K-theory of Fields and Azumaya Algebras

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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).