Abstract:  Anyone who has worked in $\beta $N will not be surprised to learn that some of the algebraically defined subsets of $\beta N$ are not topologically simple, even though their algebraic definition may be very simple.  I shall show that the following subsets of $\beta N$ are not Borel: $N^*+N^*$; the smallest ideal of $\beta N$; the set of idempotents in $\beta N$; any semiprincipal right ideal in $\beta N$; the set of idempotents in any left ideal in $\beta N$.

The "Weyl fermion" was discovered in a topological semimetal in
2015. Its mathematical characterisation turns out to involve deep and subtle
results in differential topology. I will outline this theory, and explain
some connections to Euler structures, torsion of manifolds,
and Seiberg-Witten invariants. I also propose interesting generalisations
with torsion topological charges arising from Kervaire semicharacteristics
and ``Quaternionic'' characteristic classes.

In this talk, I will talk about my current approach to model customer movements and in particular congestion inside supermarkets using queuing networks. As the research question for my project is ‘How should one design supermarkets to minimize congestion?’, I will then talk about my current progress in understanding how the network structure can affect this dynamics.

There are several conjectures in the literature suggesting that absolute Galois groups of fields tend to be "as free as possible," given their "almost-abelian" data.
This can be made precise in various ways, one of which is via the notion of "1-formality" arising in analogy with the concept in rational homotopy theory.
In this talk, I will discuss several examples which illustrate this phenomenon, as well as some surprising diophantine consequences.
This discussion will also include some recent joint work with Guillot, Mináč, Tân and Wittenberg, concerning the vanishing of mod-2 4-fold Massey products in the Galois cohomology of number fields.

We begin by reviewing the “Gravity = Gauge x Gauge” paradigm that has emerged over the last decade. In particular, we will consider the origin of gravitational scattering amplitudes, symmetries and classical solutions in terms of the product of two Yang-Mills theories. Motivated by these developments we begin to address the classification of gravitational theories admitting a “factorisation” into a product of gauge theories. Progress in this direction leads us to twin supergravity theories - pair of supergravities with distinct supersymmetries, but identical bosonic sectors - from the perspective of Yang-Mills squared. 

Mirror symmetry predicts surprising geometric correspondences between distinct families of algebraic varieties. In some cases, these correspondences have arithmetic consequences. Among the arithmetic correspondences predicted by mirror symmetry are correspondences between point counts over finite fields, and more generally between factors of their Zeta functions. In particular, we will discuss our results on a common factor for Zeta functions alternate families of invertible polynomials. We will also explore closed formulas for the point counts for our alternate mirror families of K3 surfaces and their relation to their Picard–Fuchs equations. Finally, we will discuss how all of this relates to hypergeometric motives. This is joint work with: Charles Doran (University of Alberta, Canada), Tyler Kelly (University of Cambridge, UK), Steven Sperber (University of Minnesota, USA), John Voight (Dartmouth College, USA), and Ursula Whitcher (American Mathematical Society, USA).

Recently, millions of novel examples of compact G2 holonomy manifolds have been constructed as twisted connected sums of asymptotically cylindrical Calabi-Yau threefolds. In case these are K3 fibred, they can in turn be constructed from dual pairs of tops. This is analogous to Batyrev's construction of Calabi-Yau manifolds via reflexive polytopes. For compactifications of Type II superstrings on such G2 manifolds, we formulate a construction of the mirror.