Counting critical points of Gaussian random functions in high dimensions

Several examples of random functions defined on N dimensional domains are expected to have exponentially many critical values in N as the dimension diverges. These critical values play an important role in the description of certain complex phenomena in statistical physics and algorithm performance in neural nets. In this talk, I will survey two examples where such predictions can be made rigorous, describe its connections to spin glasses and random matrix theory and share a few open questions. 

Based on joint works with Qiang Zeng (University of Macau).