In the aftermath of the Thirty Years War (1618–1648), the mining regions of Central Europe underwent numerous technical and political evolutions. In this context, the role of underground geometry expanded considerably: drawing mining maps and working on them became widespread in the second half of the seventeenth century. The new mathematics of subterranean surveyors finally realized the old dream of “seeing through stones,” gradually replacing alternative tools such as written reports of visitations, wood models, or commented sketches.

I argue that the development of new cartographic tools to visualize the underground was deeply linked to broad changes in the political structure of mining regions. In Saxony, arguably the leading mining region, captain-general Abraham von Schönberg (1640–1711) put his weight and reputation behind the new geometrical technology, hoping that its acceptance would in turn help him advance his reform agenda. At-scale representations were instrumental in justifying new investments, while offering technical road maps to implement them.

In 2013, Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees found a remarkable correspondence between SCFTs in 4d with N ≥ 2 and vertex algebras. The chiral algebras of class S, i.e. the vertex algebras associated to theories of Class S, are of particular interest as they exhibit rich algebraic structures arising from the requirement of generalised S-duality. I will explain a mathematical construction of these vertex algebras, due to Arakawa, that is remarkably uniform and requires no knowledge of the underlying SCFT. Time permitting, I will detail a recent generalisation of this construction to the case of the chiral algebras of class S with outer automorphism twist lines.

Understanding the impact of societal and economic change on the labour market is important for many causes, such as automation or the post-carbon transition. Occupational mobility plays a role in how these changes impact the labour market because of indirect effects, brought on by the different levels of direct impact felt by individual occupations. We develop an agent-based model which uses a network representation of the labour market to understand these impacts. This network connects occupations that workers have transitioned between in the past, and captures the complex structure of relationships between occupations within the labour market. We develop these networks in both space and time using rich survey data to compare occupational mobility across the United States and through economic upturns and downturns to start understanding the factors that influence differences in occupational mobility.

For almost calibrated Lagrangian mean curvature flow it is known that all singularities are of Type II. To understand the finer structure of the singularities forming, it is thus necessary to understand the structure of general ancient solutions arising as potential limit flows at such singularities. We will discuss recent progress showing that ancient solutions with a blow-down a pair of static planes meeting along a 1-dimensional line are translators. This is joint work with J. Lotay and G. Szekelyhidi.

Let $N$ be a compact Riemannian manifold of dimension 3 or higher, and $g$ a Lipschitz non-negative (or non-positive) function on $N$. In joint works with Neshan Wickramasekera we prove that there exists a closed hypersurface $M$ whose mean curvature attains the values prescribed by $g$. Except possibly for a small singular set (of codimension 7 or higher), the hypersurface $M$ is $C^2$ immersed and two-sided (it admits a global unit normal); the scalar mean curvature at $x$ is $g(x)$ with respect to a global choice of unit normal. More precisely, the immersion is a quasi-embedding, namely the only non-embedded points are caused by tangential self-intersections: around such a non-embedded point, the local structure is given by two disks, lying on one side of each other, and intersecting tangentially (as in the case of two spherical caps touching at a point). A special case of PMC (prescribed-mean-curvature) hypersurfaces is obtained when $g$ is a constant, in which the above result gives a CMC (constant-mean-curvature) hypersurface for any prescribed value of the mean curvature.