Martin Gallauer (North): "Algebraic algebraic geometry"
If a space is described by algebraic equations, its algebraic invariants are endowed with additional structure. I will illustrate this with some simple examples, and speculate on the meaning of the title of my talk.

Zhaohe Dai (South): "Two-dimensional material bubbles"
Two-dimensional (2D) materials are a relatively new class of thin sheets consisting of a single layer of covalently bonded atoms and have shown a host of unique electronic properties. In 2D material electronic devices, however, bubbles often form spontaneously due to the trapping of air or ambient contaminants (such as water molecules and hydrocarbons) at sheet-substrate interfaces. Though they have been considered to be a nuisance, I will discuss that bubbles can be used to characterize 2D materials' bending rigidity after the pressure inside being well controlled. I will then focus on bubbles of relatively large deformations so that the elastic tension could drive the radial slippage of the sheet on its substrate. Finally, I will discuss that the consideration of such slippage is vital to characterize the sheet's stretching stiffness and gives new opportunities to understand the adhesive and frictional interactions between the sheet and various substrates that it contacts.
 

Motivated by an attempt to construct a theory of quantum gravity as a perturbation around some flat background, Penrose has shown that, despite being asymptotically flat, there is an inconsistency between the causal structure at infinity of Schwarzschild and Minkowski spacetimes. This suggests that such a perturbative approach cannot possibly work. However, the proof of this inconsistency is specific to 4 spacetime dimensions. In this talk I will discuss how this result extends to higher (and lower) dimensions. More generally, I will consider examples of how the causal structure of asymptotically flat spacetimes are affected by dimension and by the presence of mass (both positive and negative). I will then show how these ideas can be used to prove a higher dimensional extension of the positive mass theorem of Penrose, Sorkin and Woolgar.

Kirigami, the relatively unheralded cousin of origami, is the art of cutting paper to articulate and deploy it as a whole. By varying the number, size, orientation and coordination of the cuts, artists have used their imagination and intuition to create remarkable sculptures in 2 and 3 dimensions. I will describe some of our attempts to quantify the inverse problem that artists routinely solve, combining elementary mathematical ideas, with computations and physical models. 

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Artificial microswimmers are an emerging field of research, attracting
interest as testing beds for physical theories of complex biological
entities, as inspiration for the design of smart materials, and for the
sheer elegance, and often quite counterintuitive phenomena of
experimental nonlinear dynamics.

Self-propelling droplets are among the most simplified swimmer models
imaginable, requiring just three components (oil, water, surfactant). In
this talk, I will show how these inherently stupid objects can make
surprisingly smart decisions based on interactions with microfluidic
structures and self-generated and external chemical fields.

Michael Coughlan (with Sam Howison, Ian Hewitt, Andrew Wells)

Arctic sea ice forms a thin but significant layer at the ocean surface, mediating key climate feedbacks. During summer, surface melting produces considerable volumes of water, which collect on the ice surface in ponds. These ponds have long been suggested as a contributing factor to the discrepancy between observed and predicted sea ice extent. When viewed at large scales ponds have a complicated, approximately fractal geometry and vary in area from tens to thousands of square meters. Increases in pond depth and area lead to further increases in heat absorption and overall melting, contributing to the ice-albedo feedback. 

Previous modelling work has focussed either on the physics of individual ponds or on the statistical behaviour of systems of ponds. In this talk I present a physically-based network model for systems of ponds which accounts for both the individual and collective behaviour of ponds. Each pond initially occupies a distinct catchment basin and evolves according to a mass-conserving differential equation representing the melting dynamics for bare and water-covered ice. Ponds can later connect together to form a network with fluxes of water between catchment areas, constrained by the ice topography and pond water levels. 

I use the model to explore how the evolution of pond area and hence melting depends on the governing parameters, and to explore how the connections between ponds develop over the melt season. Comparisons with observations are made to demonstrate the ways in which the model qualitatively replicates properties of pond systems, including fractal dimension of pond areas and two distinct regimes of pond complexity that are observed during their development cycle. 

Different perimeter-area relationships exist for ponds in the two regimes. The model replicates these relationships and exhibits a percolation transition around the transition between these regimes, a facet of pond behaviour suggested by previous studies. The results reinforce the findings of these studies on percolation thresholds in pond systems and further allow us to constrain pond coverage at this threshold - an important quantity in measuring the scale and effects of the ice-albedo feedback.

We propose a subspace Gauss-Newton method for nonlinear least squares problems that builds a sketch of the Jacobian on each iteration. We provide global rates of convergence for regularization and trust-region variants, both in expectation and as a tail bound, for diverse choices of the sketching matrix that are suitable for dense and sparse problems. We also have encouraging computational results on machine learning problems.

Let W be the Witt algebra of vector fields on the punctured complex plane, and let Vir be the Virasoro algebra, the unique nontrivial central extension of W.  We discuss work in progress with Alexey Petukhov to analyse Poisson ideals of the symmetric algebra of Vir.

We focus on understanding maximal Poisson ideals, which can be given as the Poisson cores of maximal ideals of Sym(Vir) and of Sym(W).  We give a complete classification of maximal ideals of Sym(W) which have nontrivial Poisson cores.  We then lift this classification to Sym(Vir), and use it to show that if $\lambda \neq 0$, then $(z-\lambda)$ is a maximal Poisson ideal of Sym(Vir).

Sieve methods are analytic tools that we can use to tackle problems in additive number theory. This talk will serve as a gentle introduction to the area. At the end we will discuss recent progress on a variation on the prime $k$-tuples conjecture which involves sums of two squares. No knowledge of sieves is required!

In 1983, Faltings proved Mordell's conjecture on the finiteness of $K$-points on curves of genus >1 defined over a number field $K$ by proving the finiteness of isomorphism classes of isogenous abelian varieties over $K$. The "first" major step from Mordell's conjecture to what Faltings did came 15 years earlier when Parshin showed that a certain conjecture of Shafarevich would imply Mordell's conjecture. In this talk, I'll focus on motivating and sketching Parshin's argument in an accessible manner and provide some heuristics on how to get from Faltings' finiteness statement to the Shafarevich conjecture.