My research specialty is commutative algebra, the study of solutions to polynomial systems of equations. Commutative algebra is a generalization of linear algebra, which deals with solutions to linear systems of equations.
In linear systems, variables have only numerical coefficients -- meaning they don't have powers, aren't multiplied together, and aren't part of functions like $\sin(x)$. For example:
\begin{align*}
x + y + z &= 1 \\
2x-3y+z &= 10 \\
5x+2y-2z &= -1
\end{align*}
In polynomial systems of equations, variables can have integer exponents and can be multiplied together. Below is an example of a polynomial system involving three variables:
\begin{align*}
x^2 + xyz - 2y^3 +z &= -2 \\
x +y^2-x^2y + z^3 &= 5
\end{align*}
While linear systems can easily be solved by computers using matrices, polynomial systems have no such analog. Instead, we study
ideals, or collections of sums of polynomial multiples of the left-hand sides (after moving everything to the left to get a $0$ on the right) of the equations in the system.
Since earning my PhD, my primary research has focused on ideals generated by principal minors, the topic of my thesis. These ideals have significant applications in algebraic statistics, particularly in solving the PMAP problem, though much remains unknown about them. In the summer of 2022, I mentored three students in the Georgia Tech REU (Research Experience for Undergraduates, funded by NSF) on a project that explored the toric structure of ideals generated by principal 2-minors, which also have applications in integer programming. I presented this research at the Joint Math Meetings (JMM) in January 2023.
All my papers, whether published or in progress, are available on arxiv.org. I have delivered several talks on these projects, detailed in my CV. Notably, here is a polished version of my Bertini theorem talk presented at JMM 2023, and an early version of my matroid varieties talk given at the MHC colloquium, designed for undergraduates. For the local cohomology project, I presented these slides at both JMM and KUMUNUjr (now URiCA) in 2017. Lastly, here is the talk I delivered at JMM 2014, covering the two papers that emerged from my thesis, which includes my earliest use of TikZ for creating standout slides.
.jpg){kind=link}