Project Fully Nonlinear Eigenvalue Problems Program Purdue Experimental Mathematics Lab Term Spring 2026 Status Accepted Research Area Mathematics Description A common problem in partial differential equations (PDEs) is the eigenvalue problem. The original example is to consider a vibrating string, when two endpoints are fixed (e.g. on a violin). The eigenvalues correspond to the natural frequencies at which the string "prefers" to vibrate, and can be computed by solving a second-order ODE with fixed boundary values. When you move to higher dimensions, the ODE is replaced with a PDE. For simple PDE, we can compute the eigenvalues over the unit ball by hand. For not-so-simple ones however, this is not tractable, and we need to resort to numerical methods to make progress. In this project, we will explore eigenvalues for highly non-linear PDE in large dimensions. We will consider PDE related to the Monge-Ampere operator, which plays an important role in convex and complex geometry. On the unit ball, we can use symmetry to reduce the MA equation to a non-linear ODE, making the problem more approachable. Special focus will be given to the behaviour of the eigenvalues as the dimension of the ball goes to infinity. Supervisor Thomas J Sinclair Mentor Type of work required Course-based, vertically-integrated research projects in mathematics. Each project will consist of a small research team consisting of typically 2-4 undergraduates, a graduate mentor, and a faculty mentor. The graduate mentor and undergraduates will meet on a weekly basis, with full team meetings every few weeks as determined by the faculty mentor. To apply include a brief (one page or less) statement explaining your interest in mathematics research. Additionally, list all mathematics courses you have taken with your grade in each one, as well as any other coursework or qualifications that you feel are pertinent. Undergraduates who have been accepted into a project must sign up for the 3-credit "Purdue Experimental Math Lab" course (currently listed under MA 490) and must pledge that they are able to dedicate 10 hours of effort per week to the project. Websites https://www.math.purdue.edu/pxml/join-pxml.html Qualifications Completion of MA 353 and MA 375. Some knowledge of probability and topology is strongly preferred. Ability to meet Tuesday or Wednesday, in the afternoon or evening Credit Hours 3 Weekly Hours 10 (estimated)