Professor Dr Camillo De Lellis

Research Priorities: Calculus of variations, partial differential equations, geometric measure theory, incompressible fluid dynamics
Camillo De Lellis is an Italian-Swiss mathematician, whose main fields of research are the calculus of variations and the equations of incompressible fluid dynamics. 
In the calculus of variations, one seeks the solution of a minimum problem, for instance a shape that optimizes a certain feature. A prominent example is named after the Belgian nineteenth-century physicist Joseph Plateau, who proposed to study area-minimizing surfaces, namely surfaces which minimize their area among those which span a fixed contour. It has long been known that such surfaces might have singularities, for instance, the formation of a certain type of corners, but a complete description of the type and size of the singularities is a long-standing open problem. A large proportion of Camillo De Lellis' research is dedicated to describing and understanding the specific nature of the singularities of such surfaces.
The first system of partial differential equations ever written down in fluid dynamics is given by the Euler equations which were found more than 250 years ago. The incompressible Euler equations are a limiting case of another well-known system, the Navier-Stokes equations. Whether regular solutions of the Euler and Navier-Stokes equations might form singularities in finite time is one of the biggest open problems in mathematics: for the Navier-Stokes equations, it is one of the famous millennium prize problems. Together with Hungarian mathematician László Székelyhidi, Jr., Camillo De Lellis has shown that there are very irregular solutions, many more than expected, and that they might behave in a very surprising way. Their new approach borrows from the pioneering work of 1950s American mathematician John Nash on the isometric embedding problem, a thus far completely unrelated topic in differential geometry, another branch of mathematics.
Both researchers' ideas provide the basis for recent important developments, such as the resolution by American mathematician Phil Isett of a 1949 fundamental conjecture of Norwegian physical chemist and theoretical physicist Lars Onsager in the theory of turbulent flows, and the unexpected discovery by mathematicians Tristan Buckmaster and Vlad Vicol that irregular solutions of the Navier-Stokes system are not uniquely determined by the equations.