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John Baez (Oct 18 2020 at 20:11):I'm writing a regular column for the Notices of the American Mathematical Society, and here is a draft of my second column:
It's about a really cool "random surface", sort of the 2d analogue of Brownian motion. I'd really appreciate it if anyone could read this and comment about things that are hard to understand. I guess it's supposed to assume a grad-level course or two on topology and real analysis (mainly measure theory and Hilbert spaces).
John Baez (Oct 18 2020 at 20:11):It's just two pages long.
John Baez (Oct 18 2020 at 20:13):The paragraph starting
But this result is just a small part of something much bigger: is deliberately a bit vaguer and requires knowing what a Riemann surface is - it's just supposed to give a taste of how this subject is connected to some other ideas.
Tim Hosgood (Oct 18 2020 at 20:15):I like it! I must admit, I had to google what exactly the "conformal structure" of a Riemann surface was, but maybe this is something that other people have generally heard of.
John Baez (Oct 18 2020 at 20:25):I'll think about it.
Btw, conformal field theory, which is the basis of string theory, is about quantum field theories on Riemann surfaces that are invariant under automorphisms of those Riemann surfaces (aka "conformal transformations").
John Baez (Oct 18 2020 at 20:27):Maybe saying that would be good.
David Michael Roberts (Oct 18 2020 at 22:59):Maybe just a line about how conformal transformations preserve angles, but not necessarily scale? So a sphere of radius r and a sphere of radius R are not isometric, but they are conformally equivalent? Or the good old Mercator projection being a conformal transformation from a twice-punctured sphere to a cylinder? And then the conformal structure is an equivalence class of metrics under these transformations?
David Michael Roberts (Oct 18 2020 at 23:01):Or perhaps a more concrete example? The cylinder has two simply-described conformal structures: one coming from the product of the circle and the real line, the other coming from the Mercator projection. They are the same real surface, but different Riemann surfaces-with-conformal-structure.
John Baez (Oct 19 2020 at 03:16):These columns have to be pretty short, basically two pages. So, the most I could do is add a sentence or two reminding people what a conformal structure on a Riemann surface is.