CBMS Conference: Smooth and Non-Smooth Harmonic Analysis

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This National Science Foundation award provides support for the Conference Board of the Mathematical Sciences Conference "Harmonic Analysis: Smooth and Non-smooth" to be held at Iowa State University on June 4-8, 2018.  In recent years there has been an increasing interest in understanding the harmonic analysis of non-smooth geometries that are unlike the familiar smooth Euclidean geometry, such as when nearby points are not locally connected to each other.  Real-world examples in which these types of geometry appear include large computer networks, relationships in datasets, and fractal structures such as those found in crystalline substances, light scattering, and other natural phenomena where dynamical systems are present. The proposed conference will consist of a series of ten lectures by Professor Palle Jorgensen from the University of Iowa, a leader in the fields of smooth and non-smooth harmonic analysis, who will in the course of the conference demonstrate the surprising connections between the two domains. The conference aims to bring both experienced and new researchers together to stimulate collaboration on this timely topic.  The broader impacts of the conference include the advancement of underrepresented minorities within mathematics and the development of a globally competitive STEM workforce.  Approximately thirty participants will be supported, many of whom will be current graduate students.  The conference will contribute to those graduate students' educational and professional development and hence prepare the nation's next generation of researchers to engage this increasingly important subject area.A remarkable and oft-cited result of Professor Jorgensen is that there exists a Cantor-like set with the property that the uniform measure supported on that set is spectral, meaning that there exists a sequence of frequencies for which the corresponding exponential functions form an orthonormal basis in the Hilbert space of square-integrable functions with respect to that measure.  Research that has been inspired by this stunning result includes: fractal Fourier analyses, spectral theory of Ruelle operators, representation theory of Cuntz algebras, convergence of the cascade algorithm in wavelet theory, reproducing kernels and their boundary representations, Bernoulli convolutions, and Markov processes.  The remarkable feature of this array of subjects is that they straddle both the smooth and non-smooth settings.  The lectures presented by Professor Jorgensen will unify these far-reaching research areas at the interface of smooth and non-smooth harmonic analysis, with the corresponding monograph being of great value to current and future researchers as a consolidated reference for these topical connections. Information regarding the conference is at the following URL:  http://www.public.iastate.edu/~esweber/cbms2018/