Charting a New Paradigm for Large Non-Exchangeable Multi-Agent and Many-Particle Systems

该项目将为研究相互作用的“代理人”或“粒子”的大型系统开发一个新的框架。主要目标是通过用平均场的概念取代精确的相互作用，来降低具有大量代理的系统的复杂性。现有的理论，为了证明这种近似，通常假设代理人是相同的或至少是不可区分的。这个假设在考虑物理学中的经典应用时是有效的，例如静电势下的电子动力学。研究人员将引入新的方法来消除不可区分的粒子或代理的假设，同时仍然涉及现实的奇异相互作用：这将允许考虑更复杂的应用，如物理学中的多物种模型和生物神经元网络动力学模型。该项目为研究生提供培训研究机会。从数学上讲，拥有不同的粒子或代理会改变系统的结构。首先，它使它们非对称和不可交换，这在试图定义所谓的可观测量时具有深刻的后果，例如子系统的联合定律。它还打破了一些抵消效应，例如相互作用的反对称性，许多当前方法依赖于处理奇异相互作用。该项目引入了新的方法来绕过这些问题，如新的二元公式和边际的扩展概念。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The project will develop a new framework for the study of large systems of interacting "agents" or "particles". The main goal is to obtain a reduction in complexity for systems with a very large number of agents, by replacing exact interactions with a notion of mean field. Existing theories, to justify this approximation, typically assume that the agents are identical or at least indistinguishable. This assumption can be valid when considering classical applications in Physics, such as the dynamics of electrons under an electrostatic potential. The investigator will introduce new approaches to remove the assumption of indistinguishable particles or agents while still involving realistic singular interactions: This will allow to consider more complex applications such as multi-species models in Physics and models for the dynamics of networks of biological neurons. The project provides training research opportunities for graduate students. Mathematically speaking having non-identical particles or agents changes the structure of the systems. First, it makes them non-symmetric and non-exchangeable, which has deep consequences when trying to define so-called observables, such as the joint law of a sub-system. It also breaks some of the cancellation effects, for example the anti-symmetry of the interaction, that many current methods relied on to handle singular interactions. The project introduces novel methods to bypass those issues such as new duality formulations and an extended notion of marginal.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.