16:00
The local-to-global property for Morse quasi-geodesics
Abstract
An important property of Gromov hyperbolic spaces is the fact that every path for which all sufficiently long subpaths are quasi-geodesics is itself a quasi-geodesic. Gromov showed that this property is actually a characterization of hyperbolic spaces. In this talk, we will consider a weakened version of this local-to-global behaviour, called the Morse local-to-global property. The class of spaces that satisfy the Morse local-to-global property include several examples of interest, such as CAT(0) spaces, Mapping Class Groups, fundamental groups of closed 3-manifolds and more. The leverage offered by knowing that a space satisfies this property allows us to import several results and techniques from the theory of hyperbolic groups. In particular, we obtain results relating to stable subgroups, normal subgroups and algorithmic properties.
Areas-of-areas on Hall trees generate the shuffle algebra
Abstract
We consider the coordinate-iterated-integral as an algebraic product on the shuffle algebra, called the (right) half-shuffle product. Its anti-symmetrization defines the biproduct area(.,.), interpretable as the signed-area between two real-valued coordinate paths. We consider specific sets of binary, rooted trees known as Hall sets. These set have a complex combinatorial structure, which can be almost entirely circumvented by introducing the equivalent notion of Lazard sets. Using analytic results from dynamical systems and algebraic results from the theory of Lie algebras, we show that shuffle-polynomials in areas-of-areas on Hall trees generate the shuffle algebra.
For centuries, engineers have sought to prevent structures from buckling under heavy loads or large impacts, constructing ever larger buildings and safer vehicles. However, recent advances in soft matter are redefining the way we manipulate materials. In particular, an age-old aversion to buckling is being recast in a new light as researchers find that structural instabilities can be harnessed for functionality.
Energy production is arguably one of the most important factors underlying modern civilisation. Energy allows us to inhabit inhospitable parts of the Earth in relative comfort (using heating and air conditioning), create large cities (by efficiently transporting food and pumping water), or maintain our health (providing the energy for water purification). It also connects people by allowing long-distance travel and facilitating digital communication.
Lava flows: theory, laboratory experiments and field data
Abstract
World wide, unconstrained lava flows kill people almost each year and cause extensive damage, costing millions of pounds. Defending against lava flows is possible by using topographic variations sensibly, placing buildings considerately, constructing defending walls of appropriate size and the like. Hinton, Hogg and Huppert have recently published three rather mathematical papers outlining how viscous flows down slopes interact with a variety of geometrical shapes; evaluating, in particular, the conditions under which “dry zones” form – safe places for people and belongings – and the size of a protective wall required to defend a given size building.
Following a desktop experimental demonstration, we will discuss these analyses and their consequences.