I will review how the equations of general relativity near a spacetime singularity map onto an arithmetic hyperbolic billiard dynamics. The semiclassical quantum states for this dynamics are Maaβ cusp forms on fundamental domains of modular groups. For example, gravity in four spacetime dimensions leads to PSL(2,Z) while five dimensional gravity leads to PSL(2,Z[w]), with Z[w] the Eisenstein integers. The automorphic forms can be expressed, in a dilatation (Mellin transformed) basis as L-functions. The Euler product representation of these L-functions indicates that these quantum states admit a dual interpretation as a "primon gas" partition function. I will describe some physically motivated mathematical questions that arise from these observations.

TBA

Wave energy has the theoretical potential to meet global electricity demand, but it remains less mature and less cost-competitive than wind or solar power. A key barrier is the absence of engineering convergence on an optimal wave energy converter (WEC) design. In this work, I demonstrate how geometry optimisation can deliver step-change improvements in WEC performance. I present methodology and results from optimisations of two types of WECs: an axisymmetric point-absorber WEC and a top-hinged WEC. I show how the two types need different optimisation frameworks due to the differing physics of how they make waves. For axisymmetric WECs, optimisation achieves a 69% reduction in surface area (a cost proxy) while preserving power capture and motion constraints. For top-hinged WECs, optimisation reduces the reaction moment (another cost proxy) by 35% with only a 12% decrease in power. These result show that geometry optimisation can substantially improve performance and reduce costs of WECs.

In this talk, we will explore three flow configurations that illustrate the behaviour of slow-moving viscous fluids in confined geometries: viscous gravity currents, fracturing of shear-thinning fluids in a Hele-Shaw cell, and rectangular channel flows of non-Newtonian fluids. We will first develop simple mathematical models to describe each setup, and then we will compare the theoretical predictions from these models with laboratory experiments. As is often the case, we will see that even models that are grounded in solid physical principles often fail to accurately predict the real-world flow behaviour. Our aim is to identify the primary physical mechanisms absent from the model using laboratory experiments. We will then refine the mathematical models and see whether better agreement between theory and experiment can be achieved.

When a specimen of non-trivial shape undergoes deformation under a dead load or during an active process, finite element simulations are the only technique for evaluating the deformation. Classical books describe complicated techniques for evaluating stresses and strains in semi-infinite, circular or cylindrical objects.  However, the results obtained are limited, and it is well known that elasticity (linear or nonlinear) is strongly intertwined with geometry. For the simplest geometries, it is possible to determine the exact deformation, essentially for low loading values, and prove that there is a threshold above which the specimen loses stability. The next step is to apply perturbation techniques (linear and nonlinear bifurcation theory).
 

In this talk, I will demonstrate how many aspects can be simplified or revealed through the use of complex analysis and conformal mapping techniques for shapes, strains, and active stresses in thin samples. Examples include leaves and embryonic jellyfish.

In this talk, we will explore three flow configurations that illustrate the behaviour of slow-moving viscous fluids in confined geometries: viscous gravity currents, fracturing of shear-thinning fluids in a Hele-Shaw cell, and rectangular channel flows of non-Newtonian fluids. We will first develop simple mathematical models to describe each setup, and then we will compare the theoretical predictions from these models with laboratory experiments. As is often the case, we will see that even models that are grounded in solid physical principles often fail to accurately predict the real-world flow behaviour. Our aim is to identify the primary physical mechanisms absent from the model using laboratory experiments. We will then refine the mathematical models and see whether better agreement between theory and experiment can be achieved.