The output files of a digital camera are typically only 10% the size of the raw data.
Can one design a physical process that only records the relevant information to
begin with? The recently developed field of "compressed sensing" achieves this
trick for certain signals. Building on tools from mathematical physics, we have
introduced new methods to the theory which have since found applications in
areas as diverse as face recognition, despeckling of movie frames, and large
deviation bounds for non commutative processes.
In a second, unrelated part, I will talk about the one-body quantum marginal
problem. Here, the task is to decide which set of local reduced density matrices
are compatible with a globally pure state. Known to be non-trivial since the 70s,
only recent developments in symplectic geometry and asymptotic representation
geometry have made the problem tractable. I will report on our work that connects
these methods to the study of multi-partite entanglement and fermionic many-body
systems.
Lastly, I will mention some results related to the classification of time evolutions in
discrete quantum lattice systems (otherwise known as quantum cellular
automata).