Variational Problems and Dynamics

这项研究的目的是解决数学问题，这些问题不仅是重要的数学问题，而且是由物理学和生物学中出现的问题提出的。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有精确的定量理解，这是由数学不等式提供的，通常具有几何性质。对这些过程的更好理解的探索在某种程度上是对新的、更精确的数学不等式的探索。发现和证明这种数学不等式的一种方法，是该项目的核心，是通过考虑辅助的动态过程，将系统的状态演变成一种易于分析的形式。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。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工作的广泛影响，这是进一步加强学生的参与，有助于培养下一代的研究人员。在描述物理和生物系统时出现的许多非线性进化方程中，有玻尔兹曼方程和趋化性的凯勒-西格尔方程。对于这两个问题，关于解的行为的一个重要信息来源是先验的函数不等式。例如，玻尔兹曼方程的解趋向于平衡解，而这种情况发生的速度是由一个与相对熵和进化所产生的熵有关的不等式决定的。这种泛函不等式是通过完全解决一个变分问题来建立的：找到某个泛函的最小值，确定最小化函数的完整集合，最后证明如果一个泛函的值接近最优值，那么它的参数一定接近优化器的结果。这种变分问题的完整解不仅对研究物理系统的演化有兴趣，而且，变分问题有时可以通过研究与之相关的适当动力学来最好地解决。非线性动力学和变分问题之间的这种相互作用是最近许多进展的来源。该项目侧重于变分问题和非线性进化方程，重点是那些研究者期望特别富有成效的相互作用的问题。第二个重点是量子系统的算子和迹不等式。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激发的，并且量子动力学和待研究的不等式之间再次存在密切的相互作用，除了这里的函数被算子取代和非交换性问题出现。

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.