3rd Year Candidacy Requirement/Analysis Seminar: "History and Future Directions of Wavelet Representation"

Speaker: Jeremy Cummings, Washington University in Saint Louis

Abstract: In 2007, Petermichl proved a sharp Ap bound for the Hilbert transform via a decomposition into dyadic shifts, which act on the basis of Haar functions. This sparked an interest in such representations of operators, leading to the Hytönen representation theorem for Calderón–Zygmund operators, again in terms of averages of dyadic shifts. In this talk I will outline the history of such representations, including both the Haar case and the smooth representation of Di Plinio, Wick, and Williams. I will further examine several avenues for generalizations of the latter theorem, particularly ways of extending the notion of wavelet representation to spaces of homogeneous type. In the course of this discussion we will review notions of wavelet bases in such spaces as well possible notions of Sobolev spaces.

Host: Walton Green