Research

I use tools from algebraic geometry to answer questions about resolvent degree.  Resolvent degree is a measure of complexity for field extensions, branched covers of varieties, algebraic functions, groups, etc.

My dissertation was titled Upper Bounds on the Resolvent Degree of General Polynomials and the Families of Alternating and Symmetric Groups and is available here. 

My postdoc advisor is Angélica Cueto.



My Ph.D advisor was Jesse Wolfson

Papers


On Resolvent Degree

A Summary of Known Bounds on the Essential Dimension and Resolvent Degree of Finite Groups

We summarize what is currently known about ed(G) and RD(G) for finite groups G over ℂ (i.e. in characteristic 0). In Appendix A, we also give an argument which improves the known bound on RD(PSL(2,𝔽_{11})). 

arXiv link


Generalized Versality, Special Points, and Resolvent Degree for the Sporadic Groups

Joint with Claudio Gómez-Gonzáles and Jesse Wolfson

See the full project page for this paper (including relevant code) here

Resolvent degree is an invariant measuring the complexity of algebraic and geometric phenomena, including the complexity of finite groups. To date, the resolvent degree of a finite simple group G has only been investigated when G is a cylic group; an alternating group; a simple factor of a Weyl group of type E_6, E_7, or E_8; or PSL(2,𝔽_7). In this paper, we establish upper bounds on the resolvent degrees of the sporadic groups by using the invariant theory of their projective representations. To do so, we introduce the notion of (weak) RD_k^{≤d}-versality, which we connect to the existence of "special points" on varieties.

arXiv link


Upper Bounds on Resolvent Degree via Sylvester's Obliteration Algorithm

Joint with Curtis Heberle, New York Journal of Mathematics

For each n, let RD(n) be the minimal d for which there exists a formula for the roots of the generic degree n polynomial using algebraic functions of at most d variables. In this paper, we recover an algorithm of Sylvester for determining non-zero solutions of systems of homogeneous polynomials, which we present from a modern algebro-geometric perspective. We then use this geometric algorithm to determine improved thresholds for upper bounds on RD(n).

Journal link    arXiv link     ar5iv link


Upper Bounds on Resolvent Degree and Its Growth Rate

Submitted for publication

For each n, let RD(n) be the minimal d for which there exists a formula for the roots of the generic degree n polynomial using algebraic functions of at most d variables. In this paper, we provide improved thresholds for upper bounds on RD(n). Our techniques build upon classical algebraic geometry to provide new upper bounds for small n and, in doing so, fix gaps in the proofs of A. Wiman and G.N. Chebotarev in [Wim1927] and [Che1954].

For this project, I have translated the relevant articles of Wiman and Chebotarev; see the Translations section below.

arXiv link     ar5iv link


On Betti Diagrams

Rational combinations of Betti diagrams of complete intersections 

Joint with M. Annunziata, C. Gibbons, and C. Hawkins, Journal of Algebra and Its Applications

Research conducted as part of the Willamette Mathematics Consortium REU in 2015

We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij-Söderberg theory. That is, given a Betti diagram, we determine if it is possible to decompose it into the Betti diagrams of complete intersections. To do so, we determine the extremal rays of the cone generated by the diagrams of complete intersections and provide a rudimentary algorithm for decomposition.

Journal link     arXiv link     ar5iv link

Translations

For several projects I have worked on, I have translated certain classical articles from their original language into modern English.


On the Problem of Resolvents

Originally written in Russian by G.N. Chebotarev in 1954.

arXiv link     ar5iv link


On the Application of Tschirnhaus Transformations to the Reduction of Algebraic Equations

Originally written in German by Anders Wiman in 1927.

arXiv link     ar5iv link


About the Solution of the General Equations of Fifth and Sixth Degree (Excerpt from a letter to Mr. K. Hensel) 

Originally written in German by Felix Klein in 1905.

arXiv link     ar5iv link

Talks

Invited Talks


Departmental Talks

UC Irvine


Oberlin College


Poster Presentations

Alex Sutherland's Poster for GSTGC 2022.pdf