The fermion sign problem is studied in the path integral formalism. The
standard picture of Fermi liquids is first critically analyzed, pointing out
some of its rather peculiar properties. The insightful work of Ceperley in
constructing fermionic path integrals in terms of constrained world-lines is
then reviewed. In this representation, the minus signs associated with
Fermi-Dirac statistics are self consistently translated into a geometrical
constraint structure (the {\em nodal hypersurface}) acting on an effective
bosonic dynamics. As an illustrative example we use this formalism to study
1+1-dimensional systems, where statistics are irrelevant, and hence the sign
problem can be circumvented. In this low-dimensional example, the structure of
the nodal constraints leads to a lucid picture of the entropic interaction
essential to one-dimensional physics. Working with the path integral in
momentum space, we then show that the Fermi gas can be understood by analogy to
a Mott insulator in a harmonic trap. Going back to real space, we discuss the
topological properties of the nodal cells, and suggest a new holographic
conjecture relating Fermi liquids in higher dimensions to soft-core bosons in
one dimension. We also discuss some possible connections between mixed
Bose/Fermi systems and supersymmetry.Comment: 28 pages, 5 figure