Isoperimetric inequalities and the large-scale geometry of groups

【摘要】本课题主要研究离散群和度量空间上的等周不等式问题。Sobolev和Poincare不等式是此类不等式的经典例子，非精确性或Kazhdan性质(T)等性质也可以视为广义等周条件。该项目的主要目标是扩大对C*-代数中群通过自同构作用的系数等周不等式及其几何含义的理解。特别是，寻找具有这种不平等的群体的明确的、自然的例子的问题将得到解决。这将有助于我们更好地理解诺维科夫猜想和鲍姆-康恩斯猜想背后的几何原理，在这些猜想中，精确性和a- t可操作性发挥了主要作用。等周问题是数学中最基本的问题之一，可以追溯到古代。其最简单的形式就是在一个固定长度的封闭曲线所围成的平面上求出最大的面积。这一经典案例已被广泛研究，并在分析和几何中有许多应用。因此，我们很自然地要求将“面积”和“周长”用更灵活的概念取代，从而实现从经典案例到更苛刻环境的想法。这些问题位于数学的几个主要分支的交叉点，这个项目的主要目标之一是促进基础代数，解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).