Statistical mechanics naturally lends itself to computational algorithms that use random sampling to leverage the law of large numbers. Though these methods have proven invaluable, in many cases yielding solutions to otherwise intractable problems, to an extent they also obscure the underlying physics. The two numerical methods studied in this thesis do not fit this description. The first, the conformal bootstrap, imposes symmetry constraints on the four-point correlation functions of a conformal field theory to restrict the spectrum of allowed scaling dimensions of a theory. A variation of the conformal bootstrap is implemented to treat two important non-unitary CFTs, percolation and the self-avoiding walk. The second involves a formalism which provides an integral equation with a kernel based on the two-body S-matrix. The solution of this integral equation represents a single particle's energy in the presence of interactions with the rest of a cold atom gas. The formalism is applied to Berezinskii-Kosterlitz-Thouless and upper branch phase transitions.