Cohomology, Finiteness Conditions and Classifying Spaces for Families

Leary教授将继续研究几何群论，几何拓扑和L2-上同调，无论是单独还是合作。特别是，他将构造群，其适当作用的分类空间具有令人惊讶的有限性属性。 他还将开发和研究一个函子，保持同调，但取代一个任意的拓扑空间的空间承认一个非积极弯曲度量。 这项工作应该有应用totopology，代数K-理论和群论。 第三，他将有助于研究L2-上同调，通过计算这一不变量的一大类拓扑空间，包括补的超平面promisements.A组是数学家的抽象概念的对称性：群测量对称性的方式一样，号码衡量数量。 一些最有趣的群体来自几何学，如平面平铺的对称性集合。 里尔教授的目标是构建群体，这些群体具有来自平铺的群体的许多属性，但不能从任何空间的许多“平铺”中产生。 在另一个方向上，他希望表明几何的某些方面可以在群论中建模（在某种精确的意义上），这将在几何和群论中带来新的见解。

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.