RTG: Analysis and Applications

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The University of Wisconsin Research and Training Groups (RTG) project in Analysis and Applications consists of a variety of initiatives with the aim of increasing the number of US citizens and permanent residents who study mathematics and then pursue teaching and research careers in mathematics. The project will significantly increase both the intensity and range of activities at UW Madison, and create a culture of broad education and intensive well-rounded training, for trainees at the postdoctoral, graduate and undergraduate level. The research scope of the proposal is both broad and coherent, interwoven by common themes and ideas. The harmonic analysis research topics include classical questions on Fourier multipliers, restriction of Fourier transforms and the interplay of sub-Riemannian geometry and analysis. In mathematical fluid dynamics, the research will focus on regularity and blow up problems for active scalars, on investigating realistic models of atmospheric dynamics building upon the classical Boussinesq system, and on mixing problems that lie on the interface of partial differential equations, dynamical systems and probability. In mathematical biology, we will work on reaction networks that model a broad range of processes in biochemistry, systems biology and ecology, and on applications of chemotaxis to enhancement of biological reactions. The research of this project will be focused on analysis and applications. The applied problems will come from a variety of areas and sciences. There will be applications that involve models used in atmospheric science for weather and climate studies. There will be problems on efficiency of mixing in fluids, of interest in applications ranging from internal combustion modeling to food processing. Finally, there will be problems on reaction networks of interest in biochemistry, and problems on sensing of chemicals by living organisms of interest in ecology and oceanography. The project is designed to give young participants experience and depth necessary to adapt to a variety of career paths and to pursue the most promising directions available. It offers a wide range of teaching and research activities, including summer training programs for incoming graduate students, annual research seminars for graduate students and postdocs, annual research experience programs for undergraduates, research workshops and outreach opportunities. The research experience programs for undergraduates seek to early acquaint students with the excitement of mathematics research and provide leadership training opportunities for graduate students and postdocs. Those activities will have a large innovative teaching component during the academic year, and will be followed by intensive research programs in the summer. The graduate research seminars will cover a diverse range of active topics, exposing researchers and students to a mix of advanced analysis techniques along with concrete applied problems where these techniques can be used. The research workshops will serve to broaden the perspective of the trainees and suggest new directions of research. Outreach activities sponsored by the project will include regular mathematics seminars for high-school students, with the participation of RTG trainees. The activities supported by the RTG grant will significantly strengthen the UW mathematics department's research and training programs which contribute to the effort of creating a mathematically skilled workforce. This contribution represents the national and statewide effect of this project.