Energy-driven systems: Geometry of energy landscapes and applications

能量驱动系统的特点有两个要素：被耗散的能量和耗散能量的机制。耗散机制使构型空间具有几何结构。研究者研究构型空间的几何形状和相关的能量景观，以及它们在梯度流动动力学中的应用。考虑到可以被描述为这种流动的系统的数量以及在过去十年中获得的丰富结果，Wasserstein度量中的梯度流动提供了一个主要的例子。研究者继续研究Wasserstein度量中的梯度流，但也研究了相对于其他较少研究的、自然出现在物理和生物系统中的度量的梯度流。理解能量景观的几何结构使人们能够对复杂非线性系统的动力学做出结论和预测。在这样的系统中，详细的动力学通常实际上是难以处理的，但许多系统平均量，如能量，遵循精确的标度定律。研究者研究了几个表现出粗化行为的能量驱动系统的标度定律。研究者研究与材料科学、物理学和种群生物学应用相关的复杂非线性系统。能够准确预测这些系统的大规模特征对于设计更好的人造材料、控制涉及相分离的过程、实现可控的纳米粒子自组装以及理解（并可能控制）大规模动物群体（如昆虫群）的行为非常重要。此外，他还研究几何方法来理解生物学和医学中的显微镜图像集合，特别是开发用于图像分类和配准的新工具，这可以导致生物和医学图像自动化处理的各种应用。

SlepcevDMS-0908415 Energy-driven systems are characterized by two elements: the energy being dissipated and the mechanism by which the energy is dissipated. The dissipation mechanism endows the configuration space with a geometric structure. The investigator studies the geometry of configuration spaces and relevant energy landscapes, and their applications to the dynamics of gradient flows. Gradient flows in Wasserstein metric provide a prime example, given the number of the systems that can be described as such flows and the wealth of results obtained over the last decade. The investigator continues studying gradient flows in the Wasserstein metric, but also examines gradient flows with respect to other, less studied, metrics that appear naturally in physical and biological systems. Understanding the geometry of energy landscapes enables one to make conclusions and predictions about dynamics of complex nonlinear systems. In such systems the detailed dynamics are often practically intractable, but many system-averaged quantities, such as energy, follow precise scaling laws. The investigator studies scaling laws in several energy-driven systems that display coarsening behavior. The investigator studies complex nonlinear systems relevant to applications in materials science, physics, and population biology. Being able to accurately predict the large-scale features of these systems is important for designing better man-made materials, controlling processes involving phase separation, achieving controlled self-assembly of nano-particles, and understanding (and potentially controlling) the behavior of large-scale animal groups (such as insect swarms). Additionally, he works on geometrical approaches to understanding collections of microscopy images in biology and medicine, in particular, on developing novel tools for image classification and registration, which can lead to a variety of applications to automated processing of biological and medical images.