C5

I will explain a beautiful link between differential and algebraic geometry, called the Kempf-Ness Theorem, which says that the natural notions of "quotient spaces" in the symplectic and algebraic categories can often be identified. The result will be presented in its most general form where actions are not necessarily free and hence I will also introduce the notion of stratified spaces.

We describe a locally finite graph naturally associated to each knot type K, called the Reidemeister graph. We determine several local and global properties of this graph and prove that the graph-isomorphism type is a complete knot invariant up to mirroring. Lastly (time permitting), we introduce another object, relating the Reidemeister and Gordian graphs, and briefly present an application to the study of DNA.

Snap-through buckling is a type of instability in which an elastic object rapidly jumps from one state to another, just as an umbrella flips upwards in a gust of wind. While snap-through under dry, mechanical loads has already been harnessed in engineering to generate fast motions between two states, the mechanisms underlying snapping in bulk fluid flows remain relatively unexplored. In this talk we demonstrate how elastic snap-through may be used to passively control fluid flows at low Reynolds number, in contrast to some pre-existing valves that rely on active control. We study viscous flow through a channel in which one of the bounding walls is an elastic arch. By performing experiments at the macroscopic scale, we show that snap-through of the arch rapidly changes the channel from a constricted to an unconstricted state, increasing the hydraulic conductivity by up to an order of magnitude. We also observe nonlinear pressure-flux characteristics away from snapping due to the coupling between the driving flow and elasticity. This behaviour is confirmed by a mathematical model that also shows the device may readily be scaled down for microfluidic applications. Finally, we demonstrate that such a device may be used to create a fluidic analogue of a fuse: the fluid flux through a channel may not rise above a given value. 

Many species of insects adhere to vertical and inverted surfaces using footpads that secrete thin films of a mediating fluid. The fluid bridges the gap between the foot and the target surface. The precise role of this liquid is still subject to debate, but it is thought that the contribution of surface tension to the adhesive force may be significant. It is also known that the footpad is soft, suggesting that capillary forces might deform its surface. Inspired by these physical ingredients, we study a model problem in which a thin, deformable membrane under tension is adhered to a flat, rigid surface by a liquid droplet. We find that there can be multiple possible equilibrium states, with the number depending on the applied tension and aspect ratio of the system. The presence of elastic deformation significantly enhances the adhesion force compared to a rigid footpad. A mathematical model shows that the equilibria of the system can be controlled via two key parameters depending on the imposed separation of the foot and target surface, and the tension applied to the membrane. We confirm this finding experimentally and show that the system may transition rapidly between two states as the two parameters are varied. This suggests that different strategies may be used to adhere strongly and then detach quickly.

I will give a survey of known results about when two RAAGs are quasi-isometric, and will then describe a visual graph of groups decomposition of a RAAG (its JSJ tree of cylinders) that can often be used to determine whether or not two RAAGs are quasi-isometric.

If $G$ is an irreducible lattice in a semisimple Lie group, every action of $G$ on a tree has a global fixed point. I will give an elementary discussion of Y. Shalom's proof of this result, focussing on the case of $SL_2(\mathbb{R}) \times SL_2(\mathbb{R})$. Emphasis will be placed on the geometric aspects of the proof and on the importance of reduced cohomology, while other representation theoretic/functional analytic tools will be relegated to a couple of black boxes.