Diffusion remains a central part of mathematics and physics, inspiring influential work over the last three centuries.  It has proven to be an important model of a wide variety of phenomena, including heat flow and  chemical reactions.  It is an essential part of the modeling of atmospheric chemistry, as depicted in the picture above. The Kato problem formed one part of this theory which recently has been completely resolved.

It  asks for  regularity of diffusions through an arbitrary choice of medium,  concentrating on the effects on diffusion as the medium  is allowed to  change.  Knowing this regularity, we can be more confident of the predictions made with the aid of such models.  In the Kato problem, the material  changes in a way that demands the most refined estimate of regularity of diffusion. Such paradigms test our understanding of the underlying physical process in the deepest way.

 The resolution of this question had previously only been known completely in one dimension, and that result by Coifman, Meyer and McIntosh was a key development in harmonic analysis in the 1980s.   In the intervening years, many people made essential contributions to the question although the most general formulation  remained unresolved. The complete resolution of the Kato problem in all dimensions has been achieved by mathematicians in the US, France and Australia, Auscher, Hofmann, Lacey, McIntosh and Tchamitchian. The US participants Hofmann and Lacey are supported by the NSF. The resolution  in all dimensions highlights previously hidden relationships between the key aspects of the subject. Correspondingly the analytic/synthetic  tools we can bring to questions of this type has now been enhanced.   It also points to the important role of international collaboration in the continued vitality of mathematics in the U.S.