University of Oxford

An expander is a family of finite graphs of uniformly bounded degree, increasing number of vertices and Cheeger constant bounded away from zero. They occur throughout mathematics and computer science; the most famous constructions of expanders rely on powerful results in geometric group theory and number theory, while expanders are used in everything from error-correcting codes, through disproving the strongest version of the Baum-Connes conjecture, to affine sieve theory and the twin prime, Mersenne prime and Hardy-Littlewood conjectures.

However, very little was known about how different the geometry of two expanders could be. This question was raised by Ostrovskii in 2013, and a year later Mendel and Naor gave the first example of two 'distinct' expanders.

In this talk I will construct a continuum of expanders which are, in a certain sense, geometrically incomparable. Once the existence of a single expander is accepted, the remainder of the proof is a heady mix of counting, addition, multiplication, and just for the experts, a little bit of division. Two very different - and very interesting - continuums of 'distinct' expanders have since been constructed by Khukhro-Valette and Das.

I'll be talking about my masters' research in Quantum Gravity in a way that is accessible to mathematicians.

A randomly trapped random walk on a graph is a simple continuous time random walk in which the holding time at a given vertex is an independent sample from a probability measure determined by the trapping landscape, a collection of probability measures indexed by the vertices.

This is a time change of the simple random walk. For the constant speed continuous time random walk, the landscape has an exponential distribution with rate 1 at each vertex. For the Bouchaud trap model it has an exponential random variable at each vertex but where the rate for the exponential is chosen from a heavy tailed distribution. In one dimension the possible scaling limits are time changes of Brownian motion and include the fractional kinetics process and the Fontes-Isopi-Newman (FIN) singular diffusion. We extend this analysis to put these models in the setting of resistance forms, a framework that includes finitely ramified fractals. In particular we will construct a FIN diffusion as the limit of the Bouchaud trap model and the random conductance model on fractal graphs. We will establish heat kernel estimates for the FIN diffusion extending what is known even in the one-dimensional case.

Gaussian fields are prevalent throughout mathematics and the sciences, for instance in physics (wave-functions of high energy electrons), astronomy (cosmic microwave background radiation) and probability theory (connections to SLE, random tilings etc). Despite this, the geometry of such fields, for instance the connectivity properties of level sets, is poorly understood. In this talk I will discuss methods of extracting geometric information about levels sets of a planar Gaussian field through discrete observations of the field. In particular, I will present recent work that studies three such discretisation schemes, each tailored to extract geometric information about the levels set to a different level of precision, along with some applications.

In this talk, I will talk about my current approach to model customer movements and in particular congestion inside supermarkets using queuing networks. As the research question for my project is ‘How should one design supermarkets to minimize congestion?’, I will then talk about my current progress in understanding how the network structure can affect this dynamics.

What makes cities successful? A complex systems approach to modelling urban economies

Urban centres draw a diverse range of people, attracted by opportunity, amenities, and the energy of crowds. Yet, while benefiting from density and proximity of people, cities also suffer from issues surrounding crime, congestion and density. Seeking to uncover the mechanisms behind the success of cities using novel tools from the mathematical and data sciences, this work uses network techniques to model the opportunity landscape of cities. Under the theory that cities move into new economic activities that share inputs with existing capabilities, path dependent industrial diversification can be described using a network of industries. Edges represent shared necessary capabilities, and are empirically estimated via flows of workers moving between industries. The position of a city in this network (i.e., the subnetwork of its current industries) will determine its future diversification potential. A city located in a central well-connected region has many options, but one with only few peripheral industries has limited opportunities.

We develop this framework to explain the large variation in labour formality rates across cities in the developing world, using data from Colombia. We show that, as cities become larger, they move into increasingly complex industries as firms combine complementary capabilities derived from a more diverse pool of workers. We further show that a level of agglomeration equivalent to between 45 and 75 minutes of commuting time maximizes the ability of cities to generate formal employment using the variety of skills available. Our results suggest that rather than discouraging the expansion of metropolitan areas, cities should invest in transportation to enable firms to take advantage of urban diversity.

This talk will be based on joint work with Eduardo Lora and Andres Gomez at Harvard University.

Hamilton-Jacobi-Bellman equations for dynamic pricing

I will discuss the Hamilton-Jacobi-Bellman (HJB) equation, which is a nonlinear, second-order, terminal value PDE problem. The equation arises in optimal control theory as an optimality condition.

Consider a dynamic pricing problem: over a given period, what is the best strategy to maximise revenues and minimise the cost of unsold items?

This is formulated as a stochastic control problem in continuous time, where we try to find a function that controls a stochastic differential equation based on the current state of the system.

The optimal control function can be found by solving the corresponding HJB equation.

I will present the solution of the HJB equation using a toy problem, for a risk-neutral and a risk-averse decision maker.

Cells adapt their metabolic state in response to changes in the environment.  I will present a systematic framework for the construction of flux graphs to represent organism-wide metabolic networks.  These graphs encode the directionality of metabolic fluxes via links that represent the flow of metabolites from source to target reactions.  The weights of the links have a precise interpretation in terms of probabilities or metabolite flow per unit time. The methodology can be applied both in the absence of a specific biological context, or tailored to different environmental conditions by incorporating flux distributions computed from constraint-based modelling (e.g., Flux-Balance Analysis). I will illustrate the approach on the central carbon metabolism of Escherichia coli, revealing drastic changes in the topological and community structure of the metabolic graphs, which capture the re-routing of metabolic fluxes under each growth condition.

By integrating Flux Balance Analysis and tools from network science, our framework allows for the interrogation of environment-specific metabolic responses beyond fixed, standard pathway descriptions.

The immune system finds very rare amounts of pathogens and responds against them in a timely and efficient manner. The time to find and respond against pathogens does not vary appreciably with the size of the host animal (scale invariant search and response). This is surprising since the search and response against pathogens is harder in larger animals.

The first part of the talk will focus on using techniques from computer science to solve problems in immunology, specifically how the immune system achieves scale invariant search and response. I use machine learning techniques, ordinary differential equation models and spatially explicit agent based models to understand the dynamics of the immune system. I will talk about Hierarchical Bayesian non-linear mixed effects models to simulate immune response in different species.

The second part of the talk will focus on taking inspiration from the immune system to solve problems in computer science. I will talk about a model that describes the optimal architecture of the immune system and then show how architectures and strategies inspired by the immune system can be used to create distributed systems with faster search and response characteristics.

I argue that techniques from computer science can be applied to the immune system and that the immune system can provide valuable inspiration for robust computing in human engineered distributed systems.