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We consider classical, pure Yang-Mills theory in a box. We show how a set of static electric fields that solve the theory in an adiabatic limit correspond to geodesic motion on the space of vacua, equipped with a particular Riemannian metric that we identify. The vacua are generated by spontaneously broken global gauge symmetries, leading to an infinite number of conserved momenta of the geodesic motion. We show that these correspond to the soft multipole charges of Yang-Mills theory.

We present an in-depth study of the domain walls available in the color code. We begin by presenting new boundaries which gives rise to a new family of color codes. Interestingly, the smallest example of such a code consists of just 4 qubits and weight three parity check measurements, making it an accessible playground for today's experimentalists interested in small scale experiments on topological codes. Secondly, we catalogue the twist defects that are accessible with the color code model. We give lattice representations of these twists and investigate how they interact with one another, and how they interact with the anyons of the system. Our categorisation allows us to explore new approaches for the fault-tolerant storage and manipulation of quantum information in color codes. This research combines and extends recent work with the surface code [1,2] to the color code models, whose continuous domain walls have been studied in generality in [3]. [1] Delfosse, Nicolas, Pavithran Iyer, and David Poulin. "Generalized surface codes and packing of logical qubits." arXiv preprint arXiv:1606.07116 (2016). [2] Brown, Benjamin J., et al. "Poking holes and cutting corners to achieve Clifford gates with the surface code." Physical Review X 7.2 (2017): 021029. [3] Yoshida, Beni. "Topological color code and symmetry-protected topological phases." Physical Review B 91.24 (2015): 245131.

There is a long tradition of formulating Quantum Theory in an operational manner.  In one version of this a circuit is formed by wiring together operations. Each operation has knob settings and outcomes (lights flashing, detectors clicking, ..).  The question raised and answered in this colloquium is whether we can do the same for General Relativity.  There are hurdles that have to be overcome to do this - in particular, we need to define a diffeomorphism invariant notion of operations, identify what the knob settings and outcomes are, and what the wiring corresponds to.   In this colloquium I will provide an outline of how to formulate General Relativity like this providing a possibilistic calculus (we can calculate whether operationally described situations are possible or not).  This provides the beginnings of an operational approach to the problem of Quantum Gravity.  This talk is based on arxiv:1608.06940.  

I will talk about a recent proof, joint with M. Gröchenig and D. Wyss, of a conjecture of Hausel and Thaddeus which predicts the equality of suitably defined Hodge numbers of moduli spaces of Higgs bundles with SL(n)- and PGL(n)-structure. The proof, inspired by an argument of Batyrev, proceeds by comparing the number of points of these moduli spaces over finite fields via p-adic integration.