Hi. I'm a 6th year math PhD student at Cornell University working with Allen Knutson.
Aside from that, my artistic passions include painting and songwriting. I'm also a trained reiki practitioner, and, some would say, a burgeoning seer, scryer, comedian, and juggler.
Email: pxa2 at cornell dot edu
Current research
My research is mainly in the field of Schubert calculus, and I have had a particular focus on Schubert puzzles, which are combinatorial objects that compute the structure constants of the cohomology ring of the Grassmannian, or in other cohomology theories. Much of my research and my interests have centered on observing and studying the interplay of algebraic geometry and combinatorics through the bridge of Schubert calculus.
In my paper "Commutative Properties...", I generalize triangular puzzles by considering puzzles with other convex polygonal boundary shapes with 4, 5, and 6 sides. From there I prove commutative properties for these polygonal puzzles analogous to the commutative property of classical triangular puzzles. Both geometric and combinatorial proofs are provided. Part of my ongoing research interests is to explore how these symmetries manifest in other objects that also compute Littlewood-Richardson numbers.
I've also developed an interest in generalizing "puzzles" in some sense. One can define Schubert puzzles completely algebraically as the set of sections of a certain ring homomorphism onto a path algebra. Really, it's a more general framework for talking about edge-labeled graphs and imposing relatively arbitrary conditions that the labeling must obey (e.g. for Schubert puzzles, we have the condition that each 3-cycle must be labeled with one of 5 options so that it corresponds to a valid puzzle piece). This formulation allows us to broaden and diversify the class of objects we're considering, and for me it has facilitated connections to ideas from other areas of math such as algebra, algebraic geometry, and category theory. I would like to explore this further to try to gain a deeper understanding of the properties of puzzles and their generalizations and relationships to other mathematical structures.
Papers
- Commutative Properties of Schubert Puzzles with Convex Polygonal Boundary Shapes, 2024.
- Generalized splines on graphs with two labels and polynomial splines on cycles, Electron. J. Combin. 31 (2024), no. 1, Paper No. 1.29, 52 pp., joint with Julianna Tymoczko and Jacob Matherne, 2024.
Slides from my talks
- Commutative Properties of Polygonal Schubert Puzzles, University of Minnesota Combinatorics Seminar.
- Geometric manifestations of an equality of Littlewood-Richardson numbers seen through comparing skew tableaux-counting rules, University of Minnesota Combinatorics Seminar.
- Schubert calculus and edge-swapping symmetries of Knutson-Tao puzzles, University of Minnesota Combinatorics Seminar.
- Schubert Calculus with Puzzles: a dialogue between geometry and combinatorics, Route 81 Conference.
Artwork
A few pieces I did:
Don't Ask, digital, 2020
Dam Youths, digital, 2019
Self, digital, 2019
Music
I write songs. My LP is going to drop right here. Please be patient.
