# Publications

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of### Syzygies of Curves in Products of Projective Spaces

Motivated by toric geometry, machinery for understanding syzygies of curves in projective space is lifted to the setting of products of projective spaces. Using this machinery, we derive sample multigraded analogues of influential theorems concerning the regularity of curves and the degrees of their defining equations in this generalized setting.

### Virtual Criterion for Generalized Eagon-Northcott Complexes

Any matrix gives rise to an associated family of canonical complexes associated to it, including the Eagon-Northcott and Buchsbaum-Rim. Famously, there is a simple criterion for checking the exactness of these complexes. We give generalization of this criterion for these complexes to be *virtual resolutions*, which are “exact enough” to be relevant to the geometry of toric varieties.

### Quarternion-Valued Breather Soliton, Rational, and Periodic QKdV Solutions

The KdV equation is a prototypical example of an exactly solvable PDE, due in part to its admittance of soliton solutions. This paper examines and classifies the dynamics of quaternion-valued solutions of the noncommutative KdV equation.