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In the first part of the talk, I will review gauge/gravity duality in the case where the gauge theory is just quantum mechanics of a finite number of degrees of freedom. In particular, I will review how the low temperature thermodynamics of this model can be extracted from Monte-Carlo simulations, reproducing the predictions from the dual black hole geometry. In the second part of the talk, I will discuss a particular system known as the Plane-Wave Matrix Model and describe the construction of the black hole geometry that describes its high temperature phase. This construction involves the use of spectral methods to solve the partial differential equations of general relativity.
