Fourier analysis and partial differential equations

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Fefferman has been working to find efficient algorithms to interpolate multidimensional data by an interpolating function that is "nearly as smooth as possible" (under various interpretations of the phrase in quotes); and investigating a possible self-similar breakdown scenario for the surface QG equation, an incompressible fluid equation in 2 space dimensions. Stein has been working on finding general theorems that guarantee that relevant operators arising in several complex variables are bounded on appropriate function spaces. He has also been working to develop the asymptotics of stock-price distributions for basic models involving stochastic volatility. Fefferman: Interpolation of data is important for many purposes in science and technology. (Nearly) incompressible fluids occur in nature, but there is little fundamental understanding of how they flow.Stein: Methods of harmonic analysis used in studying operators on function spaces have vast applications in understanding a variety of phenomena in science and technology. There is also great utility in better understanding the nature of solutions of stochastic differential equations.