The Habiro ring originally arose in topology as the natural home for quantum invariants of knots and 3-manifolds, but is actually a purely number-theoretical object and in fact a very interesting one, though for some reason not yet well known to number theorists.  I will give the definition of this ring and explain its main properties and some different elementary ways of looking at it, with several examples.  If time permits, I will also say something about its recent generalization to a ring associated to an arbitrary number field.  This latter direction arose from my work with Stavros Garoufalidis on quantum invariants of knots and 3-manifolds and was then developed in detail in a joint paper with him, Peter Scholze, and Campbell Wheeler. 

 

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