My work sits in an area of mathematics called Schubert calculus, which began with a classical style of question: given some geometric objects in space, how many other objects meet all of them? A famous example asks how many lines in three-dimensional space touch four given lines — and the answer, remarkably, is exactly two. Schubert calculus turns questions like this into concrete computations by translating geometry into algebra and combinatorics.
The modern, "quantum" version of the subject goes a step further. Ideas originally motivated by string theory led mathematicians to quantum cohomology, which keeps track not just of how objects intersect, but of how curves can be drawn through prescribed points and conditions. These curve counts are packaged into a rich algebraic structure — a quantum cohomology ring — whose behavior encodes deep geometric information.
Most of what is known about these rings concerns highly symmetric spaces. I focus on spaces that fall just outside that comfortable setting, such as odd symplectic Grassmannians and flag varieties, where the symmetry is slightly broken and new phenomena appear. The formula displayed below — an equivariant quantum Chevalley rule for the odd symplectic Grassmannian, proved with Leonardo Mihalcea as part of my dissertation work — computes multiplication by the Schubert divisor class, and it determines the entire ring structure. Recurring tools in my work are partitions and Maya diagrams, which turn questions about these spaces into combinatorial puzzles you can draw by hand.
Much of this research is done in collaboration with undergraduate students at Salisbury University; their names are highlighted throughout the publication list below. If you are a student who enjoys puzzles, patterns, and pictures, this is a very approachable corner of modern geometry — stop by my office.