Abstract: Spectral theory on noncompact locally symmetric spaces has profound connections to number theory, and has thus for many years been an active topic of research. Additionally, there are motivations from physics and topology for understanding L2 Hodge and signature theory over noncompact manifolds, and locally symmetric spaces are an interesting test case, because in this case these theorems are understood. Pseudodifferential operators are an important tool for analysis of partial differential operators on compact manifolds, thus it is interesting to examine what the appropriate operator calculus should be for locally symmetric spaces and manifolds modelled them. In the Q-rank 1 case, locally symmetric spaces are examples of $\phi$-manifolds and their natural geometric operators are elements of the $\phi$-pseudodifferential operator calculus studied first by Melrose and Mazzeo, and later in more generality by Grieser and Hunsicker. In the $\phi$-calculus a pair of ``symbols" describe the obstruction to the construction of a parametrix in the calculus. Operators for which both symbols are invertible are called "fully elliptic". A frustrating aspect of this calculus, however, is that most geometric operators over $\phi$-manifolds are not fully elliptic. This talk will describe how to enlarge the calculus to include parametrices for such operators, and the resulting Fredholm, mapping and regularity results for them.