  Over the years, deep insights into string theory and supersymmetric gauge
theories have come from studying geometry emerging from matrix models. In this
thesis, I study the N=1* and N=2* theories from which an elliptic curve is
known to emerge, alongside an elliptic function called the generalized
resolvent into which the physics is encoded. This is indicative of the common
origin of the two theories in N=4 SYM. The N=1* Dijkgraaf-Vafa matrix model is
intrinsically holomorphic with parameter space corresponding to the upper-half
plane. The Dijkgraaf-Vafa matrix model 't Hooft coupling S has been previously
shown to be holomorphic on the upper-half plane and quasi-modular with respect
to SL(2,Z). The allowed N=2* coupling is constrained to a Hermitian slice
through the enlarged moduli space of the holomorphic N=1* model. After
explicitly constructing the map from the elliptic curve to the eigenvalue
plane, I argue that the N=1* coupling S encodes data reminiscent of N=2*. A
collection of extrema (saddle-points) of S behave curiously like the quantum
critical points of N=2* theory. For the first critical point, the match is
exact. This collection of points lie on the line of degeneration which behaves
in a sense, like a boundary at infinity. I also show explicitly that the
emergent elliptic curve along with the generalized resolvent allow one to
recover exact eigenvalue densities. At weak coupling, my method reproduces the
inverse square root of N=2* as well as the Wigner semi-circle in N=1*. At
strong coupling in N=1*, I provide encouraging evidence of the parabolic
density arising in the neighborhood of the line of degeneration. To my
knowledge, the parabolic density has only been observed asymptotically. It is
interesting to see evidence that it may be exactly encoded in the other form of
emergent geometry: the elliptic curve with the generalized resolvent.
