Reduced density matrices (RDM) of states of multi-partite systems play a fundamental role in both quantum chemistry and quantum information theory. The quantum marginal problem asks when a set of RDM's is consistent with the existence of a multi-partite state with certain properties. The special case in which the N-particle state is required to have permutational symmetry was extensively studied by quantum chemists in the 1960's and is known as N-representability.

In the last decade, new methods have been developed which use the two-electron RDM to study N-electron systems while circumventing the N-representability problem. More recently, two striking results were obtained on this long-standing question, the solution of the pure state problem for the one-particle RDM and a proof of the existence of computationally intractable model Hamiltonians (even with a quantum computer) for the two-particle RDM.

Quantum information theory has led to developments in different directions. There are some surprising connections with representations of the symmetric group and with spin coherent states. A different type of approximation for symmetric density matrices is treated by the so-called quantum di Finetti theorems. Recent developments involving density matrix renormalization methods for treating strongly correlated systems in spin chains will be treated, with emphasis on the practical implications.

Because Schubert calculus plays a key role some recent developments, Allen Knutson will give a "crash course" on this topic during the first two days. By bringing together quantum chemists, mathematicians and quantum information scientists, we hope to identify the new mathematical challenges in this area and the practical uses of recent theoretical developments.