K-theory of Fields and Azumaya Algebras

据说数学是关于研究模式和结构的。这就把数学分成了几个学科。例如，分支拓扑学研究因连续变形(即拉伸和挤压)而保持不变的形状的属性。形状上有孔是这些特性之一。考虑一下一个甜甜圈和一个茶杯，它们都是由橡胶制成的。我们可以通过拉伸和挤压来将其中一个转换成另一个，而不需要撕裂或粘合先前分离的部分。因此，这两个对象(或拓扑空间)在拓扑上是相同的。两个形状在拓扑上是否相同的自然问题是一个非常困难的问题。分支代数在历史上是研究一个或多个代数方程的解的学科，涉及一个或多个变量的多项式。当所有多项式的次数都是一次(线性方程组)时，就产生了线性代数。在研究一个多项式的根的单一方程的情况下，引出了场论和伽罗瓦理论。几个高次方程的一般情况导致了代数几何，之所以被称为代数几何，是因为这类系统的解集通常是用几何方法来研究的。现代代数学家不仅在方程式理论中，而且在一般的数学中，越来越抽象和公理化了所遇到的论点的结构和模式。这些结构的例子包括群和环。群是一个集合和一种组合元素以获得新元素的方法(如加法或乘法，或一般定义为组合两个元素A和B的A*B)，它满足某些性质。一个群称为阿贝尔群，如果对所有元素A*B=B*A。理解群体的行为并对其进行分类是数学的一个分支，称为群论。现代数学的成就之一是认识到不同学科之间存在着结构相似之处。20世纪40年代的S和50年代的S见证了表达它们的一种正式语言--范畴理论的产生。类别是通过将类似的数学对象(如拓扑空间或群)组合在一起而自然产生的。研究“同类”对象的一些重要方法是给它们附加保持结构的不变量。例如，存在一种将阿贝尔群连接到拓扑空间的方法。在某种意义上，这个阿贝尔群探测到空间中存在空洞。一种技术方法是，从拓扑空间范畴到交换群范畴都有一个‘函子’。这里这个函数器是一种过滤器，在给定输入空格的情况下，它返回一个阿贝尔群。这个返回的群就是空间的空穴结构的表示。K-理论是通过使用范畴理论来创建的，为上述考虑提供了系统的基础。这是一门新的数学学科，涵盖了许多其他主要学科的核心概念和问题。K理论的成功在于它在其他学科的重要问题上的许多应用，以及它在获得立足点后适应数学各个领域正在进行的研究的能力。在这个过程中，K-理论吸收了大量其他学科，包括数论、代数几何和微分拓扑学。代数K-理论提供了某些函子，称为K0，K1，K2，…，将阿贝尔群附加到一个代数对象上。这些团体彼此关系很好(排成长长的精确序列，……)并在许多情况下帮助我们更多地了解原始对象。这个项目的目的是研究这些附着在某些代数对象上的K-群，即域和除法代数(参见项目技术描述的支持案例)。

It is said that mathematics is about studying patterns and structures. This divides mathematics into several disciplines. For example the branch Topology is the study of properties of a shape which are left unchanged by continuous deformations (i.e. stretching and squeezing). Having holes in a shape is one of these properties. Consider a doughnut and a teacup, both made of rubber. We can take one of these and transform it into the other by stretching and squeezing, without tearing or sticking together bits which were previously separated. Thus these two objects (or topological spaces) are topologically the same. The natural question of whether two shapes are topologically the same is a very difficult question.The branch Algebra, historically, is the study of solutions of one or several algebraic equations, involving polynomials in one or several variables. The case where all the polynomials have degree one (system of linear equations) leads to Linear Algebra. The case of a single equation, in which one studies the roots of one polynomial, leads to Field Theory and to Galois Theory. The general case of several equations of higher degree leads to Algebraic Geometry, so named because the sets of solutions of such systems are often studied by geometric methods.Modern algebraists have increasingly abstracted and axiomatized the structures and patterns of arguments encountered not only in the theory of equations, but in mathematics generally. Examples of these structures include groups and rings . A group is a set together with a method of combining the elements to get new ones (like addition or multiplication or in general A*B defined for combining two elements A and B) which satisfies certain properties. A group is called Abelian if A*B=B*A for all the elements. Understanding the behaviour of groups and classifying them is a branch of mathematics called Group Theory. One of the achievements of modern mathematics is the realisation of existance of structural similarities amongst diverse disciplines. The 1940's and 50's witnessed the creation of a formal language for expressing them, namely category theory. Categories occur naturally by grouping together mathematical objects 'of the same kind' such as topological spaces or groups. Some important ways of studying objects 'of the same kind' are to attach structure-preserveing invariants to them. For example there exists a way of linking an Abelian group to a topological space. In some sense this Abelian group detects the presence of holes in the space. A technical way to say this is, there is a 'functor' from the category of topological spaces to the category of abelian groups. Here this functor is a kind of filter, and given an input space, it spits out an Abelian group in return. This returned group is then a representation of the hole structure of the space.K-theory was created by using category theory to provide a systematic basis for the considerations above. This is a new discipline of mathematics embracing concepts and problems central to many other major discipline. The success of K-thoery rest with its many applications to important problems in other disciplines and its ability to adapt to ongoing research in various areas of mathematics after obtaining a foothold there. In this process, K-theory has assimilated large tracts of other disciplines including number theory, algebraic geometry and differential topology to name a few. Algebraic K- theory provides certain functors, called K0,K1,K2,..., attaching Abelian groups to an algebraic object. These groups have nice relations with each other (sit in a long exact sequence,...) and in many instances help us to understand more about the original object.The aim of this project is to study these K-groups attached to certain algebraic objects, namely fields and division algebras (see the case of support for technical description of the project).