New Paper on Rational Cubics by Candace Bethea

We’re excited to announce that Candace has just published a new paper: “The Equivariant Degree and an Enriched Count of Rational Cubics.” Dr. Bethea is a part of StemJazz’s 2024 Provost Cohort as well as a National Science Foundation postdoctoral researcher in the mathematics department. Outside of research, Dr. Bethea’s interests are in broadening diversity in mathematics in part through mentorship. She is particularly interested in increasing participation in advanced mathematics by underrepresented demographic groups through middle and high school outreach, as evidenced by her work serving on charter school boards and with Bridge to Enter Advanced Mathematics (BEAM). 

In the best STEMJazz broad-thinking multidisciplinary style, she’s tied a seemingly distant subject (equivalent homotopy) to the original problem.

Candace, of course, says it best:

One of the central questions in algebraic geometry is how to count rational curves, or curves defined by degree d polynomial equations, that lie on a surface. There are infinitely many in general, but finitely many once we impose the natural condition that rational curves must pass through a prescribed number of points on the surface. These are called enumerative questions, and algebraic geometry is also interested in understanding symmetries, or orbits, of solutions to enumerative questions under the presence of a group action. Studying symmetries of solutions has applications to counting real rational embedded curves fixed under complex conjugation on a complex manifold, of interest in Gromov-Witten theory.  The paper develops new tools in equivariant homotopy theory, a seemingly distant subject, to obtain concrete counts of orbits of rational plane cubic curves through points on a surface, producing the first formula counting both real and complex rational curves in complex projective space in a single equation. This recovers both a Gromov-Witten and Welschinger invariant, of interest in string theory, in a single formula, providing evidence for the conjecture that counts of real and complex rational curves are related via equivariant topological invariants.

 

So, congratulations Candace!