(This seminar will be held jointly with Applied Mathematics Seminar.)
Frameworks for the analysis of nonlinear dynamics such as statistical mechanics or geometric methods focus on providing conceptual understanding and were developed before computation and direct use of data became ubiquitous in science and engineering. I will argue that the theory of the above mentioned frameworks is too rich, especially in the context of characterization of dynamics based on finite data sets.
This suggests the need for new frameworks and motivates the introduction of homological dynamics.
The name homological dynamics is meant to invoke an analogy: homological dynamics is to the geometric theory of dynamics as homological algebra is to algebraic topology. I will define a homological dynamical system and discuss the comparison between different homological dynamical systems and highlight obvious open problems. I will also describe how data driven dynamics can be translated into homological dynamics and how information from homological dynamics can be translated to the setting of continuous dynamics.