Title:  Identifying Global Dynamics with Sparse Information
Abstract:  Ultimately understanding systems biology is a question of understanding dynamics, but there are at least two perspectives to this challenge: one, given an observed phenotype we can ask what network of genetic/biochemical interactions can produce such behavior; and two, given a regulatory network what types of phenotypes can be produced as a function of variations in parameters. From a mathematical perspective an overarching conundrum is that answering these questions require us to understand the dynamics of multi-parameter nonlinear multiscale problems for which the nonlinearities cannot be derived from first principles and the parameters are at best heuristic quantities. This motivates the philosophy adopted for this talk: since precision is not possible we should focus on accuracy, which in this context involves developing tools by which we can efficiently identify robust characterizations of dynamics that are valid over large ranges of parameter values. I will present a combinatorial/homological approach to nonlinear dynamics that can be rigorously associated with the dynamics of multi-parameter differential equations. I will argue that because this approach is robust the results are valid even without explicit analytic representations of the differential equations. I will also present examples of how this approach can be applied to both of the above mentioned perspectives.