Variational Problems and Dynamics

该研究旨在解决不仅与数学一样重要的数学问题，而且还由物理学和生物学中出现的问题所暗示的数学问题。对控制物理和生物过程的方程的分析通常需要对这些过程中涉及的各种量的相对大小进行精确的定量理解，而这是由数学不等式（通常具有几何性质）提供的。寻求更好地理解这些过程的部分原因是寻求新的、更精确的数学不等式。 发现和证明这种数学不等式的一种方法是该项目的核心，是通过考虑辅助动力学过程，将系统的状态演变为易于分析的形式。这一研究领域取得了丰硕的成果，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。学生。这项研究的智力价值在于，它不仅会产生重要的新数学，而且还会产生与物理和生物科学相关的结果。这些在其他领域的应用保证了这项工作的广泛影响，而学生的参与进一步增强了这种影响，有助于培养下一代研究人员。在描述物理和生物系统时出现的许多非线性演化方程中，有玻尔兹曼方程和趋化性凯勒-西格尔方程。 对于这两者，有关解的行为的重要信息来源是先验函数不等式。 例如，玻尔兹曼方程的解趋向于平衡解，并且这种情况发生的速率由与相对熵和进化所强制的熵产生相关的不等式控制。这种函数不等式是通过完全解决变分问题来建立的：找到某个函数的最小值，确定最小化函数的全套，最后证明结果，断言如果函数的值接近最优值，那么它的参数一定接近优化器。变分问题的这种完整解决方案不仅对研究物理系统的演化感兴趣，而且有时可以通过研究与其相关的适当动力学来最好地解决变分问题。非线性动力学和变分问题之间的相互作用是许多最新进展的源泉。该项目侧重于变分问题和非线性演化方程，重点关注研究人员期望特别富有成效的相互作用的那些问题。 第二个焦点是量子系统的算子不等式和迹不等式。 对这些问题的研究是由量子统计力学和量子信息论中的问题推动的，并且量子动力学和要研究的不等式之间也存在密切的相互作用，只是这里函数被算子取代并且出现了非交换性问题。

The research is aimed at solving mathematical problems that are not only significant as mathematics, but that have been suggested by problems arising in physics and biology. The analysis of the equations governing physical and biological 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, often of a geometric nature. The quest for a better understanding of these processes is in part quest for new and more precise mathematical inequalities. One way of discovering and proving such mathematical inequalities, which is central to the project, is through the consideration of auxiliary dynamical processes that evolve the state of a system into a form that is amenable to analysis. 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 physical and biological sciences as well. These applications 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. Among the many nonlinear evolution equations that arise in the description of physical and biological systems are the Boltzmann equation and the Keller-Segel equations for chemotaxis. For both of these, an essential source of information on the behavior of solutions is a priori functional inequalities. For example, solutions of the Boltzmann equation tend towards equilibrium solutions, and the rate at which this happens is governed by an inequality relating relative entropy and the entropy production forced by the evolution. Such functional inequalities are established by completely solving a variational problem: finding the minimum value of some functional, determining the full set of minimizing functions, and finally, proving results that assert that if the value of a functional is close to the optimal value, then its argument must be close to an optimizer. Such complete solutions of variational problems are not only of interest for studying the evolution of physical systems, but also, variational problems can sometimes be best solved by studying an appropriate dynamics associated with them. This interplay between nonlinear dynamics and variational problems has been the source of much recent progress. This project focuses on variational problems and on nonlinear evolution equations, with emphasis on those problems in which the investigator expects a particularly fruitful interplay. A second focus is on operator and trace inequalities for quantum systems. Investigation of these is motivated by problems in quantum statistical mechanics and quantum information theory, and again there is close interplay between quantum dynamics and the inequalities to be investigated, except that here functions are replaced by operators and non-commutativity issues arise.