A mathematical analysis of learning in both biological and artificial neural networks

An exciting area of intellectual activity in this century may well revolve around a synthesis of machine learning, theoretical physics, and neuroscience.  The unification of these fields will likely enable us to exploit the power of complex systems analysis, developed in theoretical physics and applied mathematics, to elucidate the design principles governing neural systems, both biological and artificial, and deploy these principles to develop better algorithms in machine learning.  We will give several vignettes in this direction, including:  (1) determining the best optimization problem to solve in order to learn regression in high dimensions;  (2) finding exact solutions to the dynamics of generalization error in deep linear networks; (3) analyzing how neural networks can learn semantic concepts like infants.