Here is my suggestion for an important research question :

in our opinion, there is a strong need for a " new" numerical many-body  
technique
capabel of going to essentially zero temperature ,while at the same  
time treating perturbation-free the kinetic energy- and  
interaction-energy parts of similar magnitude.

QMC is a finite-(high) temperature technique,additionally limited at  
lower temperatures
,because of the infamous minus-sign problem ,to small system sizes.ED  
is clearly limited by the latter.the promising DMFT( dyn.  
mean-field-th.) uses often again the QMC method.

in my group, a very good postdoc (m.potthoff) has recently proposed a  
new idea, which is called the "selfenergy -functional approach  
(SFA).this idea has in the mean-time been taken up not only by our  
group but also by some established groups elsewhere,such as the group  
of tremblay  and his canadian coworkers.both groups studied with this  
formalism the competition of d-sc and AF in the high-Tc cuprates.by  
first comparing the single-particle spectra of hubbard models for both  
el. and hole-doping with exper., we verify that this approach correctly  
treats the low- energy excitations in a strongly correlated system.the  
cluster calculations reproduce the overal ground-state phase diagram  
(for the cluster solution,which follows from the SFA,is using as the  
"cluster-solver"  exact diagonalization) of the high-Tc cuprates.

one crucial question,which is left open,and which in my opinion is  
decesive, is the correct description of two-particle correlation  
functions.they really contain most important information on collective  
excitations (magnons,phonons,etc.),but also excitons,with the latter  
showing up as a shift between optical data and single-particle  
photoemission-type excitations.i  am particularly interested in this  
two-part. description in the context of quantum criticality and would  
be very happy about interactions with other participants.