樓理論及其在幾何和拓撲中的應用 內容簡介

本書的內容是關于樓（building）理論及其在幾何和拓撲中的應用。樓作為一種組合和幾何結構由Jacques Tits引入，作為理解任意域上保距還原線性代數群結構的一種方法，Tits因此項工作獲得2008年Abel獎。樓理論是研究代數群及其表示的必要工具，在幾個相當不同的領域中具有重要應用。本書的**部分是作者專為國內學生學習樓理論準備的導讀資料，其中特別注重利用例子說明問題，可讀性很強；第二部分則綜述了樓理論在幾何與拓撲方面的應用，不僅總結了近些年樓理論研究的成就，還提出了未來的研究方向。本書是一本觀點較高、極具學術價值的數學學習資料，可供我國高等院校代數及相關專業作為教學參考書使用。 Symmetry is an essential concept in mathematics, science and daily life, and an effective mathematical tool to describe symmetry is the notion of groups. For example, the symmetries of the regular solids (or Platonic solids) are described by the finite subgroups of the rotation group SO(3). Therefore, finding the symmetry group of a geometric object or space is a classical and important problem. On the other hand, given a group, how to find a natural geometric space which realizes the group as its symmetries is also interesting and fruitful. One of the most useful or beautiful class of groups consists of algebraic groups, and their corresponding geometric spaces are given by Tits buildings. Originally introduced by Tits to give a geometric description of exceptional simple algebraic groups, buildings have turned out to be extremely useful in a broad range of subjects in contemporary mathematics, including algebra, geometry, topology, number theory, and analysis etc. Since the theory of algebraic groups is complicated, the theory of buildings can be technical and demanding by itself. This book gives an accessible approach by using elementary and concrete examples and by emphasizing many applications in many seemingly unrelated subjects. The reader will learn from this book what buildings are, why they are useful, and how they can be used.本書的內容是關于樓（building）理論及其在幾何和拓撲中的應用。樓作為一種組合和幾何結構由Jacques Tits引入，作為理解任意域上保距還原線性代數群結構的一種方法，Tits因此項工作獲得2008年Abel獎。樓理論是研究代數群及其表示的必要工具，在幾個相當不同的領域中具有重要應用。本書的**部分是作者專為國內學生學習樓理論準備的導讀資料，其中特別注重利用例子說明問題，可讀性很強；第二部分則綜述了樓理論在幾何與拓撲方面的應用，不僅總結了近些年樓理論研究的成就，還提出了未來的研究方向。本書是一本觀點較高、極具學術價值的數學學習資料，可供我國高等院校代數及相關專業作為教學參考書使用。 Symmetry is an essential concept in mathematics, science and daily life, and an effective mathematical tool to describe symmetry is the notion of groups. For example, the symmetries of the regular solids (or Platonic solids) are described by the finite subgroups of the rotation group SO(3). Therefore, finding the symmetry group of a geometric object or space is a classical and important problem. On the other hand, given a group, how to find a natural geometric space which realizes the group as its symmetries is also interesting and fruitful. One of the most useful or beautiful class of groups consists of algebraic groups, and their corresponding geometric spaces are given by Tits buildings. Originally introduced by Tits to give a geometric description of exceptional simple algebraic groups, buildings have turned out to be extremely useful in a broad range of subjects in contemporary mathematics, including algebra, geometry, topology, number theory, and analysis etc. Since the theory of algebraic groups is complicated, the theory of buildings can be technical and demanding by itself. This book gives an accessible approach by using elementary and concrete examples and by emphasizing many applications in many seemingly unrelated subjects. The reader will learn from this book what buildings are, why they are useful, and how they can be used.