What is your research about?
My research is in a field of mathematics called partial differential equations. These equations describe the evolution with time of various physical systems, ranging from the motion of water in the ocean or of air in the atmosphere, to the strength of an electromagnetic signal, and all the way to the motion of galaxies according to Einstein's equations of general relativity.
Partial differential equations have fundamental importance in everyday applications. For example, they have allowed us over the past 50 years to make more accurate weather predictions, improve how we handle turbulence in the atmosphere in relation to air travel, and invent faster ways to transfer information by electromagnetic signals (like using fiber optics for fast speed internet communication). Partial differential equations also appear in finance, where they are used to model changes in stock prices.
The study of partial differential equations is broad because of the various applications. I work on a particular class called nonlinear dispersive equations. This class includes equations governing ocean and atmospheric sciences, plasma physics, nonlinear optics (including fiber optics), and Einstein's equations of general relativity.
The first question when analyzing such equations is whether a solution exists, or not. The second question is whether we can describe the properties of this solution. In most cases, this solution cannot be written explicitly, so we have little to no way of inspecting its properties beyond analyzing the equation itself. This leads to a very nice and delicate study, in which we try to figure out the properties of the solution without having a formula for it, but only by looking at the equation that it solves.