NSCPHD1129 Algebraic structures of Hochschild complexes
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There are several related but different perspectives on Hochschild complexes. They show up for example in
- deformation complexes of algebras
- algebraic models for loop space homology (string topology)
- approximations to K-theory
The course will present some aspects of the theory arising in these different contexts, with an emphasis on algebraic structures and graph complexes.
One of the important aspects of the theory is the noncommutative differential calculus. This is a theory which replaces the ring of smooth functions on a manifold by an arbitrary associative algebra; it defines basic algebraic structures of differential calculus on manifolds in terms of the algebra of functions, in such a way that it works for any associative algebra. Within this framework, the role of noncommutative differential forms is played by Hochschild chains, and the role of noncommutative multivector fields is played by Hochschild cochains. The main result in the subject is the theorem, due to Tamarkin and Tsygan, that asserts that Hochschild chains and cochains possess a certain algebraic structure called "a calculus up to homotopy". This is a major instrument for proving a formality theorem for Hochschild and cyclic chains and an index theorem for deformation quantizations.
One of major tools in concrete computations turn out to be the graph complexes and their homology.
The graph complexes in its various guises are some of the most mysterious and fascinating objects in mathematics. They are combinatorially very simple to define, as linear combinations of graphs of a certain kind, with the operation of edge contraction or dually, vertex splitting as differential. Still, their cohomology is very hard to compute and largely unknown at present.
There are various versions of graph cohomology, each playing a central role in one or more fields of algebra, topology or mathematical physics.
- Ordinary (non-decorated) graph cohomology describes the deformation theory of the En operads and plays a central role in many quantization problems.
- Ribbon graph complexes describe the cohomology of the moduli spaces of curves. This example gives the TQFT used in the study of string topology.
- Lie decorated graph complexes describe the cohomology of the automorphisms of a free group and play a central role in many results in low dimensional topology.
Knowledge: overview of the current state of research
Skills: proof techniques in the different contexts
Competences: ability to read research papers in the subject
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- 2,5 ECTS
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