Energy-driven systems: Geometry of energy landscapes and applications

SepcevDMS-0908415能量驱动系统的特征有两个：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。研究人员研究了构型空间和相关能量景观的几何，以及它们在梯度流动力学中的应用。瓦瑟斯坦度量中的梯度流提供了一个主要的例子，给出了可以描述为这种流的系统的数量和在过去十年中获得的大量结果。研究人员继续研究沃瑟斯坦度量中的梯度流，但也检查了相对于其他自然出现在物理和生物系统中的较少研究的度量的梯度流。了解能量景观的几何结构使人们能够对复杂的非线性系统的动力学做出结论和预测。在这样的系统中，详细的动力学通常是难以处理的，但许多系统平均的量，如能量，遵循精确的标度定律。这位研究人员研究了几个表现出粗化行为的能量驱动系统的标度律。研究人员研究与材料科学、物理和种群生物学中的应用相关的复杂的非线性系统。能够准确地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、实现纳米粒子的受控自组装以及了解(并潜在地控制)大型动物群体(如昆虫群)的行为是重要的。此外，他致力于几何方法来理解生物学和医学中的显微图像集合，特别是开发图像分类和配准的新工具，这可以导致生物和医学图像的自动处理的各种应用。

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.