This article discusses the challenges in climate science from the emerging viewpoint of stochastic-statistical properties of turbulent dynamical systems. The mathematical topics discussed here include empirical information theory, fluctuation-dissipation theorems, reduced-order stochastic modeling, and the development of mathematically unambiguous exactly solvable test models for climate science that capture crucial features of vastly more complex scientific problems. The applied mathematics topics include the emerging development of multiscale algorithms for filtering/data assimilation and superparametrization for climate science and other problems in science and engineering, as well as suitable unambiguous mathematical test problems for their behavior. Interesting contemporary research directions and specific open problems are mentioned throughout the article. The perspective here should also be useful for applications to other complex dynamical systems arising in neural science, material science, and environmental/mechanical engineering. © 2011 Wiley Periodicals, Inc.
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