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.)

PDEs are ubiquitous in scientific fields and finance while machine learning (ML), like deep neural networks (NNs), has made tremendous progress over the last couple of years. In the past two years, the question whether NN can effectively learn to solve PDEs has been revisited. A potentially powerful application is for NNs to learn a mapping from any functional parametric dependence, to the solution. Thus, the network learns to solve an entire family of PDEs, in contrast to classical approaches that solve just one instance of the equation. This work introduces this idea, describes recent results and outlines further research opportunities.

One of the very unique properties of AdS_3 spacetimes is that we can introduce a finite number of massless higher spin fields without yielding an inconsistent theory. In this talk, we would like to comment on what the spectrum of these theories looks like: from the known contribution of the light spectrum, that corresponds to the vacuum character of the W_N algebra, we can use modular invariance to constraint the heavy spectrum of the theory. However, in doing so, we find negative norm states, inconsistent with unitarity. We propose a possible cure by adding light states that can be interpreted as massive particles with a conical defect associated to them, and study what scenario we are left with. The results that we will revisit are those presented in 2009.01830. 

Magnitude homology is a homology theory for enriched categories, first defined for graphs by Hepworth and Willerton in 2015, and extended to a much wider class of categories in 2017 by Leinster and Shulman. Magnitude homology is designed to categorify magnitude, a numerical invariant of enriched categories which generalizes topological Euler characteristic. In the case of metric spaces - regarded as categories enriched in the poset of nonnegative reals - magnitude has been shown to capture a rich variety of geometric data related to volume or size; magnitude homology deepens this picture, encoding in its first few groups subtle information about the convexity of a space.

This talk is about what happens when we enrich in a category of enriched categories. Enriching in the category of locally small categories gives us 2-categories, and iterating that step leads to the definition of a strict n-category. Other forms of second-order enrichment occur in the wild: for instance, locally posetal 2-categories arise naturally in theoretical computer science and elsewhere, while metrically enriched categories are important in the study of 'approximate categorical structures', in which diagrams may be said to commute 'up to epsilon'.

I'll be describing some work-in-progress concerning the magnitude homology of such iterated enriched categories, emphasising examples that may be familiar to the non-category-theorist. After an introduction to the theory of magnitude and magnitude homology, I'll summarise recent findings regarding the homology of 2-categories, including 2-groups, and of certain strict n-categories for higher values of n; if time permits I may also describe the low-dimensional homology of certain Met-enriched categories.

The concept of numerical index was introduced by Lumer in 1968 in the context of the study and classification of operator algebras. This is a constant relating the norm and the numerical range of bounded linear operators on the space. More precisely, the numerical index of a Banach space X, n(X), is the greatest constant k0$ such that \(k\|T\|\leq \sup\bigl\{|x^\ast(Tx)|\colon x^\ast\in X^\ast,\,x\in X,\,\|x^\ast\|=\|x\|=x^\ast(x)=1\bigr\}\) for every \(T\in\mathcal{L}(X)\).

Recently, Ardalani introduced new concepts of numerical range, numerical radius, and numerical index, which generalize in a natural way the classical ones and allow to extend the setting to the context of operators between possibly different Banach spaces. Given a norm-one operator \(G\in \mathcal{L}(X,Y)\) between two Banach spaces \(X\) and \(Y\), the numerical index with respect to \(G\), \(n_G(X,Y)\), is the greatest constant \(k\geq 0\) such that $k|T|_{>0} {|y^(Tx)|yY^,,xX,,|y|=|x|=1,, y^(Gx)>1-} $ for every \(T\in \mathcal{L}(X,Y)\).

In this talk, we will give an overview of the topic, analysing differences and similarities between these concepts, and presenting some classical and recent results in the area.

The Cuntz semigroup is a powerful invariant in the classification theory of C-algebras that generalizes the construction of the Murray-von Neumann semigroup of projections. As shown by Coward, Elliott and Ivanescu, the Cuntz semigroup of a C-algebra is always a positively ordered monoid that satisfies four additional continuity conditions. Such conditions led to the definiton of abstract Cuntz semigroups, or Cu-semigroups for short.

In this talk I will introduce Cuntz semigroups together with their abstract counterpart and explain how the study of Cu-semigroups can provide insights on the structure of Cuntz semigroups. I will also give a short list of open problems in the field.