  Nonextensive Statistical Mechanics has developed into an important framework
for modeling the thermodynamics of complex systems and the information of
complex signals. Upon the 80th birthday of the field's founder, Constantino
Tsallis, a review of open problems that can stimulate future research is
provided. Over the thirty-year development of NSM a variety of criticisms have
been published ranging from questions about the justification for generalizing
the entropy function to interpretation of the generalizing parameter q. While
these criticisms have been addressed in the past and the breadth of
applications has demonstrated the utility of the NSM methodologies, this review
provides insights on how the field can continue to improve the understanding
and application of complex system models. The review starts by grounding
q-statistics within scale-shape distributions and then frames a series of open
problems for investigation. The open problems include using the degree of
freedom to quantify the difference between entropy and its generalization;
clarifying the physical interpretation of the parameter q; improving the
definition of the generalized product using multidimensional analysis; defining
a generalized Fourier transform applicable to signal processing applications;
and re-examination of the normalization of nonextensive entropy. The review
concludes with a proposal that the shape parameter is a candidate for defining
the statistical complexity of a system.
