University of Oxford

We study the evolution of solid surfaces and pattern formation by
surface diffusion. Phase field models with degenerate mobilities are
frequently used to model such phenomena, and are validated by
investigating their sharp interface limits. We demonstrate by a careful
asymptotic analysis involving the matching of exponential terms that a
certain combination of degenerate mobility and a double well potential
leads to a combination of both surface and non-linear bulk diffusion to
leading order. If time permits, we will discuss implications and extensions.

We consider the Plebanski-Demianski family of solutions of minimal gauged supergravity in D=4, which describes an accelerating, rotating and charged black-hole in AdS4. The 4d metric has conical singularities, but we show that it can uplifted to a completely regular solution of D=11 supergravity. We focus on the supersymmetric and extremal case, where the near-horizon geometry is AdS2x\Sigma, where \Sigma is a spindle, or weighted projective space. We argue that this is dual to a d=1, N=(2,0) SCFT which is the IR limit of a 3d SCFT compactified on a spindle. This, in turn, should be realized holographically by wrapping a stack of M2-branes on a spindle. Such construction displays two interesting features: 1) supersymmetry is realized in a novel way, which is not the topological twist, and 2) the R-symmetry of the d=1 SCFT mixes with the U(1) isometry of the spindle, even in the absence of rotation. A similar idea also applies to a class of AdS3x\Sigma solutions of minimal gauged supergravity in D=5.

There is a connection between certain smooth representations of a reductive p-adic group and the representations of the Iwahori-Hecke algebra of this p-adic group. This Iwahori-Hecke algebra is a specialisation of a more general affine Hecke algebra. In this talk, we will discuss affine Hecke algebras and graded Hecke algebras. We will state a result from Lusztig (1989) that relates the representation theory of an affine Hecke algebra and a particular graded Hecke algebra and we will present a simple example of this relation.

Whether one uses the sales, the number of employees or any other proxy for firm "size", it is well known that this quantity is power-law distributed, with important consequences to aggregate macroeconomic fluctuations. The Gibrat model explained this by proposing that firms grow multiplicatively, and much work has been done to study the statistics of their growth rates. Inspired by past work in the statistics of financial returns, I present a new framework to study these growth rates. In particular, I will show that they follow approximately Gaussian statistics, provided their heteroskedastic nature is taken into account. I will also elucidate the size/volatility scaling relation, and show that volatility may have a strong sectoral dependence. Finally, I will show how this framework can be used to study intra-firm and supply chain dynamics.

Joint work with JP Bouchaud and Angelo Secchi.

A quantum circuit defines a discrete-time evolution for a set of quantum spins/qubits, via a sequence of unitary 'gates’ coupling nearby spins. I will describe how random quantum circuits, where each gate is a random unitary matrix, serve as minimal models for various universal features of many-body dynamics. These include the dynamical generation of entanglement between distant spatial regions, and the quantum "butterfly effect". I will give a very schematic overview of mappings that relate averages in random circuits to the classical statistical mechanics of random paths. Time permitting, I will describe a new phase transition in the dynamics of a many-body wavefunction, due to repeated measurements by an external observer.

Will a large economy be stable? In this talk, I will present a model for a network economy where firms' productions are interdependent, and study the conditions under which such input-output networks admit a competitive economic equilibrium, where markets clear and profits are zero. Insights from random matrix theory allow to understand some of the emergent properties of this equilibrium and to provide a classification for the different types of crises it can be subject to. After this, I will endow the model with dynamics, and present results with strong links to generalised Lotka-Volterra models in theoretical ecology, where inter-species interactions are modelled with random matrices and where the system naturally self-organises into a critical state. In both cases, the stationary points must consist of positive species populations/prices/outputs. Building on these ideas, I will show the key concepts behind an economic agent-based model that can exhibit convergence to equilibrium, limit cycles and chaotic dynamics, as well as a phase of spontaneous crises whose origin can be understood using "semi-linear" dynamics.

A topology optimization problem for Stokes flow finds the optimal material distribution of a fluid in Stokes flow that minimizes the fluid’s power dissipation under a volume constraint. In 2003, T. Borrvall and J. Petersson [1] formulated a nonconvex optimization problem for this objective. They proved the existence of minimizers in the infinite-dimensional setting and showed that a suitably chosen finite element method will converge in a weak(-*) sense to an unspecified solution. In this talk, we will extend and refine their numerical analysis. In particular, we will show that there exist finite element functions, satisfying the necessary first-order conditions of optimality, that converge strongly to each isolated local minimizer of the problem.

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Fully nonlinear PDEs involving the eigenvalues of matrix-valued differential operators (such as the Hessian) have been the subject of intensive study over the last few decades, since the seminal work of Caffarelli, Kohn, Nirenberg and Spruck. In this talk I will discuss some recent joint work with Luc Nguyen on the regularity theory for a large class of these equations, with a particular emphasis on a special case known as the sigma_k-Yamabe equation, which arises in conformal geometry. 

[1] T. Borrvall, J. Petersson, Topology optimization of fluids in Stokes flow, International Journal for Numerical Methods in Fluids 41 (1) (2003) 77–107. doi:10.1002/fld.426.

 An important property of a Gromov hyperbolic space is that every path that is locally a quasi-geodesic is globally a quasi-geodesic. A theorem of Gromov states that this is a characterization of hyperbolicity, which means that all the properties of hyperbolic spaces and groups can be traced back to this simple fact. In this talk we generalize this property by considering only Morse quasi-geodesics.

We show that not only does this allow us to consider a much larger class of examples, such as CAT(0) spaces, hierarchically hyperbolic spaces and fundamental groups of 3-manifolds, but also we can effortlessly generalize several results from the theory of hyperbolic groups that were previously unknown in this generality.
 

Abstract: In many situations, one needs to decide between acting to reveal data about a system and acting to generate profit; this is the trade-off between exploration and exploitation. A simple situation where we face this trade-off is a multiarmed bandit problem, where one has M ‘bandits’ which generate reward from an unknown distribution, and one must choose which bandit to play at each time. The key difficulty in the multi-armed bandit problem is that the action often affects the information obtained. Due to the curse of dimensionality, solving the bandit problem directly is often computationally intractable.

In this talk, we will formulate a general class of the multi-armed bandit problem as a relaxed stochastic control problem. By introducing an entropy premium, we obtain a smooth asymptotic approximation to the value function. This yields a novel semi-index approximation of the optimal decision process, obtained numerically by solving a fixed point problem, which can be interpreted as explicitly balancing an exploration–exploitation trade-off.  Performance of the resulting Asymptotic Randomised Control (ARC) algorithm compares favourably with other approaches to correlated multi-armed bandits.

As an application of the multi-armed bandit, we also consider a multi-armed bandit problem where the observation from each bandit arrive from a Generalised Linear Model. We then use such model to consider a dynamic online pricing problem. The numerical simulation shows that the ARC algorithm also performs well compared to others.
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One of the very unique properties of AdS_3 spacetimes is that we can introduce a finite number of massless higher spin fields without yielding an inconsistent theory. In this talk, we would like to comment on what the spectrum of these theories looks like: from the known contribution of the light spectrum, that corresponds to the vacuum character of the W_N algebra, we can use modular invariance to constraint the heavy spectrum of the theory. However, in doing so, we find negative norm states, inconsistent with unitarity. We propose a possible cure by adding light states that can be interpreted as massive particles with a conical defect associated to them, and study what scenario we are left with. The results that we will revisit are those presented in 2009.01830.