## Research

I wrote my doctoral dissertation under Manjul Bhargava at Princeton University on a way of generalizing certain aspects of Galois theory to non-separable ring extensions, and then spent three years as a postdoc at the Leiden University in the Netherlands. My current research focuses on discriminant algebras (ways of converting arbitrary rank-n algebras into rank-2 algebras) and the operadic structure of gluing together small networks into bigger networks.

Here’s a timeline of the highlights of my mathematical activity since my PhD: (for preprints, only the most recent publishing status is shown)

• Mar. 2021: “A principle for converting Lindström-type lemmas to Stembridge-type theorems, with applications to walks, groves, and alternating flows” (arxiv) to appear in Graphs and Combinatorics.
• Mar. 2021: “Isomorphisms of Discriminant Algebras” (arxiv) with Alberto Gioia, under revision for the International Journal of Algebra and Computation.
• June 2020: “Fine compactified moduli of enriched structures on stable curves” (arxiv) with David Holmes, to appear in Memoirs of the American Mathematical Society.
• May 2020: Seminar “Toward a Universal Theory of Discriminant Algebras” (slides) for the Dartmouth College algebra and number theory seminar.
• May 2019: Conference talk “Duality for Algebras of the Connected Planar Wiring Diagrams Operad” (slides) at the Fourth Symposium on Compositional Structures (SYCO 4) at Chapman University.
• Feb. 2019: Seminar “A Fine Moduli Space of Enriched Structures” (pdf) for the University of South Carolina algebraic geometry seminar.
• Nov. 2018: Seminar “G-closures and discriminant algebras” for the University of Waterloo number theory seminar.
• Dec. 2017: “Néron Models and the Height Jump Divisor” (arxiv) with David Holmes and Robin de Jong, published in Transactions of the American Mathematical Society.
• July 2017: “Projective modules of locally constant finite rank are finitely generated” (pdf), a letter to Hendrik Lenstra.
• July 2017: “Galois closure data for extensions of rings” (arxiv) published in Transformation Groups.
• Mar. 2017: Seminar “Paths that never cross” for Carleton College.
• Dec. 2016: Seminar “A new discriminant algebra construction” for the University of Minnesota’s algebraic geometry seminar.
• Dec. 2016: Seminar “The tropical Grassmannian” for the University of Minnesota’s algebraic geometry reading group.
• Nov. 2016: Seminar “The boundary complex of $\bar{\mathcal{M}}_{g,n}$” (ditto).
• Oct. 2016: Seminar “Introduction to Tropicalization” (ditto).
• Sep. 2016: “A new discriminant algebra construction” (pdf) with Alberto Gioia, published in Documenta Mathematica.
• June 2016: “Non-existence of a functorial resolvent cubic algebra” (pdf), a letter to Manjul Bhargava.
• Mar. 2016: Seminar “What’s inside the black box?” for Carleton College.
• Jan. 2016: Attended the AMS joint mathematics meetings in Seattle.
• Jul. 2015: Attended the AMS summer research institute on algebraic geometry in Salt Lake City.
• Apr. 2015: “Uniqueness of discriminant algebras in ranks ≤ 3” (pdf), a letter to Alberto Gioia.
• Aug. 2014: Article “The Gibbs Phenomenon: Why JPEGs Look Yucky” (pdf) for Leiden University’s Eureka! Magazine.
• May 2014: Seminar “Solvability by Radicals and Solvable Galois Groups” (pdf) for the University of Vienna’s algebraic geometry seminar.
• Mar. 2014: Seminar “A New Definition of Discriminant Algebra” (pdf) for Leiden University’s algebra, geometry, and number theory seminar.
• June 2013: “Galois Closures for Rings” (pdf), PhD Thesis with Manjul Bhargava

(Last updated March 2021.)

Some undergraduate work on resistor networks with L. Peter Mannisto:

• “On local recoverability of circular planar electrical networks” (pdf)
• “The connection-determinant formula, with applications to electrical networks” (pdf)
• “The tree diagrams formula” (pdf)
• “Open subgraphs of a graph with boundary” (pdf)