Broadly speaking my research focuses on geometric analysis and partial differential equations. As a student I started working in the calculus of variations (more specifically in geometric measure theory) under the supervision of Luigi Ambrosio. With time my interests have branched towards more classic partial differential equations (systems of conservation laws, transport equations, Hamilton-Jacobi equations, the incompressible Euler equations) and minimal surfaces in Riemannian manifolds. In all these fields I am typically interested in the formation and behavior of singularities (or, in some lucky cases, in the absence of singularities).
Let me list three very broad subjects in which the presence (or absence) of singularities seem to be the most compelling issue.
Consider the famous Plateau's problem: given a surface N of dimension k in a certain ambient space, we look for the surface(s) of dimension k+1 which span N and have the least area possible. It is well known that, in general, these least area (or area-minimizing) surfaces are singular.
Hyperbolic systems of conservation laws
Several classical models in continuum physics end up in formulating partial differential equations describing the evolution of a given physical systems. Often these systems are the mathematical description of conservation laws for some relevant physical quantities. Several of these systems of partial differential equations (especially in compressible fluid dynamics) fall in a class called "hyperbolic systems of conservation laws". For these systems a typical phenomenon is the creation of singularities which travel in time, called shock waves.
Incompressible fluid dynamics
Consider the Euler equations for incompressible fluids (derived by Euler more than 250 years ago) or the Navier-Stokes equations (roughly speaking a modification of the Euler equations proposed in the nineteenth century to take into account the effect of viscosity). It was proved in the last century that in 2 space dimensions there is no formation of singularities for these equations*. It is presently not known whether this is true also in 3 dimensions. For the Navier-Stokes equation this is one of the famous "millennium problems" and hence regarded as one of the major challenges in mathematics.
*I am obviously speaking of the Cauchy problem in the whole space: it is well known that for more complicated boundary conditions the description of boundary layers for the Euler equations is one of the formidable problems of fluid-dynamics.