Variational Questions, Stability, and Dynamics

变分问题、稳定性和动力学

• 批准号: 2055282
• 负责人: Eric Carlen
• 金额: $ 28.83万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2024-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055282&HistoricalAwards=false
• 关键词: Variational; Questions; Stability; Dynamics

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.

寻找对许多物理过程以及控制它们的演化方程的更好理解，部分是寻求新的，更精确的数学不平等现象。诸如“此过程能快的速度？”之类的问题的答案？和“通过此渠道传达信息的速度？”经常打开发现新的数学不平等现象，这是项目的大部分主题。要研究的问题在纯数学中也具有重大关注。但是，由于它们是由于其他领域（尤其是物理学和量子信息理论）出现的问题的动机，因此它们的解决方案将在纯数学之外具有影响和应用。学生将参与该项目，为下一代研究人员的培训做出贡献。数学不平等与物理过程之间的联系均两种方式运行。因此，在试图证明特定的不平等现象时，人们可能会尝试将其与简单且知名的进化方程联系起来。这一研究领域不仅在产生更广泛的科学界感兴趣的结果方面富有成果，而且还吸引了博士学位的兴趣。学生。该项目特别强调以敏锐的形式证明功能不平等。这意味着完全解决以下变异问题：找到某些功能的最小值并确定整个平等案例。某些问题进一步发展，并寻求这种急剧不平等的稳定性结果，即当某个功能几乎在功能不平等中产生平等时，定理的定理，那么该功能在某些平等案例中就接近了。这种稳定结果具有许多重要的含义。他们的证明和申请将由PI及其合作者进行。这项研究不仅会产生重要的新数学，而且还会产生与物理科学甚至工程相关的结果。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，认为值得通过评估值得支持。

项目成果

Monotonicity versions of Epstein's Concavity Theorem and related inequalities

爱泼斯坦凹性定理和相关不等式的单调性版本

• DOI: 10.1016/j.laa.2022.09.001
• 发表时间: 2022
• 期刊: Linear Algebra and its Applications
• 影响因子: 1.1
• 作者: Carlen, Eric A.;Zhang, Haonan
• 通讯作者: Zhang, Haonan

A monotonicity version of a concavity theorem of Lieb

Lieb 凹性定理的单调性版本

• DOI: 10.1007/s00013-022-01774-6
• 发表时间: 2022
• 期刊: Archiv der Mathematik
• 影响因子: 0.6
• 作者: Carlen, Eric A.

A trace inequality of Ando, Hiai, and Okubo and a monotonicity property of the Golden–Thompson inequality

Ando、Hiai 和 Okubo 的微量不等式以及 Golden–Thompson 不等式的单调性

• DOI: 10.1063/5.0091111
• 发表时间: 2022
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Carlen, Eric A.;Lieb, Elliott H.
• 通讯作者: Lieb, Elliott H.

Characterizing Schwarz maps by tracial inequalities

通过痕迹不等式表征施瓦茨地图

• DOI: 10.1007/s11005-023-01636-4
• 发表时间: 2023
• 期刊: Letters in Mathematical Physics
• 影响因子: 1.2
• 作者: Carlen, Eric;Müller-Hermes, Alexander
• 通讯作者: Müller-Hermes, Alexander

A Kac model with exclusion

具有排除的 Kac 模型

• DOI: 10.1214/22-aihp1276
• 发表时间: 2023
• 期刊: Probabilités et Statistiques
• 作者: Carlen, Eric;Wennberg, Bernt
• 通讯作者: Wennberg, Bernt