Cohomology, Finiteness Conditions and Classifying Spaces for Families

Leary教授将继续单独和合作研究几何群论、几何拓扑学和L2上同调，特别是他将构造适当行动的分类空间具有惊人有限性质的群。他还将发展和研究一种保持同调的函子，但用允许非正曲线度量的空间来取代任意拓扑空间。这项工作将在拓扑学、代数K-理论和群论中得到应用。第三，他将通过计算包括超平面排列的补在内的一大类拓扑空间的不变量，对L2上同调的研究做出贡献。群是数学家对对称性概念的抽象：群衡量对称性的方式与数字衡量数量的方式相同。一些最有趣的群产生于几何学，例如平面平铺的对称性的集合。Leary教授的目标是构建具有许多来自瓷砖的小组的属性的小组，但这不能因任何空间的多次“瓷砖”而产生。在一个不同的方向上，他希望证明几何学的某些方面可以在群论中建模(在某种精确的意义上)。这应该会在几何学和群论中产生新的见解。

Professor Leary will continue research on geometric group theory,geometric topology and L2-cohomology, both alone and in collaboration.In particular, he will construct groups for which the classifyingspace for proper actions has surprising finiteness properties. Hewill also develop and study a functor that preserves homology butreplaces an arbitrary topological space by a space admitting anon-positively curved metric. This work should have applications totopology, algebraic K-theory and group theory. Thirdly, he willcontribute to the study of L2-cohomology, by computing this invariantfor a large class of topological spaces, including the complements ofhyperplane arrangements.A group is the mathematician's abstraction of the notionof symmetry: groups measure symmetry in the same way thatnumbers measure quantity. Some of the most interestinggroups arise from geometry, such as the collection ofsymmetries of a tiling of the plane. Professor Learyaims to construct groups that share many of the propertiesof groups coming from tilings, but that cannot arise fromany `tiling' of any space. In a different direction, hehopes to show that some aspects of geometry can be modeled(in a certain precise sense) within the theory of groups.This should lead to new insights within both geometry andgroup theory.