Many physical phenomema lead to tracking moving fronts whose speed depends on the curvature. The "level set method" has been tremendously succesful for this, but the solutions are typically only continuous. We will discuss results that show that the level set flow has twice differentiable solutions. This is optimal.
These analytical questions crucially rely on understanding the underlying geometry. The proofs draws inspiration from real algebraic geometry and the theory of analytical functions.
Tobias Colding's research studies problems in differential geometry, geometric analysis, PDEs and low-dimensional topology. His works covers several different fields, with strong implications on all of them. Early in his career he introduced new ideas coming from analysis to study the geometry of manifolds with bounds on Ricci curvature. This started a collaboration with Jeff Cheeger, that gave a new perspective on the study of the Gromov-Hausdorff closure of such manifolds. Soon after that, he worked with William Minicozzi, first on harmonic functions of polynomial growth over manifolds (solving a conjecture of S.T.Yau on the dimension of the space of such functions), and later in groundbreaking work on the structure of minimal surfaces. As a result of this work, they were awarded the 2010 AMS Oswald Veblen prize in Geometry; the prize statement for that year states:
"The 2010 Veblen Prize in Geometry is awarded to Tobias H. Colding and William P. Minicozzi II for their profound work on minimal surfaces. In a series of papers they have developed a structure theory for minimal surfaces with bounded genus in 3-manifolds, which yields a remarkable global picture for an arbitrary minimal surface of bounded genus. This contribution led to the resolution of long-standing conjectures and initiated a wave of new results.
Nowadays he and Minicozzi continue with their work on mean curvature flow of hypersurfaces in Euclidean space.
He received his Ph.D. in mathematics in 1992 at the University of Pennsylvania under the direction of Chris Croke. He was on the faculty at the Courant Institute of New York University in various positions from 1992 to 2008. He has also been a visiting professor at MIT (2000–01) and at Princeton University (2001–02) and a postdoctoral fellow at MSRI (1993–94).
He is a foreign member of the Royal Danish Academy of Science and Letters, and received an honorary professorship at the University of Copenhagen in 2006. He was elected Fellow of the American Academy of Arts & Sciences in 2008. He was selected by the MIT Mathematics Department as the holder of the Norman Levinson Professorship, 2009-2014. He was appointed Senior Scholar of the Clay Mathematics Institute, 2011-2012. In 2013, he was recognized for his commitment, service and scholarship when appointed the Cecil and Ida B. Green Distinguished Professorship of Mathematics. In 2016, Colding received the Carlsberg Foundation Research Prize for ground-breaking research in differential geometry and geometric analysis. For 2015-16, he received a second Senior Scholar appointment by the Clay Mathematics Institute. In 2017, he received the Simons Fellowship in Mathematics.