Noncommutative Algebraic Geometry

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Algebraic geometry is the study of solution 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 polynomial equations one is solving. In noncommutative algebra, applying operations in different orders can yield different results. For example, putting on one's shoes and then putting on one's socks is not the same as doing it in the conventional order. More formally, we have that x times y is not necessarily equal to y times x. Noncommutative algebra, just as in the study of the algebra of polynomials, is often motivated by geometric problems and intuition. In this setting, the field is called noncommutative algebraic geometry and 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, which describes the quantum mechanical relation between position and momentum, has no matrix solutions, but the equation yx+xy=0 has many. In general, an important goal of noncommutative algebraic geometry is to understand the space of solutions to polynomial equations where the variables need not commute. In addition to having immediate applications in noncommutative algebra, there are applications to algebraic geometry and physics. 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. This requires the work of undergraduate and graduate students, and postdoctoral fellows. They develop technical expertise in algebra and computation and will learn how to collaborate, disseminate results, and solve multi-layered technical problems. Their highly developed skill sets will make them valuable additions to academic research in various fields, or allow them to work in industry in diverse areas such as cryptography, or coding.