Topics that I am currently interested in:

-The computational complexity of quantum many body systems: when can they be simulated classically or quantumly?

-The relation of that complexity with physical features and phenomena.

-The structure of correlations between particular in dynamical and equilibrium settings: area laws, tensor network approximations,..

-The structure of quantum thermal states: correlations, area laws and tensor network approximations.

-Non-equilibrium dynamics of quantum many-body systems: ergodicty, ETH, ergodicity breaking (scars)…

-Quantum information theory, quantum Shannon theory, resource theories, and their relation with quantum many-body physics.

Here is a list of hard problems that I would like to see solved, in case you want to give it a go:

-A rigorous proof of the Eigenstate Thermalization Hypothesis, or some non-trivial weaker version of it.

-A meaningful definition of quantum chaos that none would disagree with. (I guess some might say this already exists?)

-A Rényi version of the quantum conditional mutual information, and of quantum Markov states. Ideally related to the cost single-shot quantum state redistribution or something of the like.

-Proving that an area law for quantity X implies an efficient MPO/MPDO approximation to a mixed state. It would be great if the solution to this was related to the point above.

-An optimal MPO approximation for large classes of mixed states: thermal, fixed points of Lindbladians, NESS, thermalized states at late times,…

-A direct, fundamental and important physics application of the resource theoretic frameworks of thermodynamics and asymmetry, in a way similar to how tensor networks are an application of entanglement theory.