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, Journal of Algebra

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.

Journal Link    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

Talks

Invited Talks

1. Solving Polynomials: An Ever-Evolving Discipline, Math/Stats Colloquium, Carleton College, Sep 2023.

2. RD_k^{≤d}-Versality and Resolvent Degree for the Sporadic Groups, KOALA 2023: Workshop of the Kentucky-Ohio ALgebra Alliance, The Ohio State University, May 2023.

3. Solving Polynomials, Resolvent Degree, and Geometry, Algebraic & Number Theory Seminar, UC Santa Cruz, May 2022.

4. Solving Polynomials, Resolvent Degree, and Geometry, Algebraic Geometry Seminar, The Ohio State University, Feb 2022.

5. What is... resolvent degree?, What is... a seminar?, September 2021. (Virtual)

6. Solving Polynomials, Resolvent Degree, and Geometry, Mathematics Teacher-Scholar Symposium, Reed College, May 2021. (Virtual)

7. On the Geometry of Solutions of the Sextic in Two Variables, Algebra Seminar, UC San Diego, Feb 2020.

8. On Klein and Fricke’s Modular Solution of the Sextic, AMS Western Sectional Meeting, UC Riverside, Nov 2019.

9. Mathematics & Music: Application of Groups in Music Theory and Composition, Occidental College Mathematics Department, Sep 2019.

Departmental Talks

UC Irvine

1. Embracing Geometry: Or How I Learned to Stop Worrying and Solve Polynomials, MGSC and AMS Math Grad Student Conference, Apr 2020.

2. On the Geometry of Solutions of the Sextic in Two Variables, Mathematics Graduate Student Colloquium, Jan 2020.

3. On Klein and Fricke’s Modular Solution of the Sextic, Mathematics Graduate Student Colloquium, Nov 2019.

4. Solving Polynomials After Klein: The Theory of Resolvent Degree, Mathematics Graduate Student Colloquium, May 2019.

5. Mathematics & Music: Applications of Groups in Music Theory and Composition, Future Faculty Program, Feb 2018.

Oberlin College

6. Mathematics & Music: Applications of Groups in Music Theory and Composition, Mathematics Senior Lecture, May 2016.

7. Rational Combinations of Betti Diagrams of Complete Intersections, Summer Research Showcase, Nov 2015.

Poster Presentations

1. A. Sutherland, Solving Polynomials, Resolvent Degree, and Geometry, Combinatorial, Computational, and Applied Algebraic Geometry, Seattle, WA, Jun 2022.

2. A. Sutherland, Solving Polynomials, Resolvent Degree, and Geometry, Graduate Student Topology and Geometry Conference, Atlanta, GA, Apr 2022 (Virtual).

3. A. Sutherland, Special Points on Fano Varieties of Complete Intersections and Applications to Resolvent Degree, Graduate Student Topology and Geometry Conference, Bloomington, IN, Apr 2021 (Virtual).

4. M. Annunziata, C. Hawkins and A. Sutherland, Rational combinations of Betti diagrams of complete intersections, MAA Undergraduate Poster Session at Joint Mathematics Meetings, Seattle, WA, Jan 2016.