Deformation theory is a collection of techniques from homological algebra which enable one to understand the local topology of moduli spaces of mathematical objects.
The standard example of a moduli space is the space of associative algebra structures m: V x V -> V on a fixed vector space V. The moduli space can be thought of as an algebraic variety, defined by equations in the structure constants which correspond to associativity of m, and quotiented by the action of the group of automorphisms of V. It is a highly singular space.
A curve in this space through a given point m can be thought of as a 1-parameter family {m_t} of deformations of m, and can be written using a Taylor expansion
m_t(a,b) = m(a,b) + t m_1(a,b) + t^2 m_2(a,b) + ...
The Zariski tangent space to the variety at the given point m parametrises possible first-order deformations m_1 which cause
m_t(a,b) = m(a,b) + t m_1(a,b)
to be associative modulo t^2. It turns out to be the 2nd Hochschild cohomology group HH^2(A;A) of the algebra A corresponding to m. But not all such first-order deformations extend to second (or higher) order ones: the first obstruction comes from a Lie bracket HH^2 x HH^2 -> HH^3.
The general principle is that the deformation problem to all orders is encoded in a differential graded Lie algebra (in the above case, the Hochschild cochain complex) and that homological algebra can be used to solve it. The most famous example of this method is Kontsevich's solution of the problem of deformation quantization of Poisson manifolds.
This term we will try to learn about these homological techniques (and related algebraic structures such as operads and infinity algebras)
References
There are lots of useful papers on the web. In particular here are
some of the ones I like (sorry, too lazy to link):
Bernhard Keller "Notes for an introduction to Kontsevich's
quantization" (on his webpage), also notes on A-infinity algebras on
the arxiv
Kontsevich "Deformation quantization I", on arxiv
Markl "Notes on deformation theory", on arxiv
Markl, Shnider, Stasheff book "Operads"
Jan 15: Justin Roberts Introduction
Jan 22: Justin Roberts Basic deformation theory
Jan 29: Ben Wilson Hochschild cohomology
Feb 5: John Foley Operads
Feb 12: Dan Budreau Differential graded Lie algebras
Feb 19: More on dglas
Feb 26: Mark Gross Deformations of complex manifolds
Mar 3: Deformation quantization of Poisson manifolds
justin@math.ucsd.edu
