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

这个I-Corps项目的更广泛的影响/商业潜力是开发微分几何和优化运输的应用，以解决复杂的优化问题。这个项目的重点是了解在运动训练和招募领域，哪些生物力学参数捕捉了复杂的人类运动活动的关键特征。所提出的技术可能对职业运动以及有抱负的和业余运动员有许多应用。此外，它可能有利于减少运动员受伤的风险，减少运动员从受伤中恢复的时间。拟议中的技术也可用于球员技能的发展，这在各级体育运动中起着核心作用。人才评估和球员招募也可能受到影响，这在专业和大学水平上都需要大量的资源。其他可能受益的领域包括天气预报、最优网络设计、经济学、物理学和光学。这个I-Corps项目是基于数学工具和算法的发展，灵感来自于完全非线性的蒙日-安培方程的深层理论，这是微分几何和最优运输的广泛领域的共同分母。微分几何结合了来自机器学习和最先进的生物力学研究的数据分析工具，以深入分析和开发涉及多个肢体同时使用的某些复杂人类运动活动的新的生物力学理解。拟议的技术侧重于需要高度技能和协调的活动，并且对培训和学习具有挑战性。了解一个特定的方程是否可以用来为优化复杂的人类活动提供数学见解，可以为这个方程和用于研究它的深层理论提供另一个现实世界的应用。微分几何中的迭代过程，如利玛窦迭代和流，为无限维构型空间中的训练模型提供了灵感，这些模型可以模拟复杂的人类活动和优化算法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.