CAREER: Properties of Solutions to Singular Stochastic Partial Differential Equations from Quantum Field Theory

这个数学研究项目的重点是研究物理学中一些量子场论模型的小尺度行为。该研究旨在准确描述和理解紫外发散现象。例如，在基本粒子标准模型的研究中，物理学家经常使用蒙特-卡罗模拟;在这个项目中进行的研究旨在揭示为什么离散模拟可以准确地预测连续量子场的行为，动力学算法如何快速收敛到所观察的量，以及均匀化近似在多大程度上能够有效地描述多体问题的宏观性质。该项目还包括两个教育部分，重点是研究生概率会议（GSPC）的复兴，该会议完全由主要来自美国的研究生运行，以及随机偏微分方程（SPDE）教育计划，该计划垂直整合了整个学术研究领域。GSPC和SPDE教育计划旨在为概率界的后代提供持续的长期利益。本项目主要研究一类奇异随机偏微分方程解的性质。近年来，在这些方程的解的构造方面取得了迅速进展，该项目旨在研究已构造的解的性质。具体来说，研究者将研究随机杨米尔斯方程的普遍极限，并探索随机偏微分方程，量子场论中的大N问题和平均场极限理论之间的联系。最近开发的解决方案理论将发挥核心作用。PI旨在为做出实际预测的结果提供严格的证明。这些研究有望与物理学产生深远的联系，甚至影响其他自然科学。该奖项反映了NSF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。

This mathematics research project focuses on studying the small-scale behavior of some quantum field theory models in physics. The research aims to produce an accurate description and understanding of the ultraviolet divergence phenomenon. In the study of the standard model of elementary particles, for example, physicists often use Monte-Carlo simulation; the research carried out in this project aims to reveal why discrete simulation can accurately predict the behavior of a continuous quantum field, how quickly dynamical algorithms will converge to the quantity being observed, and to what degree a homogenized approximation can effectively describe the macroscopic nature of a many-body problem. The project also includes two educational components, focusing on rejuvenation of the Graduate Student Probability Conference (GSPC), which is entirely run by graduate students primarily from the US, and a Stochastic Partial Differential Equations (SPDE) education program that is vertically integrated across the full spectrum of academic research. The GSPC and the SPDE education program seek sustained long-term benefit to the future generations of the probability community. The project focuses on studying the properties of the solutions of some singular stochastic partial differential equations. In recent years, there has been rapid progress in the construction of the solutions of these equations, and the project aims to study the properties of the solutions that have been constructed. Specifically, the investigator will study the universal limits of the stochastic Yang-Mills equation and explore the connections between stochastic partial differential equations, large-N problems in quantum field theory, and the theory of mean field limits. Recently developed theory of solutions will play a central role. The PI aims to develop rigorous proofs for results that make practical predictions. These studies are anticipated to result in profound connections with physics and even to influence other natural sciences.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.