  We prove that the coefficients of certain weight -1/2 harmonic Maass forms
are traces of singular moduli for weak Maass forms. To prove this theorem, we
construct a theta lift from spaces of weight -2 harmonic weak Maass forms to
spaces of weight -1/2 vector-valued harmonic weak Maass forms on Mp_2(Z), a
result which is of independent interest. We then prove a general theorem which
guarantees (with bounded denominator) when such Maass singular moduli are
algebraic. As an example of these results, we derive a formula for the
partition function p(n) as a finite sum of algebraic numbers which lie in the
usual discriminant -24n+1 ring class field. We indicate how these results
extend to general weights. In particular, we illustrate how one can compute
theta lifts for general weights by making use of the Kudla-Millson kernel and
Maass differential operators.
