As we saw in earlier talks, a rigid framework (G,p) (by which we mean continuous rigidity) has the property that any equivalent (edge-length preserving) framework (G,q) in Rd which can be reached through a continuous motion of the vertices must also be congruent to (G,p). In other words, the only such continuous motions are isometries. Global rigidity takes this a step further by removing the condition that such a migration from (G,p) to (G,q) should need to be continuous. A framework is globally rigid if any equivalent framework is necessarily congruent. As such, global rigidity is a much more challenging prospect. Several rigidity results are well-known in lower dimensions such as the Laman-Pollaczek-Geringer conditions for rigidity in the plane (E = 2V – 3) and other dimensions of real-space but what happens when we impose a further condition that the framework must live on a 2-dimensional surface in 3-space? The objective of this talk is to outline the conjecture of the believed necessary and sufficient conditions to ensure (generic) Global Rigidity on the Elliptical Cylinder: namely, 2-connectedness and redundant rigidity.

It is 40 years since the appearance of N.G. de Bruijn's highly influential paper on the pentagrid method for the construction of Penrose tilings. I will consider, more generally, parallelogram tilings P and their bar-joint frameworks G_P and characterise the linearly-localised flexes of G_P. This leads to notions of zero mode wave vector spectra for general quasicrystals, with particular significance for multigrid quasicrystals G_P. The main method (and my focus in this talk) is somewhat elementary "combinatorial linear algebra" to obtain an infinite "free basis" for the flex space of G_P. This also leads quickly to a Bolker-Crapo type characterisation of rigidity for braced parallelogram tilings.

The study of the interaction between matter and radiation by means of Quantum Field Theories (QFT) represents a challenge due to the lack of a general, rigorous mathematical frame and the difficulty of the calculations involved. A successful strategy to overcome these obstacles is the derivation of effective models, which retain the important physical features of the microscopic ones, but are simpler to analyze. This is possible, for example, when the field alone is really intense or when both the number of particles and the excitations of the field is high, namely in the quasi-classical and semi-classical regimes, respectively. In the present talk we show how in these regimes, by introducing a semi-classical algebraic formulations for the QFT, the dynamics and the ground state energy of the QFT models converge in suitable topologies to the dynamics and ground state energy of effective Schrödinger models. 

There are several existing results showing some relationships between amenable operator algebras and C-algebras. For example, in 2013, the first example of an amenable operator algebra which is not "similar" to a C-algebra (where "similar" has a precise definition) was found by Choi, Farah, and Ozawa; said example was a subalgebra of ^(N;M_2). Also in 2013, Marcoux and Popov showed that all abelian amenable operator algebras are similar to C-algebras, and back in 1997, Gifford showed that all amenable subalgebras of K(H) are similar to C-algebras. (This last result was republished in 2006.)

In this talk, however, we will aim to be as accessible as possible, assuming no prior knowledge of amenability, and go through the thought process behind proving a result which lies somewhere between the Choi-Farah-Ozawa result and Gifford's result. Namely, that all amenable subalgebras A of the algebra of all convergent sequences of 2 x 2 matrices are similar to C*-algebras. (Which will be included in the speaker's thesis upon eventual completion.)