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Applications of physics to geometry have deep historical roots going back at least to Archimedes, while recent decades have seen structures of classical and quantum gauge theory lead to enormous advances in the theory of three and four manifolds. Meanwhile, geometry provides conceptual tools for physics as foundational  as variational principles, the kinematics of spacetime, and the topological  classification of matter. This talk will describe some possibilities for bringing number theory, more specifically, *arithmetic geometry* into this interaction. After giving some examples of the suggestive interplay of  number theory and physics, I will speculate somewhat on how ideas of quantum field theory may elucidate some of the deepest and hardest problems of number theory, especially the effective Mordell conjecture and the Hasse-Weil conjecture.

I will give a holographic argument in favor of the AdS Penrose inequality, which conjectures a lower bound on the total mass in terms of the area of apparent horizons. This inequality is often viewed as a test of cosmic censorship. Time permitting, I’ll also discuss a generalization to charged black holes and connections with a quasi-local energy and the second law for apparent horizons.  

I will introduce my recent works on the phenomenology of solutions to the strong CP problem, QCD axion and Parity. I will first describe the production of the QCD axion in the early universe and show that the dark matter abundance is naturally reproduced for a wide range of the parameter space. I will then show a tight relation between the Parity solution, dark matter direct detection, the proton decay, and the precise measurements of the standard model parameters.

Quasars are among the most powerful light sources in the universe and, as such, can be seen at cosmological distances. Is some rare occasions (although not that rare), a massive galaxy on their line of sight can act as a gravitational lens and produce multiple images of distant quasars. These can be used both for cosmology and astrophysics by measuring the so-called time delays between the lensed images from photometric monitoring, a quantity directly related to the Hubble-Lemaître parameter H0. The same data, affected by microlensing by stars in the lensing galaxy, can be used to measure the size and energy profile of quasar accretion disks on spatial scales of the order of the light-day, i.e. inaccessible to even the largest interferometers, including the SKA. The results in this talk are on behalf of the COSMOGRAIL and H0LiCOW collaborations.