Isoperimetric inequalities and the large-scale geometry of groups

AbstractNowak提出的项目集中在离散群和度量空间上的等周不等式的一些问题。Sobolev不等式和Poincare不等式是这类不等式的经典例子，非精确性或Kazhdan性质（T）也可以看作是广义等周条件。该项目的主要目标是扩大理解的等周不等式的系数在C*-代数上的群行为的自同构及其几何含义。特别是，找到明确的问题，自然的例子进行这种不平等的群体将得到解决。这将导致更好地理解几何背后的最近进展诺维科夫和鲍姆-康纳斯的几何，其中精确性和a-T-menability发挥了重要作用。等周问题自古以来就是数学中最基本的问题之一。在其最简单的形式，它相当于找到平面上的最大面积所包围的封闭曲线的固定长度。这个经典的例子已经被广泛研究，并在分析和几何中有许多应用。因此，自然会要求用更灵活的概念来取代“面积”和“周长”，以便将经典案例中的想法应用到更苛刻的环境中。这些问题位于数学的几个主要分支的交叉点，该项目的主要目标之一是促进基础代数，分析和几何结构之间的相互作用。该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。

AbstractNowakThe proposed project focuses on a number of problems related to isoperimetric inequalities on discrete groups and metric spaces. Sobolev and Poincare inequalities are classical examples of such inequalities, properties like non-exactness or Kazhdan's property (T) can also be viewed as generalized isoperimetric conditions. The main objective of the project is to broaden the understanding of isoperimetric inequalities with coefficients in a C*-algebra on which the group acts by automorphisms and of their geometric implications. In particular, the problem of finding explicit, natural examples of groups carrying such inequalities will be addressed. This will lead to a better understanding of the geometry which lies behind recent progress towards the Novikov and the Baum-Connes conjectures in which exactness and a-T-menability played major roles. The isoperimetric problem is one of the most fundamental problems in mathematics dating back to antiquity. In its simplest form it amounts to finding the largest area on the plane enclosed by a closed curve of fixed length. This classical case has been studied extensively and has numerous applications in analysis and geometry. It is therefore natural to ask for generalizations in which "area" and "perimeter" are replaced by more flexible notions, allowing to implement ideas from the classical case to a more demanding setting. These problems lie at the intersection of several major branches of mathematics and one of the main goals of this project is to foster interactions between the underlying algebraic, analytic and geometric structures.This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).