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Symmetric monoidal categories provide a convenient and enlightening framework within which to compare and contrast physical theories on a common mathematical footing. In this talk we consider two theories: stabiliser qubit quantum mechanics and the toy bit theory proposed by Rob Spekkens. Expressed in the categorical framework the two theories look very similar mathematically, reflecting their common physical features. There are differences though: in particular a finite Abelian group emerges naturally in the categorical framework, and this group is different in each case ($Z_4$ for the stabiliser theory and $Z_2 \times Z_2$ for the toy bit theory). It turns out that this mathematical difference corresponds directly with a key physical difference between the theories: the stabiliser theory cannot be modelled by local hidden variables, while the toy bit theory can. This analysis can be extended to other Abelian groups yielding a group-theoretic criterion for determining the possibility of local hidden variable interpretations for other physical theories.

In the picture of eternal inflation, our observable universe resides inside a single bubble nucleated from an inflating false vacuum. Some of the theories giving rise to eternal inflation predict that we have causal access to collisions with other bubble universes, providing an opportunity to confront these theories with observation. In this talk, I will outline progress on the theoretical description of eternal inflation and bubble collisions, and present results from the first search for the effects of bubble collisions in the WMAP 7-year data.

N=8 supergravity in 4 dimensions exhibits a surprisingly favorable UV behavior -- it is known from explicit computations that the 4-point amplitudes in N=8 supergravity are finite up to 4-loop order. I explain how a "matrix-element approach" can be used to study candidate counterterms for UV divergences in this theory. This approach both demystifies the finiteness found in previous computations, and predicts finiteness of arbitrary n-point amplitudes in N=8 supergravity below the 7-loop order. It also points to the 7-loop 4-point amplitude as the first amplitude whose finiteness is not guaranteed by any known symmetry of the theory.

Quantum mechanics does not allow us to measure all possible combinations of observables on one system. Even in the simplest case of two observables we know, that measuring one of the observables changes the system in such way, that the other measurement will not give us desired precise information about the state of the system. Prominent examples of such observables are measurement of position and momentum of a particle, or measuring spin along two orthogonal directions. However, once we accept the possibility of imprecise measurements, we can consider to perform such measurement within one experiment. This is the basis of the notion of coexistence. I will present the basics of coexistence by showing how to perform the spin measurement in two directions, while considering the imprecision of the measurement described by POVMs. We can also go a little further and consider coexistence of instruments, i.e. measurements, where on the output besides classical information we are also left out with quantum post-measurement state.

Recent observations first made in string theory led to the realization that introducing various ``deformations" (finite volume, appropriate double-trace operators...) allows for an analytic study of many nonperturbative aspects of arbitrary four-dimensional gauge theories. Notably, the confinement mechanism, the generation of mass gap, and, in some cases, chiral symmetry breaking can be studied in a theoretically controlled manner. In this talk, I will review: i.) the ideas and tools involved, ii.) the main results obtained so far in this approach (and its limitations), iii.) our understanding of its relation to other approaches to (non-)supersymmetric gauge dynamics, iv.) ongoing and possible future work.