Automorphic forms arise naturally when studying scattering amplitudes in toroidal compactifications of string theory. In this talk, I will summarize the conditions on four-graviton amplitudes from the literature required by U-duality, supersymmetry and string perturbation theory, which are satisfied by certain Eisenstein series on exceptional Lie groups. Physical information, such as instanton effects, are encoded in their Fourier coefficients on parabolic subgroups, which are, in general, difficult to compute. I will demonstrate a method for evaluating certain Fourier coefficients of interest in string theory. Based on arXiv:1511.04265, arXiv:1412.5625 and work in progress.
 

A conformal field theory is characterised by the CFT data, namely the spectrum of scaling dimensions and OPE coefficients. The idea of the conformal bootstrap is to use associativity of the operator algebra together with the symmetries of the theory to constraint the CFT data. For the sector of operators with large spin one can actually use these ideas to obtain analytical results. It was recently understood how to set up a systematic expansion around this sector, leading to analytic results to all orders in inverse powers of the spin. We will show how to use this large spin perturbation theory to obtain analytic results for vast families of CFTs. Some of the applications include vector models, weakly coupled gauge theories and the computation of loops for scalar theories in AdS.

The ambitwistor string of Mason and Skinner has been very successful in describing field theory amplitudes, at both loop and tree-level for a variety of theories. But the original action given by Mason and Skinner is already partially gauge-fixed, which obscures some issues related to modular invariance and the connection to conventional string theories. In this talk I will argue that the Null string is the ungauge-fixed version of the Ambitwistor string. This clarifies the geometry of the original Ambitwistor string and gives a road map to understanding modular invariance, and gives new formulas for loop amplitudes in which we expect that UV divergences will be easier to analyse.

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.

Averaging, either spatial or temporal, is a powerful technique in complex multi-scale systems.

However, in some situations it can be difficult to justify.

For example, many real-world networks in technology, engineering and biology have a function and exhibit dynamics that cannot always be adequately reproduced using network models given by the smooth dynamical systems and fixed network topology that typically result from averaging. Motivated by examples from neuroscience and engineering, we describe a model for what we call a "functional asynchronous network". The model allows for changes in network topology through decoupling of nodes and stopping and restarting of nodes, local times, adaptivity and control. Our long-term goal is to obtain an understanding of structure (why the network works) and how function is optimized (through bifurcation).

We describe a prototypical theorem that yields a functional decomposition for a large class of functional asynchronous networks. The result allows us to express the function of a dynamical network in terms of individual nodes and constituent subnetworks.

Consider the following particle system. We are given a uniform random rooted tree on vertices labelled by $[n] = \{1,2,\ldots,n\}$, with edges directed towards the root. Each node of the tree has space for a single particle (we think of them as cars). A number $m \le n$ of cars arrive one by one, and car $i$ wishes to park at node $S_i$, $1 \le i \le m$, where $S_1, S_2, \ldots, S_m$ are i.i.d. uniform random variables on $[n]$. If a car wishes to park at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Let $A_{n,m}$ denote the event that all $m$ cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Set $m = \lfloor \alpha n \rfloor$. Then if $\alpha \le 1/2$, $\mathbb{P}(A_{n,\lfloor \alpha n \rfloor}) \to \frac{\sqrt{1-2\alpha}}{1-\alpha}$, whereas if $\alpha > 1/2$ we have $\mathbb{P}(A_{n,\lfloor \alpha n \rfloor}) \to 0$. In this talk, we will give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method.

Joint work with Christina Goldschmidt.