Geometry of groups and group actions

PI的长期目标是理解被视为几何对象的群体，建模这些群体的空间的几何和拓扑，以及群体在这些空间上的行为的拓扑方面。具体目标包括分析和确定群的高维填充不变量，建立这些不变量的基本结果，并通过组合莫尔斯理论发展关系间隙问题的方法。PI还建议研究促销图及其性质，以及开放问题的应用。群论是对称理论，在数学中起着基础性的作用。对称模式（或“群”）在几何中非常自然地出现，但在其他更抽象的数学领域也是如此。在几何群论中，人们通过构建几何空间来研究抽象群，为手头的群量身定制，以便“看到”对称性并更好地理解它。该项目涉及对由此构建的几何空间的详细研究。特别是，它涉及几何空间的形状和相关性质的决定基于他们所拥有的对称性的性质。

The long-term objective of the PI is to understand groups considered as geometric objects, the geometry and topology of spaces modeling these groups, and topological aspects of the group actions on such spaces. Specific objectives include the analysis and determination of higher dimensional filling invariants of groups, the establishment of foundational results for these invariants, and development of an approach to the relation gap problem via combinatorial Morse theory. The PI also proposes to study promotion maps and their properties, as well as applications to open problems.Group theory is the theory of symmetry, which plays a fundamental role in mathematics. Patterns of symmetries (or "groups") arise very naturally in geometry, but also in other, more abstract, fields of mathematics. In geometric group theory one studies abstract groups by constructing geometric spaces, tailor-made for the groups at hand, in order to "see" the symmetry and better understand it. This project involves the detailed study of the geometric spaces thus constructed. In particular, it concerns the determination of shape and related properties of geometric spaces based on the nature of the symmetries they possess.