Rectifiability: A Survey

Rectifiable sets, measures, currents and varifolds are foundational concepts in geometric measure theory. The last four decades have seen the emergence of a wealth of connections between rectifiability and other areas of analysis and geometry, including deep links with the calculus of variations and complex and harmonic analysis. This short book provides an easily digestible overview of this wide and active field, including discussions of historical background, the basic theory in Euclidean and non-Euclidean settings, and the appearance of rectifiability in analysis and geometry. The author avoids complicated technical arguments and long proofs, instead giving the reader a flavour of each of the topics in turn while providing full references to the wider literature in an extensive bibliography. It is a perfect introduction to the area for researchers and graduate students, who will find much inspiration for their own research inside.

可矯正集合、測度、流和變異體是幾何測度論中的基礎概念。過去四十年間，矯正性與分析和幾何學的其他領域之間出現了大量的聯繫，包括與變分計算和複雜和調和分析的深層關聯。這本短小的書提供了對這個廣泛且活躍的領域的易於理解的概述，包括對歷史背景、歐幾里得和非歐幾里得設定中的基本理論以及矯正性在分析和幾何中的出現的討論。作者避免了複雜的技術論證和冗長的證明，而是依次給讀者呈現每個主題的風味，同時在廣泛的參考文獻中提供完整的引用。這是研究人員和研究生對這個領域的完美介紹，他們將在其中找到自己研究的許多靈感。