Sidman proposes two main lines of investigation. The first involves lexicographic Groebner bases and the other has to do with properties of secant varieties. The goal of her work on lexicographic Groebner bases is to use the notion of Castelnuovo-Mumford regularity to produce concrete statements about the complexity of the lexicographic term ordering and the geometry of projections that would explicitly confirm long-held beliefs about their behavior. She also proposes to work on several problems in which computational and combinatorial methods are applied to the study of secant varieties. Interest in secant varieties has arisen in diverse areas including algebraic geometry, coding theory, computational complexity, and algebraic statistics. Recent applications of secant varieties in algebraic statistics have brought both new techniques and new questions to the forefront.
In the undergraduate mathematics curriculum the objects of so-called "modern" algebra, groups, rings, fields, and modules, have a reputation for being very abstract. Indeed, the names "modern algebra" and "abstract algebra" are often used interchangeably. The power of the abstraction only becomes evident when one sees how these ideas enable us to discuss an increasingly large variety of applications in a common framework. Computational algebra is an area that is useful to pure mathematicians in algebraic geometry and commutative algebra. It is also able to bridge the divide between abstract algebra and many concrete applications including integer programming, enumeration problems in statistics, and geometric modeling. Sidman's first project connects computational algebra with algebraic geometry. The questions she proposes to study related to secant varieties are also related to algebraic geometry, but were motivated by work in the emerging field of algebraic statistics. Thus, the proposed work reflects the ways in which pure and applied mathematics influence each other, often in unexpected ways.
