Powers in Commutative Algebra: Approaches, Properties, and Applications

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My research explores the intersections of Commutative Algebra, Combinatorics and Geometry. The overlap of these areas allows us to solve problems using different viewpoints. Putting the areas together empowers us to develop new techniques and questions. A simple example of the intersection of Algebra and Geometry is the use of equations to describe lines in the plane. Similarly, there is an overlap between Algebra and Graph Theory. A graph is a collection of nodes with edges connecting some of the nodes. We assign to each node a variable and to each edge a product of variables called a monomial. Doing so leads to the study of monomial ideals, a collection of combinations of monomials. ******When we look at points where polynomials evaluate to zero, we need to capture the complexity of the vanishing. For example, 0 is a vanishing point of the polynomial functions f(x) = x and g(x) = x^2. However, the vanishing at g(x) = x^2 is considered more "complicated". To study the complexity, we look at collections of polynomials called ideals and their regular and symbolic powers. At the heart of many problems in Algebra and Geometry is the difference between these powers. The long-term goals of the proposal are to develop approaches to understand the properties of and measure the differences between these powers and to explore applications.******An especially important family of ideals is the monomial ideals. These non-trivial ideals provide the base case for testing hypotheses and building arguments for more involved ideals. Also, there are tools that allow us to use monomial ideals to better understand properties of general ideals. I will use techniques from Graph Theory and Linear Programming to study invariants of symbolic powers of monomial ideals. I will also exploit a polyhedron to obtain data about the symbolic power of a monomial ideal. ******Underlying many conjectures of the relationships between regular and symbolic powers are ideals connected to geometric objects called fat points. Despite arising in numerous contexts, properties of these objects are difficult to compute. I will generalize a procedure that will use geometric information about the points to obtain algebraic information about the ideals.******Work from this proposal will expand and deepen known tools while developing new techniques. Applications include coding theory, Hadamard products, statistics and physics. As such, the impact is anticipated to be wide-ranging.******Highly qualified personnel (HQP) will master material related to the proposal, develop the ability to generate sound questions, learn computer skills via computer algebra systems, learn best practices in giving presentations, and strengthen writing skills by publishing. These skills will serve them well in a wide range of careers in fields such as mathematics, physics, statistics and computer science. Moreover, HQP will form collaborations by growing together within the research group and networking.