NCCC is by no means a popular tool in physics, and certainly the possibilities of its application have not been fully exhausted. The spinor equivalent of NCCC appears in the literature as a local twistor connection. Thus, using NCCC, spacetimes admitting covariantly constant twistor fields were once found and identified as conformal metrics constructed earlier by Fefferman and used in the geometric approach to the Cauchy-Riemann structures. A new application of NCCC is the construction of conformally invariant symplectic potential on the space of Bach flat geometries. These include all Einstein spacetimes. NCCC provides new charges and currents well-defined both at points in spacetime and at its conformal boundary. Relevant examples will be demonstrated in the Graham-Fefferman coordinates.

We define generalised higher Whitehead maps in polyhedral products and study their properties and the relations among them. By investigating the interplay between the homotopy theoretic properties of polyhedral products and the combinatorial properties of simplicial complexes, we describe new families of relations among these maps, while recovering and generalising known identities among Whitehead products.

Zoom Meeting ID: 93798234275, Passcode: 520011

(Coffee @ 3:00) Legendre pairs were introduced in 2001 by Seberry and her students, as a means to construct Hadamard matrices via a two-circulant core construction. Legendre pairs of every odd prime length exist, via a simple construction using the Legendre symbol. We will review known constructions for Legendre pairs. We will discuss various results on Legendre pairs during the past 20+ years, including the concept of compression, introduced in a joint paper with D. Z. Djokovic, as well as the computational state-of-the-art of the search for Legendre pairs. Finally, we will discuss a Coding Theory reformulation of the problem of enumerating Legendre pairs for a given odd length ell, via computing the weight enumerator of a binary code, based on work of S. Eliahou. The importance of Legendre pairs lies in the fact that they constitute a promising avenue to the Hadamard conjecture.