Queen's College

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.

Kazhdan introduced property (T) for locally compact topological groups to show that certain lattices in semisimple Lie groups are finitely generated. This talk will give an introduction to property (T) along with some first consequences and examples. We will finish with a classic application of property (T) due to Margulis: the first known construction of expanders.

I will discuss what it means to compactify complex Lie groups and introduce the so-called "Wonderful Compactification" of groups having trivial centre. I will then show how the wonderful compactification of PGL(n) can be described in terms of complete collineations. Finally, I will discuss how the new perspective provided by complete collineations provides a way to construct compactifications of arbitrary semisimple groups.

(A simplified version of) Ax-Grothendieck Theorem states that every injective polynomial map from some power of complex numbers into itself is surjective. I will present a simple model-theoretical proof of this fact. All the necessary notions from model theory will be introduced during the talk. The only prerequisite is basic field theory.

A large class of links in $S^3$ has the property that the complement admits a complete hyperbolic metric of finite volume. But is this volume understandable from the link itself, or maybe from some nice diagram of it? Marc Lackenby in the early 2000s gave a positive answer for a class of diagrams, the alternating ones. The proof of this result involves an analysis of the JSJ decomposition of the link complement: in particular of how does it appear on the link diagram. I will tell you an outline of this proof, forgetting its most technical aspects and explaining the underlying ideas in an accessible way.

I will look at the classical constructions that can be made using a straight edge and compass, I will then look at the limits of these constructions. I will then show how much further we can get with Origami, explaining how it is possible to trisect an angle or double a cube. Compasses not supplied.

I shall outline a procedure for efficiently approximating arbitrary elements of certain topological groups by words in a finite set. The method is suprisingly general and is based upon the assumption that close to the identity, group elements may be easily expressible as commutators. Time permitting, I shall discuss some applications to uniform diameter bounds for finite groups and to quantum computation.

I will give a gentle introduction to uniformly finite homology. The highlight application will be showing existence of aperiodic tilings of the hyperbolic plane.