Subgroups of simple rings/theory of coagulation

这个项目的前半部分涉及简单环的结构和性质，这些数学对象在纯数学的许多分支中以及在其他科学领域的应用中都是基本重要的。例如，这个项目的目标之一是精确理解基团的投影表示--这是研究原子和晶体结构以及固体物理等科学领域的重要工具。这个项目的后半部分涉及凝聚/碎裂理论。这一理论是近一个世纪前发现的，是对聚合物形成的化学研究的结果。它在科学和工程中发现了许多实际应用，从石油开采、塑料制造、危险材料的分散和血液学，到行星和星系形成的理论。我们的团队一直在研究通过聚集或分裂来增长和下降的粒子系统的行为(例如损坏的反应堆内的放射性尘埃，或地下油表中形成的气泡)。这些系统随着时间的推移而发展的方式，特别是外部因素影响它们演变的方式，是我们研究的重点。实际应用不仅包括预测此类系统的行为的能力，而且还包括控制该行为的能力。

The first half of this project deals with the structure and properties of simple rings, mathematical objects that are of fundamental importance in many branches of pure mathematics, as well as in applications to other areas of science. One of the goals of this project, for example, is to gain a precise understanding of projective representations of groups - important tools in such areas of science as the study of the structure of atoms and crystals, and solid-state physics.  The second half of this project deals with the theory of coagulation/fragmentation. This theory was discovered almost a century ago as a result of the chemical study of the formation of polymers. It has found numerous practical practical applications in science and engineering, ranging from oil extraction, manufacture of plastics, dispersion of hazardous materials, and hematology, to theories of planet and galaxy formation. Our group has been investigating the behavior of systems of particles which grow and decline by clustering or splitting apart (such as radioactive dust inside a damaged reactor, or gas bubbles forming in underground oil tables). The way such systems develop over time, and in particular the way external factors influence their evolution, is the focus of our research. The practical applications include the ability not only to predict the behavior of such systems, but to control that behavior.