I-Corps: Optimization Applications of Differential Geometry and Optimal Transport

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The broader impact/commercial potential of this I-Corps project is the development of applications of differential geometry and optimal transportation toward solving complex optimization problems. The focus of this project is understanding which bio-mechanical parameters capture the key features of complex human motor activities in the area of sports training and recruitment. The proposed technology may have numerous applications for professional sports as well as for aspiring and amateur athletes. Moreover, it may be beneficial for reducing injury risk for athletes and reducing recovery times for athletes returning from injury. The proposed technology also may be used for player skill development, which plays a central role in sports at all levels. Talent evaluation and player recruitment, which also receives significant resources both at the professional and collegiate levels may also be impacted. Other fields that may benefit include weather forecasting, optimal network design, economics, physics, and optics.This I-Corps project is based on the development of mathematical tools and algorithms inspired by the deep theory of the fully nonlinear Monge-Ampere equation that is a common denominator for the vast fields of Differential Geometry and Optimal Transportation. Differential Geometry is combined with data-analytic tools from Machine Learning and state-of-the-art bio-mechanical research in order to analyze in-depth and develop a new bio-mechanical understanding of certain complex human motor activities that involve the simultaneous use of multiple limbs. The proposed technology is focused on activities that require a high degree of skill and coordination and that are challenging to train and learn. Understanding whether a specific equation may be used to give mathematical insights on optimizing complex human activities could provide yet another real-world application of this equation and the deep theories used to study it. Iterative processes from Differential Geometry such as the Ricci iteration and flow provide inspiration for training models in infinite-dimensional configuration spaces that may model complex human activities and optimization algorithms.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.