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Abstract

This is an important expository paper based on recent work of \\it K. Bringmann and \\it K. Ono [Ann. Math. (2) 171, No. 1, 419--449 (2010; Zbl 05712731)] and \\it S. P. Zwegers [Contemp. Math. 291, 269--277 (2001; Zbl 1044.11029), ``Mock theta functions.'' Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Thesis) (2002; Zbl 1194.11058), Ramanujan J. 20, No. 2, 207--214 (2009; Zbl 1207.11053), and Bull. Lond. Math. Soc. 42, No. 2, 301--311 (2010; Zbl 1198.11047)] on Ramanujan's mock theta functions. These are certain q-series which belong to at least one (and presumably to all) of the following three families of functions: Appell-Lerch sums, quotients of indefinite binary theta series by unary theta series, and Fourier coefficients of meromorphic Jacobi forms. After giving some background, the author briefly reviews Zwegers' study of the transformation properties of these three families of functions and his construction of non-holomorphic correction terms used to make these functions modular. Motivated by Zwegers' work, the author then introduces the notion of a mock modular form and its shadow, discusses the relation between mock modular forms and the harmonic weak Maass forms of Bruinier and Funke, and collects a number of new examples. Finally, the author touches on some of Bringmann and Ono's applications of these ideas to the study of ranks of partitions.