Greetings. I'm a 5th 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 burgeoning seer, scryer, comedian, and juggler.
Email: pxa2 at cornell dot edu
Current research
My research so far has mainly been focused on Schubert calculus and Knutson-Tao puzzles, with a focus on looking at how symmetries that are observable puzzles can reveal unexpected geometric symmetries and vice versa. Using only geometric or cohomological means, I've discovered and proved certain edge-swapping symmetries for puzzles (computing the Schubert basis structure constants in ordinary cohomology, K-theory, and T-equivariant cohomology) with hexagonal, trapezoidal, or parallelogram-shaped boundaries. These are symmetries generalize the commutativity property found in the usual puzzles with triangular boundary. For example in ordinary cohomology, we find that we can swap the labels on two boundary edges if they have the same content (number of 1s and 0s), and the number of puzzles filling the new boundary is the same.
I've also developed an interest in generalizing "puzzles" in some sense. I came up with a way to 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
Generalized splines on graphs with two labels and polynomial splines on cycles, with Julianna Tymoczko and Jacob Matherne.Slides from my talks
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, me, digital, 2020
Dam Youths, me, digital, 2019
Self, me, digital, 2019
Music
I write songs. My LP is going to drop right here. Please be patient.
