Noncommutative Algebra and Algebraic Geometry

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We propose to study the interaction between algebraic geometry and noncommutative algebra. Algebraic geometry is an old subject where one studies geometric shapes described by polynomials and the related algebra of polynomials. We call these shapes varieties. A graph of a polynomial is a simple example of a variety. We study higher dimensional versions of these described by several polynomials in several variables. In noncommutative algebra we assume that we have variables where xy is not equal to yx. The simplest example of a noncommutative algebra is given by matrices. In noncommutative algebraic geometry we take the methods and results of regular algebraic geometry and try to extend them to the setting where our variables do not commute. The proposed research is to study orders over varieties. These are noncommutative algebras where we combine the varieties described by polynomials and matrices. An order will give us an algebra of matrices whose entries are polynomials. Since our matrices are of some fixed finite size, there is much interaction with usual algebraic geometry. This allows us to extend deep non-trivial results to the noncommutative setting. One central problem in algebraic geometry is classification. We propose to extend what is known of classification in one and two dimensions to higher dimensions. We also are proposing to study regular algebras and singularities of algebras. A variety is regular if it has no bumps or kinks called singularities. There are purely algebraic ways to determine if a variety is regular, or it has singularities there are many ways to measure the singularity. We propose to study these properties of not having any singularities, and to study the singularities in noncommutative algebraic geometry.