Variational Problems, Stability and Dynamics

该研究项目旨在解决不仅是数学重要的数学问题，而且是物理学，量子信息理论和生物学中出现的问题所提出的数学问题。分析控制物理过程的方程常常需要对这些过程中所涉及的各种量的相对大小有精确的定量理解，而这是由数学不等式提供的。一个公众熟悉的例子可以解释为在物理过程中“熵永远不会减少”;所讨论的不等式说，稍后时间的熵至少与较早时间的熵一样大。对于更具体的物理系统，人们可以说得更多，而寻求更好地理解物理过程以及支配它们的演化方程，在某种程度上就是寻求支配这些过程的新的、更精确的数学不等式。数学不等式和物理过程之间的联系是双向的，因此，在试图证明这样一个不等式时，人们可能会试图将它与一个简单而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。学生这项研究的智力价值在于，它不仅将产生重要的新数学，而且还将产生与物理科学和工程相关的结果。这种在其他领域的适用性保证了工作的广泛影响，这是通过学生的参与进一步增强，有助于培养下一代研究人员。该奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的知识价值和更广泛的影响审查标准。

This research project is aimed at solving mathematical problems that are not only significant as mathematics, but that have been suggested by problems arising in physics, quantum information theory and biology. The analysis of the equations governing physical processes often requires a precise quantitative understanding of the relative sizes of various quantities involved in these processes, and this is provided by mathematical inequalities. An example familiar to the general public can be paraphrased as saying that ``entropy never decreases'' in physical processes; the inequality in question says that the entropy at a later time is at least as large as the entropy at an earlier time. For more specific physical systems one can say much more, and the search for a better understanding of physical processes, and the evolution equations that govern them, is in part a quest for new and more precise mathematical inequalities governing these processes. The connection between mathematical inequalities and physical processes runs both ways, so that in trying to prove such an 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 intellectual merit of the research is that it will produce not only significant new mathematics, but results that are relevant to the physical sciences and engineering as well. This applicability in other fields guarantee a broad impact of the work, which is further enhanced by the involvement of students, contributing to training of the next generation of researchers.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.