Noncommutative algebraic geometry

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Algebraic geometry is the study of the geometry of solutions spaces of polynomials in several variables.  This includes the geometry of familiar shapes like parabolas, spheres, and curves in the plane defined by a polynomial.  One tries to study the solutions by relating them to other spaces via mappings or parametrizations.  This subject is highly controlled by the algebra of polynomials.  Geometric statements about the space of solutions correspond directly to algebraic statements about the polynomials one is solving.  In noncommutative algebra the order of the product yields different answers.  So xy is not equal to yx.  However, one can do the same type of algebra that is done with polynomials for geometric purposes.  This algebra is motivated by geometric problems and intuition.  It is called noncommutative algebraic geometry.  It uses the techniques and ideas of commutative algebraic geometry applied to noncommutative algebra.  For example we may try to solve noncommutative equations with matrices and try to understand that space of solutions.  More concretely, the equation yx-xy=1 has no matrix solutions, but the equation yx+xy=0 has many.  We wish to understand the space of solutions.  In addition to having immediate applications in noncommutative algebra, there are applications to algebraic geometry.  Important spaces that arise in algebraic geometry can be described as solutions of noncommutative equations with matrices.  The proposed research deals with the spaces were this interaction occurs.  One important problem is to know when we can parametrize the solutions spaces.  For example, the points on the parabola y equal x squared are parametrized by x=t, y equals t squared, but it is impossible to parametrize the solutions of y squared equals x cubed minus one, with polynomials.  We have a geometric conditions that should characterize when one can parametrize the matrix solutions of certain types of noncommutative equations.  We will try to prove that these conditions do indeed hold true.