Variational Questions, Stability, and Dynamics

对许多物理过程以及控制它们的演化方程的更好理解的探索，部分是对新的、更精确的数学不等式的探索。对诸如“这一进程能有多快？以及“通过这种渠道，信息的传递速度有多快？“经常打开新的数学不等式的发现，这是该项目的大部分中心主题。要调查的问题是显着的兴趣在纯数学以及。但是，由于他们的动机是其他领域出现的问题，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，人们可以尝试将其与一个简单且易于理解的演化方程联系起来。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，而且还吸引了博士的兴趣。学生该项目特别强调以尖锐的形式证明函数不等式。这意味着完全解决以下变分问题：找到某些泛函的最小值并确定等式的全部情况。有些问题走得更远，寻求这种急剧不平等的稳定性结果，即，这些定理断言，当某个函数几乎在函数不等式中产生相等时，则该函数在某些度量中接近于相等的情况之一。这种稳定性的结果有许多重要的意义。他们的证明和应用都将由PI和他的合作者进行。这项研究不仅将产生重要的新数学，而且还将产生与物理科学甚至工程学相关的成果。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。

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.