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This talk will focus on hypermultiplet moduli spaces of various N=2 supersymmetric gauge theories in (3+1)d. In the first part of the talk, we discuss the moduli space of instantons on C^2. For the classical groups, the ADHM construction of the moduli space can be realised on the Higgs branch of N=2 gauge theories on D3-branes probing D7-branes. No known construction is available for exceptional groups. We go over the computation of Hilbert series for the one instanton moduli space and show that it is possible to count all chiral operators on the moduli space even though a Lagrangian is not known for exceptional gauge groups. In the second part, we discuss a class of N=2 gauge theories on two M5-branes wrapping Riemann surfaces. This talk will go over Hilbert series for the hypermultiplet moduli space of such theories and show that it is possible to count all chiral operators on the hypermultiplet moduli space for any genus and any number of punctures of the Riemann surface.

The question of the existence of gravitational stress-energy in general relativity has exercised investigators in the field since the very inception of the theory. Folklore has it that no adequate definition of a localized gravitational stress-energetic quantity can be given. Most arguments to that effect invoke one version or another of the Principle of Equivalence. I argue that not only are such arguments of necessity vague and hand-waving but, worse, are beside the point and do not address the heart of the issue. Based on an analysis of what it may mean for one tensor to depend in the proper way on another, I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. Along the way, I prove a result of some interest in own right about the structure of the associated jet bundles of the bundle of Lorentz metrics over spacetime.

Over the last decade there has been strong interest in the theory and phenomenology of particle propagation in quantum spacetime. The main results concern possible Planck-scale modifications of the "dispersion" relation between energy and momentum of a particle. I review results establishing that these modifications can be tested using observations of gamma rays from sources at cosmological distances. And I report recent progress in the understanding of the implications of spacetime expansion for such studies. I also discuss recent preliminary results suggesting that the same Planck-scale modifications of the dispersion relation might have an unexpected role in gravitational collapse.