Topics in Algebra MATH 991 Algebraic Deformation Theory

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MATH 991. Algebraic Deformation Theory. Instructor: Prof. D. N. Yetter. Fall 2007. Mathematical structures on an given underlying space often organize them- …

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Topics in Algebra
MATH 991
Algebraic Deformation Theory
Instructor: Prof. D. N. Yetter
Fall 2007
Mathematical structures on an given underlying space often organize themselves
into families cut out by equations. Deformation theory systematically
studies the structures ‘nearby’ a given mathematical structure in such a space.
As one example, the associative k-algebra structures on a given vector space
V over k are described by equations which cut out a subset of Homk(V
V, V ).
In the case of k = R or C, this set has a natural manifold structure, in other
cases it has the natural structure of a scheme over k.
As another, the complex structures on a surface organize themselves into
space, called the moduli space (named from the original example of the torus,
on which complex structures correspond to the modulus-the other complex
number besides 1 spanning the lattice by which C was quotiented).
Quite remarkably, in both cases, as well as others to be covered in the course,
the infinitesimal structure of the space has the same structure: tangent vectors
(called first order infinitesimal deformations) can be identified with classes in a
cohomology group naturally constructed from…