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Over the years, deep insights into string theory and supersymmetric gaugetheories have come from studying geometry emerging from matrix models. In thisthesis, I study the N=1* and N=2* theories from which an elliptic curve isknown to emerge, alongside an elliptic function called the generalizedresolvent into which the physics is encoded. This is indicative of the commonorigin of the two theories in N=4 SYM. The N=1* Dijkgraaf-Vafa matrix model isintrinsically holomorphic with parameter space corresponding to the upper-halfplane. The Dijkgraaf-Vafa matrix model 't Hooft coupling S has been previouslyshown to be holomorphic on the upper-half plane and quasi-modular with respectto SL(2,Z). The allowed N=2* coupling is constrained to a Hermitian slicethrough the enlarged moduli space of the holomorphic N=1* model. Afterexplicitly constructing the map from the elliptic curve to the eigenvalueplane, I argue that the N=1* coupling S encodes data reminiscent of N=2*. Acollection of extrema (saddle-points) of S behave curiously like the quantumcritical points of N=2* theory. For the first critical point, the match isexact. This collection of points lie on the line of degeneration which behavesin a sense, like a boundary at infinity. I also show explicitly that theemergent elliptic curve along with the generalized resolvent allow one torecover exact eigenvalue densities. At weak coupling, my method reproduces theinverse square root of N=2* as well as the Wigner semi-circle in N=1*. Atstrong coupling in N=1*, I provide encouraging evidence of the parabolicdensity arising in the neighborhood of the line of degeneration. To myknowledge, the parabolic density has only been observed asymptotically. It isinteresting to see evidence that it may be exactly encoded in the other form ofemergent geometry: the elliptic curve with the generalized resolvent.

Emergent Geometry Through Holomorphic Matrix Models