Cohomology, curvature, classifying spaces and symmetry

Leary教授将继续在几何群论和几何拓扑学方面的研究，无论是单独的还是合作的。特别地，他将构造具有强不动点性质的无限群。这有望应用于Kropholler群体层次理论的研究。他还将研究他开发的一个函子（从简单配合物到局部CAT(0)立方配合物），该函子保持同源性。这项工作将在拓扑学、k理论和群论中得到应用。他还会对所有可以通过以某种方式粘合多边形来构建的空间进行分类，这些空间具有“最大可能”的对称群。群是数学家对对称概念的抽象：群以数字衡量量的方式衡量对称。群论和几何之间有着令人着迷的联系。Leary教授的目标是构建具有惊人性质组合的群，这将在几何和群论中产生新的结果和见解。他还将对一类几何物体进行分类，这些物体概括了柏拉图立体，这是五种对称的形状，已经被研究了至少四千年。

Professor Leary will continue research on geometric group theory, and geometric topology, both alone and in collaboration. In particular, he will construct infinite groups having strong fixed point properties. This is expected to have application to the study of Kropholler's hierarchy of groups. He will also study a functor that he has developed (from simplicial complexes to locally CAT(0) cubicalcomplexes) that preserves homology. This work should have applications to topology, K-theory and group theory. He will also classify all spaces that can be built by gluing polygons in a certain way that have the `largest possible' groups of symmetries.A group is the mathematician's abstraction of the notion of symmetry:groups measure symmetry in the same way that numbers measure quantity.There are fascinating connections between group theory and geometry.Professor Leary aims to construct groups with surprising combinations of properties, which should lead to new results and insights within both geometry and group theory. He will also classify a family of geometrical objects which generalize the platonic solids, five symmetrical shapes that have been studied for at least four thousand years.