In 2016, Habegger and Pila published a proof of the Zilber-Pink conjecture for curves in abelian varieties (all defined over $\mathbb{Q}^{\rm alg}$). Their article also contained a proof of the same conjecture for a product of modular curves that was conditional on a strong arithmetic hypothesis. Both proofs were extensions of the Pila-Zannier strategy based in o-minimality that has yielded many results in this area. In this talk, we will explain our generalisation of the strategy to the Zilber-Pink conjecture for any Shimura variety. This is joint work with J. Ren.

 Locality is not expected to be a fundamental aspect of a full theory of quantum gravity; it should be emergent in an appropriate semiclassical limit.  In the context of general holography, I'll define a new construct - the causal state - which provides a necessary and sufficient condition for a boundary state to have a holographic semiclassical dual causal geometry (and thus be "local").  This definition illuminates some general features of holographic quantum gravity: for instance, I'll show that the emergence of locality is "all or nothing" in the sense that it exhibits features of quantum error correction and quantum secret sharing.  In the special case of AdS/CFT, I'll also argue that the causal state is the natural boundary dual to the so-called causal wedge of a region. 

For every $\epsilon>0$ does there exist some $n\in\mathbb{N}$ and a bijection $f:\mathbb{Z}_n\to\mathbb{Z}_n$ such that $f(x+1)=2f(x)$ for at least $(1-\epsilon)n$ elements of $\mathbb{Z}_n$ and $f(f(f(f(x))))=(x)$ for all $x\in\mathbb{Z}_n$? I will discuss this question and its relation to an important open problem in the theory of countable discrete groups.

Multi-agent systems in nature oftentimes exhibit emergent behaviour, i.e. the formation of patterns in the absence of a leader or external stimuli such as light or food sources. We present a non-local two species crossinteraction model with cross-diffusion and explore its long-time behaviour. We observe a rich zoology of behaviours exhibiting phenomena such as mixing and/or segregation of both species and the formation of travelling pulses.