Undergraduate Student Research Projects

I maintain a list of problems which are suitable for undergraduate research projects. The problems that I am currently most interested in making progress on are listed below, with some background and prerequisites listed. Please e-mail me if you would like to discuss working on one of these projects with me.

• Classifying weighted Einstein manifolds in dimension two: Give a complete classification of all weighted Einstein manifolds in dimension two. This involves reducing a geometric problem to an ODE and solving that ODE.
  Prerequisites: A basic ODE course and a proof-based real analysis course. Familiarity with Riemannian geometry is useful, but not necessary.

• Constructing conformally covariant polydifferential operators: The GJMS operators and curved Ovsienko–Redou operators are conformally covariant operators formed from powers of the Laplacian. This project asks you to construct "higher rank" analogues of these operators, and perhaps give a classification thereof.
  Prerequisites: Multivariable calculus and some experience with basic combinatorics.

• Constructing CR covariant polydifferential operators: The curved Ovsienko–Redou operators are constructed by understanding the properties of the Laplacian acting on homogeneous polynomials in Euclidean space. This project asks you to consider the Laplacian on bihomogeneous polynomials on complex Euclidean space.
  Prerequisites: Multivariable calculus and some experience working with complex numbers and doing basic combinatorics.

• Enumeration of local conformal invariants: This project asks to find a good basis for the vector space of local conformal invariants of weight -8. The analogous classification of local conformal invariants of weight -6 has interesting applications for closed six-manifolds.
  Prerequisites: Abstract linear algebra. Familiarity with Mathematica or a rough equivalent may be useful.

• "Primed" pseudohermitian invariants: These are a new class of pseudohermitian invariants defined in a similar way as is the Q'-curvature. Very few such invariants are known; we want to construct and study new such invariants.
  Prerequisites: Abstract linear algebra.

Some Past Research Projects and Their Outcomes

Below is a list past undergraduate research projects I supervised which resulted in publications, Honor's theses, and/or presentations:

• Weiyu Luo: Studied conformally covariant boundary operators associated to the sixth-order GJMS operator.
  Outcomes:

  • Schreyer Honor's Thesis in 2018. Co-winner of the Douglas G. and Regina C. Evans Award for Research Achievement.

  • Published research paper: J.S. Case, W. Luo, Boundary operators associated to the sixth-order GJMS operator, Int. Math. Res. Not., 14:10600–10653, 2021.

• Sarah Baisely: Studied a fast diffusion equation on Eucidean space with interesting links between conformal geometry and sharp Sobolev inequalities.
  Outcomes:

  • Schreyer Honor's Thesis in 2019.

• Zhengyang Shan: Studied the nonuniqueness of the σ4-curvature prescription problem on closed manifolds with boundary.
  Outcomes:

  • Published research paper: Z. Shan, Nonuniqueness for a fully nonlinear, degenerate elliptic boundary-value problem in conformal geometry, Differential Geom. Appl., 74:101688, 2021.

• Yiyang Wang: Studied scattering for the Laplacian on hyperbolic space and the spatial AdS–Schwarzschild manifold.
  Outcomes:

  • Named a Women in Math Scholar in Summer 2019. This included giving a research presentation in Spring 2020.