My research lies generally in the area of geometric analysis where I use analysis of PDEs to study geometric properties of conformal manifolds. More specifically I work on Yamabe-type problems searching for conformal metrics that admit constant curvature, including higher order, singular, and fractional order problems.
You can read more about my research here.
For an introduction to the Yamabe Problem, I recommend the 1987 survey The Yamabe Problem by Lee and Parker.
Preprints
A. Sophie Aiken, Rayssa Caju, Jesse Ratzkin, and Almir Silva Santos. An end to end gluing construction for metrics of constant Q-curvature. 2025. arXiv: 2503.09234 [math.DG]. URL: https://arxiv.org/abs/2503.09234.
In Preparation
A. Sophie Aiken. A note on the classical and fractional Yamabe problem.Â
I am very interested in project based learning, and from 2022-2025 I worked with a team of graduate students under the direction of Nandini Bhattacharya and Pedro Morales-Almazan to design summative assessments for the Calculus for the Life Sciences (formerly Calculus with Applications) series and College Algebra at UCSC.
Every project is based on a real scientific question, often sourced from STEM faculty at USCS and from recent publications in earth and planetary science, chemistry, biology, ecology, and psychology.
The project designed for College Algebra has been released as an OER. You can view it here.