Graded Symmetry in Algebra and Analysis

这个项目将通过将数学中的分级对称建模为群胚和逆半群来研究它们。Groupoid长期以来一直是数学研究的中心，但近年来作为代数、分析和动力学的焦点而特别突出。大多数群胚可以自然地进行分级。该项目引入了群胚的分次组合不变量(例如分次同调)，并将它们与它们的Steinberg和C*-代数对应的(例如分次K-理论)联系起来。其结果是给出了连接代数和分析的广受欢迎的统一不变量，并用尽了这类更丰富的不变量将为其提供完整分类的群胚。

This project will study graded symmetries in mathematics by modelling them as groupoids and inverse semigroups. Groupoids have been at the centre of mathematical interest for a long time, but have gained special prominence in recent years as a focal point for algebra, analysis and dynamics. The majority of groupoids can be naturally graded. The project introduces graded combinatorial invariants for groupoids (such as graded homology) and relates them to their Steinberg and C*-algebra counterparts (such as graded K-theory). The outcome is to give sought-after unified invariants bridging algebra and analysis, and to exhaust the class of groupoids for which these much richer invariants will furnish a complete classification.