Mathematics Research Assistant Position
This project focuses on local energy estimates for wave equations and their applications. Wave equations have a conserved energy, but as this does not change in time, it fails to capture the dispersive nature wave equations. Instead, if one focuses on the energy within a bounded set, we expect the disturbance to eventually vacate that set leading to decay. This is certainly the case when there is trivial background geometry and waves follow rays. But in the case of nontrivial background geometry, such as a black hole where light can orbit the black hole, the possibilities of trapping occurs and can necessitate finer analyses. Precisely quantifying this is one direction of investigation. Another is the application of these estimates to nonlinear wave equations, and in particular trying to identify how nonlinear structures affect the lifespan of solutions.
Students will be introduced to the wave equation and guided through proofs of some background work. Through a series of exercises, they will be led to newer and newer material and introduced to an open problem. The goal will be to apply these techniques to provide a proof to solve the conjecture. In the process, basic techniques of partial differential equations, positive commutator arguments, local energy estimates, etc. will be introduced.
Students should have at least completed Math 233 and Math 383, though experience with Math 521 is beneficial.
Please send CV, application letter explaining interest, and a list of relevant course work to Dr. Jason Metcalfe.