The heat and particle transport mechanisms determining the performance of magnetic confinement fusion reactors are highly nonlinear, and interact over a wide range of temporal and spatial scales. A sound approach to study these mechanisms computationally is to separately develop codes describing the phenomena at each given length and time scale, and then couple them in a self-consistent way.
In my group, we develop fast, high order solvers to compute plasma equilibria in toroidally axisymmetric fusion experiments (tokamak, spherical torus). These solvers are specifically designed to be coupled with state-of-the-art micro-turbulence and transport solvers. They need to be fast, so the computational time is mostly spent on the expensive parts of the simulations (i.e. the microturbulence simulations). The equilibrium solvers also need to be accurate, so errors in the calculated equilibrium do not propagate to other elements of the simulations.
Toroidally axisymmetric equilibria are described by the Grad-Shafranov equation, a second order, nonlinear, elliptic partial differential equation. In order to satisfy the speed and accuracy requirements, we use integral equation methods to construct our solvers. We collaborate with Leslie Greengard's and Mike O'Neil's research group on these topics.

In high intensity cyclotrons, the self electric force due to the charges of the accelerated particles is strong enough that it can have a significant effect on the particle dynamics during the accelerating phase. As a result, as the beam of charged particles rotates around the cyclotron and gets accelerated, it can also spiral on itself (see figure on the left) or even break up because of this self electric field. Understanding these mechanisms is crucial if one wants to design efficient cyclotrons capable of producing clean intense beams.
In my research group, we use continuum kinetic theory for nonneutral plasmas to develop models and design numerical methods that yield new analytic insight into these phenomena and have the potential to significantly accelerate existing simulation codes.
As an alternative and complementary approach, we also investigate noise reduction strategies to improve the speed and accuracy of Particle-in-Cell (PIC) codes, which are the current codes of choice for the beam dynamics community. Presently, we are particularly interested in the potential gains obtained by combining the PIC scheme with sparse grids approaches.

Stellarators are promising magnetic confinement fusion devices that are not toroidally axisymmetric but instead intrisically three-dimensional (see a typical magnetic surface in the figure on the left). Computing three-dimensional plasma equilibria is a notoriously hard problem, both from a mathematical point of view and a numerical point of view. At the Courant Institute, Harold Grad, Paul Garabedian, and Harold Weitzner made significant contributions to solve that compicated problem.
In my group, we identify small parameters in existing and designed experiments to derive asymptotic models that reduce the complexity of the MHD equilibrium equations, and we develop numerical methods to solve these reduced equations, often based on integral equations, and also in collaboration with Leslie Greengard and Mike O'Neil.

I am interested in a variety of other problems in which methods of applied mathematics can help improve the accuracy and/or reduce the computational time of large scale simulations for fusion plasma physics. For example, Andy Majda, Di Qi and I recently derived a new reduced fluid model for magnetized plasma turbulence, inspired by the Hasegawa-Wakatani model, which we will use to investigate uncertainty quantification and reduced modeling strategies which could be applied to a broader class of turbulence models for magnetized plasmas.
I also plan to investigate ways to use the Fast Multipole Method to develop grid-free particle simulation codes for the study of beam dynamics in particle accelerators and electron guns.
Finally, I am interested in listening to any idea you might want to suggest or discuss with me. Feel free to contact me!