We present models of a vapour bubble produced during ureteroscopy and laser lithotripsy treatment of kidney stones. This common treatment for kidney stones involves passing a flexible ureteroscope containing a laser fibre via the ureter and bladder into the kidney, where the fibre is placed in contact with the stone. Laser pulses are fired to fragment the stone into pieces small enough to pass through an outflow channel. Laser energy is also transferred to the surrounding fluid, resulting in vapourisation and the production of a cavitation bubble.

While in some cases, bubbles have undesirable effects – for example, causing retropulsion of the kidney stone – it is possible to exploit bubbles to make stone fragmentation more efficient. One laser manufacturer employs a method of firing laser pulses in quick succession; the latter pulses pass through the bubble created by the first pulse, which, due to the low absorption rate of vapour in comparison to liquid, increases the laser energy reaching the stone.

As is common in bubble dynamics, we couple the Rayleigh-Plesset equation to an energy conservation equation at the vapour-liquid boundary, and an advection-diffusion equation for the surrounding liquid temperature.1 However, this present work is novel in considering the laser, not only as the cause of nucleation, but as a spatiotemporal source of heat energy during the expansion and collapse of a vapour bubble.
 

Numerical and analytical methods are employed alongside experimental work to understand the effect of laser power, pulse duration and pulse pattern. Mathematically predicting the size, shape and duration of a bubble reduces the necessary experimental work and widens the possible parameter space to inform the design and usage of lasers clinically.

Given a fixed poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-free if it does not contain an (induced) copy of $\mathcal P$. And we say that $F$ is $\mathcal P$-saturated if it is maximal $\mathcal P$-free. How small can a $\mathcal P$-saturated family be? The smallest such size is the induced saturation number of $\mathcal P$, $\text{sat}^*(n, \mathcal P)$. Even for very small posets, the question of the growth speed of $\text{sat}^*(n,\mathcal P)$ seems to be hard. We present background on this problem and some recent results.

I will discuss the regularity of solutions to quasilinear systems satisfying a Legendre-Hadamard ellipticity condition. For such systems it is known that weak solutions may which fail to be C^1 in any neighbourhood, so we cannot expect a general regularity theory. However if we assume an a-priori regularity condition of the solutions we can rule out such counterexamples. Focusing on solutions to Euler-Lagrange systems, I will present an improved regularity results for solutions whose gradient satisfies a suitable BMO / VMO condition. Ideas behind the proof will be presented in the interior case, and global consequences will also be discussed.

Nonlinear wave equations are ubiquitous in physics, and in three spatial dimensions they can exhibit a wide range of interesting behaviour even in the small data regime, ranging from dispersion and scattering on the one hand, through to finite-time blowup on the other. The type of behaviour exhibited depends on the kinds of nonlinearities present in the equations. In this talk I will explore the boundary between "good" nonlinearities (leading to dispersion similar to the linear waves) and "bad" nonlinearities (leading to finite-time blowup). In particular, I will give an overview of a proof of global existence (for small initial data) for a wide class of nonlinear wave equations, including some which almost fail to exist globally, but in which the singularity in some sense takes an infinite time to form. I will also show how to construct other examples of nonlinear wave equations whose solutions exhibit very unusual asymptotic behaviour, while still admitting global small data solutions.

The aim of this talk is to present some recent developments of Geometric Measure Theory on non smooth spaces with lower Ricci Curvature bounds, mainly related to the first and second variation formula for the area, and their applications to the isoperimetric problem on non compact manifolds. The reinterpretation of some classical results in Geometric Analysis in a low regularity setting, combined with the compactness and stability theory for spaces with lower curvature bounds, leads to a series of new geometric inequalities for smooth, non compact Riemannian manifolds. The talk is based on joint works with Andrea Mondino, Gioacchino Antonelli, Enrico Pasqualetto and Marco Pozzetta.

Non-shearing congruences of null geodesics on four-dimensional Lorentzian manifolds are fundamental objects of mathematical relativity. Their prominence in exact solutions to the Einstein field equations is supported by major results such as the Robinson, Goldberg-Sachs and Kerr theorems. Conceptually, they lie at the crossroad between Lorentzian conformal geometry and Cauchy-Riemann geometry, and are one of the original ingredients of twistor theory.
 
Identified as involutive totally null complex distributions of maximal rank, such congruences generalise to any even dimensions, under the name of Robinson structures. Nurowski and Trautman aptly described them as Lorentzian analogues of Hermitian structures. In this talk, I will give a survey of old and new results in the field.

This talk aims to be a rigorous introduction to Social Choice Theory, a sub-branch of Game Theory with natural applications to economics, sociology and politics that tries to understand how to determine, based on the personal opinions of all individuals, the collective opinion of society. The goal is to prove the three famous and pessimistic impossibility theorems: Arrow's theorem, Gibbard's theorem and Balinski-Young's theorem. Our blunt conclusion will be that, unfortunately, there are no ideally fair social choice systems. Is there any hope yet?