Seminar on Hochschild Cohomology

The notions of (co)homology were introduced in the context of Topology by Poincaré in the late XIX century. These provided, for the first time, powerful algebraic invariants used to distinguish between non-homeomorphic spaces. Since then, homological methods have found applications in a wide range of areas of Mathematics, including differential geometry, algebra, representation theoryand mathematical physics. This seminar will focus on a cohomology theory for associative algebras, Hochschild cohomology, introduced by Hochschild in 1945, and which remains a very active topic of modern research.

We will study this object from three main perspectives:

Higher structures: these are generalizations of usual algebraic structures (associative, Lie or Poisson) to graded and differential graded contexts that allow us to borrow some intuition from (algebraic) Topology and study structures “up to homotopy”.
Noncommutative geometry: we try to find replacements for notions of (commutative) algebraic geometry to non-commutative setting.
Deformation theory: this is the study of "near by" structures; the name is pretty self explanatory :)

Practicalities

There will be a preliminairy preparatory meeting for interested students on the 5^th of March at 14:00 in the Seminar Room 414.

Every student is expected to give a 1 hour talk and produce an accompanying hand-out. I would like to receive the hand-out 2 weeks before the talk and meet with the student a week before the talk.
The contents of the seminar are subject to change depending on the number and interest of the students. I will provide good references but the students are expected to fill in some details
This is a good seminar for students of Mathematics and Mathematical Physics. Advanced Algebra, Algebraic Topology and Algebraic Geometry can be good background courses to have for this seminar, but there will be topics for any background! The only pre-requisite is some contact with Homological Algebra ( (co)chain complexes, (co)homology, homotopy, quasi-isomorphisms, derived functors, Tor and Ext).
You can check the proposed Syballus and References in more detail here (final?).