Cyclic and simplicial cohomology of Banach algebras

当我们看一个日常用品，比如一个杯子时，我们可以数出它上面的洞。对于某些数学工作来说，人们所考虑的物体的精确形状并不太重要：重要的是它有多少个孔。这些孔在性质上各不相同：在杯子中，把手上的孔与你倒咖啡的孔不同。你可以想象一下，如果杯子是由一些可拉伸的材料制成的，那么它会被压扁，但是没有粘合或切割，手柄中的整体将始终保持可见。 数学工具已经发展到处理这些概念：它们是多种多样的，不仅适用于类似杯子的物体，而且适用于空间。在我们的研究中，我们着眼于函数空间和这些空间的特征。对于数学家来说，这是代数和分析的聚会场所。这是一项基础研究，内容丰富，引人入胜，形式多样。

When looking at an everyday object, like a cup, one can count the holes in the object. For some mathematical work, the precise shape of the object one is considering does not matter too much: what matters is how many holes it has. These holes vary in nature: in a cup, the hole in the handle is different from the hole you pour coffee in. You can imagine flattening a cup if it was made of some stretchable material, but without gluing or cutting, the whole in the handle will always remain visible. Mathematical tools have been developed to deal with such concepts: they are varied and apply not only to objects similar to a cup but also to spaces. In our research, we look at spaces of functions and at characteristics of these spaces. For a mathematician, this is a meeting place for algebra and analysis. It is fundamental research, rich, intriguing and varied.