The geometry of continued fractions

I’m enjoying my post-qual liberty – I passed! – by giving a talk later today for Women in Math at Berkeley on continued fractions, and the elegant efficiency they offer for performing computations in the geometry of surfaces. Here are my notes.

I suspect that, quite rightly, many of the talks in an expository forum like this are what my undergraduate advisor would call ‘propaganda talks’: a rough sketch of some of the main ideas and motivations of a field or topic, probably accompanied by some central examples.

Partly out of philosophy, partly out of curiosity, and partly because I tied myself to it in my title and abstract, I’m not going to buck this trend. Instead, I’m going to try to communicate something of the everyday activity of a geometer (at least, in my experience of course). Some big ideas will naturally be introduced into the story as required – toric geometry, intersection theory, resolution of singularities,… – but I’m hoping to drape the talk on the many beautiful and elementary examples of things I’d like to compute (for instance, invariant monomials under a group action). I anticipate that it won’t be the easiest talk to deliver or receive, but I’m intrigued to see what happens!

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