bacteria are equipped with multiple helical flagella, slender appendages whose rotation in viscous fluids allow the cells to self-propel. We highlight in this talk two consequences of hydrodynamics for bacteria. We first show how the swimming of cells with multiple flagella is enabled by an elastohydrodynamic instability. We next demonstrate how interactions between flagellar filaments mediated by the fluid govern the ability of the cells to reorient. 

How do organisms cope with cellular variability to achieve well-defined morphologies and architectures? We are addressing this question by combining experiments with live plants and analyses of (stochastic) models that integrate cell-cell communication and tissue mechanics. During the talk, I will survey our results concerning plant architecture (phyllotaxis) and organ morphogenesis.

Network models may be applied to describe many complex systems, and in the era of online social networks the study of dynamics on networks is an important branch of computational social science.  Cascade dynamics can occur when the state of a node is affected by the states of its neighbours in the network, for example when a Twitter user is inspired to retweet a message that she received from a user she follows, with one event (the retweet) potentially causing further events (retweets by followers of followers) in a chain reaction. In this talk I will review some simple models that can help us understand how social contagion (the spread of cultural fads and the viral diffusion of information) depends upon the structure of the social network and on the dynamics of human behaviour. Although the models are simple enough to allow for mathematical analysis, I will show examples where they can also provide good matches to empirical observations of cascades on social networks.

Phytoplankton moving in the ocean, spermatozoa making their way  through the female reproductive tract and harmful bacteria that form biofilms on implanted medical devices interact with a surrounding fluid. Their length scales are small enough so that viscous effects dominate inertial effects allowing the resulting fluid dynamics to be described by the linear Stokes equations. However,  nonlinear behavior can occur because these structures are flexible and their form evolves with the flow. In addition, the fluid environment may also  be complex because of embedded microstructures that further complicate the dynamics.  We will discuss recent successes and challenges in describing these elastohydrodynamic systems.

Railway traffic management is the combination of monitoring the progress of trains, forecasting of the likely future progression of trains, and evaluating the impact of intervention options in near real time in order to make traffic adjustments that minimise the combined delay of trains when measured against the planned timetable.

In a time of increasing demand for rail travel, the desire to maximise the usage of the available infrastructure capacity competes with the need for contingency space to allow traffic management when disruption occurs. Optimisation algorithms and decision support tools therefore need to be increasingly sophisticated and traffic management has become a crucial function in meeting the growing expectations of rail travellers for punctuality and quality of service.

Resonate is a technology company specialising in rail and connected transport solutions. We have embarked on a drive to maximise capacity and performance through the use of mathematical, statistical, data-driven and machine learning based methods driving decision support and automated traffic management solutions.

In certain business environments, it is essential to the success of the business that workers stick closely to their plans and are not distracted, diverted or stopped. A warehouse is a great example of this for businesses where customers order goods online and the merchants commit to delivery dates.  In a warehouse, somewhere, a team of workers are scheduled to pick the items which will make up those orders and get them shipped on time.  If the workers do not deliver to plan, then orders will not be shipped on time, reputations will be damaged, customer will be lost and companies will go out of business.

StayLinked builds software which measures what these warehouse workers do and measures the factors which cause them to be distracted, diverted or stopped.  We measure whenever they start or end a task or process (e.g. start an order, pick an item in an order, complete an order). Some of the influencing factors we measure include the way the worker interacts with the device (using keyboard, scanner, gesture), navigates through the application (screens 1-3-4-2 instead of 1-2-3-4), the performance of the battery (dead battery stops work), the performance of the network (connected to access point or not, high or low latency), the device types being used, device form factor, physical location (warehouse 1, warehouse 2), profile of worker, etc.

We are seeking to build a configurable real-time mathematical model which will allow us to take all these factors into account and confidently demonstrate a measure of their impact (positive or negative) on the business process and therefore on the worker’s productivity. We also want to alert operational staff as soon as we can identify that important events have happened.  These alerts can then be quickly acted upon and problems resolved at the earliest possible opportunity.

In this project, we would like to collaborate with the maths faculty to understand the appropriate mathematical techniques and tools to use to build this functionality.  This product is being used right now by our customers so it would also be a great opportunity for a student to quickly see the results of their work in action in a real-world environment.