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Quantized lattices, or q-lattices, appear naturally through categorification constructions (for example from zigzag-algebras), but they haven't been studied from a lattice theory point of view. After establishing the necessary background, we'll explain the q-versions of various lattice theory concepts illustrated with examples from combinatorics and graph theory, and we'll list a number of open problems.

I will first outline an effective field theory for cosmology (EFTC) that is based on the Standard Model coupled to General Relativity and improved with Weyl symmetry. Any version of quantum gravity (QG), including string theory, must include the same improvement, otherwise QG will not be geodesically complete.  Based on arguments of an underlying QG theory, this effective field theory banishes higher curvature terms in the action,  thus making this EFTC  mathematically sufficiently well behaved to make unique predictions for the classical solutions and the quantum wavefunction at the singularity, as well as past the singularity, for a complete cosmology that includes the past, future and detailed quantitative solution at the big bang.
Using this EFTC, I will show its predictions of surprising behavior of the universe at the singularity that cosmologists usually "conveniently" ignore, but they should not, because this predicts the initial conditions. I will illustrate this behavior with detailed formulas and plots of the classical and quantum solutions. The solutions are given in the geodesically complete mini-superspace that has the flavor of the extended spacetime of a black hole or extended Rindler spacetime. The analytic continuation of the quantum wavefunction across the horizons describes the passage through the singularities. The mathematical solution of this analytic continuation solves a long-standing problem of the singular 1/r^2 potential in quantum mechanics that dates back to Von-Neumann. The analytic properties of the wavefunction  also reveals an infinite stack of universes sewn together at the horizons of the geodesically complete space. Remaining open questions and problems will be outlined.

The entanglement entropy, while being under the spotlight of theoretical physics for more than ten years now, remains very challenging to compute, even in free quantum field theories, and a number of issues are yet to be explored.

One such issues concerns boundary effects on entanglement entropy, which is important both for theoretical explorations of entanglement and for applications of entanglement entropy to lattice simulations, condensed matter systems, etc.. During this talk, I will show how the presence of spacetime boundaries affects the entanglement entropy, with emphasize on universal (boundary-induced) logarithmic terms, using field theoretic, lattice, and holographic methods.  

Burst phenomena are ubiquitous in astrophysics. Understanding the origin of bright and rapid bursts, like FRBs, is an important goal of contemporary astrophysics.  We apply Dicke's superradiance, a coherent quantum mechanical radiation mechanism, to explain these burst phenomena. We show that bursts lasting from a few milliseconds (FRBs) to a few years (e.g. OH masers) can be produced by very large groups of entangled atoms/molecules. This is in contrast with the common assumption that, in the interstellar medium, the atoms/molecules in a radiating gas act independently from each other. Superradiance, a well-known and intensely studied phenomenon in the physics community, was first discussed by R. H. Dicke in 1954. I will present our superradiance models developed to explain some maser flares and FRBs, and discuss our results for the 6.7-GHz methanol, 1612-MHz OH, and 22-GHz water spectral lines. Our analyses suggest that the aforementioned groups of entangled atoms/molecules, developing over distances of up to a few kilometers for maser flares and 1000 AU for FRBs, can reproduce the observed light curves