Research

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.

Career

• since 2019 IMB von Neumann Professor, Institute for Advanced Study, Princeton, USA
• 2018‐2019 Professor, Institute for Advanced Study (IAS), Princeton, USA
• 2005‐2018 Professor, University of Zurich (UZH), Zurich, Switzerland
• 2004‐2005 Assistant Professor, University of Zurich (UZH), Zurich, Switzerland
• 2003‐2004 Postdoctoral Fellow, Swiss Federal Institute of Technology Zurich (ETH), Zurich, Switzerland
• 2002 Postdoctoral Fellow, Max Planck Institute for Mathematics in the Sciences (MiS), Leipzig, Germany
• 2002 PhD in Mathematics, Scuola Normale Superiore (SNS), Pisa, Italy
• 2000 Diploma in Mathematics, Scuola Normale Superiore (SNS), Pisa, Italy
• 1995‐1999 Degree in Mathematics, Scuola Normale Superiore (SNS), Pisa, Italy

Projects

• 2012‐2017 Principal Investigator, project “Regularity theory for area minimizing currents”, European Research Council (ERC)
• 2009‐2012 Head, subproject “Topology of partitioning surfaces (A03)”, Transregio (TRR) 71, German Research Foundation (DFG)
• 2009‐2012 Head, subproject “Min‐Max‐constructions of minimal surfaces (A04)”, Transregio (TRR) 71, German Research Foundation (DFG)
• 2009‐2012 Head, subproject “The Euler equations as a differential inclusion (C04)”, Transregio (TRR) 71, German Research Foundation (DFG)
• since 2019 Principal Investigator, Focused Research Groups (FRG) “New Challenges in Geometric Measure”, University of Texas, Austin, USA
• since 2019 Co‐Principal Investigator, Grant DMS‐1854147, Focused Research Groups (FRG) “New Challenges in Geometric Measure”, Princeton University, Princeton, USA

Honours and Memberships

• 2022 Maryam Mirzakhani Prize in Mathematics, National Academy of Sciences, USA
• 2022 Plenary Speaker, International Congress of Mathematicians (ICM), St. Petersburg, Russia
• since 2021 Member, German National Academy of Sciences Leopoldina
• 2021 Feltrinelli Prize in Mathematics, Mechanics and Applications, Accademia Nazionale dei Lincei, Italy
• 2020 Bôcher Memorial Prize, American Mathematical Society (AMS), USA
• since 2016 Member, Academia Europaea
• 2015 Amerio Prize, Académie des sciences et des lettres de l'institut lombard, Italy
• 2014 Caccioppoli Prize, Italian Mathematical Union (UMI), Italy
• 2013 Fermat Prize (together with Martin Hairer), Toulouse Mathematics Institute, University of Toulouse, Toulouse, France
• 2013 SIAM Activity Group on Analysis of Partial Differential Equations (SIAG/APDE) Prize (together with László Székelyhidi), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA
• 2012 Plenary Speaker, European Congress of Mathematics (ECM), Kraków, Poland
• 2010 Guest Speaker, International Congress of Mathematicians (ICM), Hyderabad, India
• 2009 Stampacchia Gold Medal, Italian Mathematical Union (UMI), Italy
• 2000 Benedetto Sciarra Prize for Laurea Theses in Mathematics, Scuola Normale Superiore (SNS), Pisa, Italy