K-theory of Fields and Azumaya Algebras

有人说，数学是关于研究模式和结构。这将数学分为几个学科。例如，分支拓扑学是研究形状的属性，这些属性通过连续变形（即拉伸和挤压）保持不变。形状中有洞就是这些属性之一。考虑一个甜甜圈和一个茶杯，两者都是由橡胶制成的。我们可以通过拉伸和挤压将其中一个转变为另一个，而不需要撕裂或粘在一起先前分离的碎片。因此，这两个对象（或拓扑空间）在拓扑上是相同的。两个形状是否拓扑相同的自然问题是一个非常困难的问题。代数的分支，历史上，是研究一个或几个代数方程的解，涉及一个或几个变量的多项式。所有多项式都有一阶的情况（线性方程组）导致线性代数。在研究一个多项式的根的单一方程的情况下，导致了场论和伽罗瓦理论。代数几何学是由几个高次方程的一般情况而产生的，之所以这样命名是因为这类系统的解的集合经常用几何方法来研究。现代代数学家不仅在方程理论中，而且在一般数学中，越来越多地抽象和公理化了论证的结构和模式。这些结构的实例包括基团和环。一个群是一个集合，它有一个组合元素得到新元素的方法（比如加法或乘法，或者一般来说，定义为组合两个元素A和B的A*B），它满足某些性质。如果一个群的所有元素都满足A*B=B*A，则称该群为阿贝尔群。理解群的行为并将其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结构相似性。20世纪40年代和50年代见证了一种表达它们的形式语言，即范畴理论的创造。范畴是通过将“同类”的数学对象（如拓扑空间或群）分组而自然产生的。研究"同类“对象的一些重要方法是给它们附加结构--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，这个阿贝尔群检测到空间中存在洞。一种技术上的说法是，有一个从拓扑空间范畴到阿贝尔群范畴的“函子”。这里这个函子是一种过滤器，给定一个输入空间，它返回一个阿贝尔群。这个返回的群就是空间孔结构的表示。K理论是利用范畴论创建的，为上述考虑提供了系统的基础。这是一门新的数学学科，包含许多其他主要学科的核心概念和问题。K-thoery的成功在于它在其他学科中的许多重要问题上的应用，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等。代数K-理论提供了某些函子，称为K 0，K1，K2，.，将阿贝尔群附加到代数对象上。这些群体彼此之间有很好的关系（坐在一个很长的精确序列中，...）在许多情况下，帮助我们了解更多关于原来的对象。这个项目的目的是研究这些K-群附加到某些代数对象，即领域和司代数（见支持该项目的技术说明的情况）。

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).