Nash, Nyquist and Next

Standing tall among the major mathematical achievements of 20th century are two theorems whose subsequent impacts far outweighed their original intent. One such theorem is due to John Nash, whose proof of the existence of equilibrium in a non-cooperative game gave rise to the concept of the eponymous Nash Equilibrium, which in many ways revolutionized the field of economics. Another is due to Harry Nyquist, whose Nyquist-Shannon sampling theorem, which states that every time-varying band-limited signal can be perfectly reconstructed from an infinite sequence of samples acquired at the twice rate of its maximum frequency, laid the foundation of the modern information and communication theory. I will describe some recent progress in extending these results - in dynamic game theory, where the rules of the game change over time, and in the theory of compressive sensing, which guarantees perfect reconstruction of signals from far fewer number of samples than required by the Nyquist theorem, if the signals are sparse in some appropriate domain. I will then describe some applications of these extensions for cyber security and wireless security, respectively and finally a potential surprising connection between these two important theorems.