Variational Questions, Stability, and Dynamics

为了更好地理解许多物理过程，以及控制这些过程的演化方程，在某种程度上是对新的、更精确的数学不等式的探索。“这个过程能有多快？”和“通过这个渠道信息传递的速度有多快？”这些问题的答案往往取决于新的数学不等式的发现，这是该项目的中心主题。要研究的问题在纯数学中也具有重要的意义。但由于他们是受到其他领域出现的问题的激励，特别是物理学和量子信息理论，他们的解决方案将在纯数学之外产生影响和应用。学生将参与该项目，为培养下一代研究人员做出贡献。数学不等式和物理过程之间的联系是双向的。因此，在试图证明一个特定的不等式时，人们可能会尝试将其与一个简单且易于理解的进化方程联系起来。这一领域的研究成果丰硕，不仅产生了更广泛的科学界感兴趣的结果，而且还引起了博士生的兴趣。该项目特别强调以尖锐的形式证明功能不平等。这意味着要彻底解决以下变分问题：求某个泛函的最小值，并确定相等情况的全部集合。有些问题更进一步，寻找这种尖锐不等式的稳定性结果，即，当一个函数在一个泛函不等式中几乎产生相等时，那么这个函数在某个度规上接近于相等的情况之一。这样的稳定性结果有许多重要的意义。他们的证明和应用都将由PI和他的合作者来完成。这项研究不仅会产生重要的新数学，而且会产生与物理科学甚至工程相关的结果。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 search for a better understanding of many physical processes, and the evolution equations that govern them, is in part a quest for new and more precise mathematical inequalities. The answers to such questions as "How fast can this process go?" and "How rapidly can information be communicated through this channel?" often turn on the discovery of new mathematical inequalities, which are the central theme of much of the project. The problems to be investigated are of significant interest within pure mathematics as well. But because they are motivated by problems arising in other fields, especially physics and quantum information theory, their solution will have impact and applications outside of pure mathematics. Students will be involved in the project, contributing to training of the next generation of researchers.The connection between mathematical inequalities and physical processes runs both ways. So, in trying to prove a particular inequality, one may try to relate it to a simple and well understood evolution equation. This area of research has been fruitful not only in producing results that are of interest to a wider scientific community, but also in engaging the interest of Ph.D. students. The project places particular emphasis on proving functional inequalities in sharp form. This means to completely solve the following variational problem: find the minimum value of some functional and determine the full set of equality cases. Some of the questions go further and seek stability results for such sharp inequalities, i.e., theorems that assert that when a certain function almost yields equality in a functional inequality, then that function is close in some metric to one of the cases of equality. Such stability results have many important implications. Both their proofs and applications will be worked on by the PI and his collaborators. This research will produce not only significant new mathematics, but results that are relevant to the physical sciences and even engineering as well.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.