Hochschild homology
Supervisor: Morten Brun, email: morten.brun math.uib.no
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Prerequisites: MAT220 is essential. Parts of MAT224, which can be picked up by following the course in parallel with the project.
Description: Homology is an important tool in algebraic topology that tells os about a space's qualitative properties. Loosely speaking, homology tells us how many "holes" there are in a space. Hochschild homology is a type of homology for rings that can be used to create homology theory for something called non-commutative space. In short, every nice space is defined by the commutative ring of continuous real functions in the space. The Hochschild homology of this ring can be used to compute the space's homology. A non-commutative ring that is similar to a ring of real functions is sometimes called a non-commutative space, and the Fields medal winner Allan Connes has used Hochschild homology to define the homology of such non-commutative spaces.
This project is to first understand Connes cyclic homology, and then calculate the Hochschild homology of simple commutative rings. The project concludes with varying the definitions of Hochschild homology and repeating the calculations for these variations.
What you will learn:
- simplicial methods,
- homological algebra.
References:
[1] , . Second edition. [Fundamental Principles of Mathematical Sciences], 301. , Berlin, 1998.