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The near-horizon noncommutative geometry of black holes, given byAdS^2_{\theta} x S^2_N, is discussed and the phase structure of thecorresponding Yang-Mills matrix models is presented. The dominant phasetransition as the system cools down, i.e. as the gauge coupling constant isdecreased is an emergent geometry transition between a geometric noncommutativeAdS^2_{\theta} x S^2_N phase (discrete spectrum) and a Yang-Mills matrix phase(continuous spectrum) with no background geometrical structure. We also find apossibility for topology change transitions in which space or time directionsgrow or decay as the temperature is varied. Indeed, the noncommutativenear-horizon geometry AdS^2_{\theta} x S^2_N can evaporate only partially to afuzzy sphere S^2_N (emergence of time) or to a noncommutative anti-de Sitterspacetime AdS^2_{\theta} (topology change).

The AdS^2_{\theta}/CFT_1 Correspondence and Noncommutative Geometry III: Phase Structure of the Noncommutative AdS^2_{\theta} x S^2_N