proj geom = dimensional analysis 8

the title of my talk is ... (pointing to it on whiteboard) a lie ... or at least, an exaggeration ... in fact, neither side of this equation is precisely enough defined to be exactly equal to itself, let alone to anything else. for example, projective geometry comes in at least two different flavors, "synthetic" and "analytic", and both of these flavors are important in the study of group representations as it happens, and they're related to each other in a funny way ... although analytic projective geometry is perhaps more commonly known these days as _algebraic_ projective geometry (ot at least, those are roughly the same thing ...), and it's that form of projective geometry, namely projective _algebraic _ geometry, that i'm going to be connecting to (/ equating with ??) dimensional analysis in this talk...

but all of this vagueness and fuzziness is pretty typical when you're ... living in this universe, or more particularly when you're trying to set up one of these secret dictionaries, or "cryptomorphisms" as they're sometimes called, that translate between two different branches of mathematics or two different ways of thinking. typically there'll be some words in each of the two languages that sound funny when you translate them into the other language, which might make you doubt the validity of the dictionary... but that's actually supposed to be one of the benefits of such a dictionary, that it forces you to stretch your conceptual view of each side of the equation, trying to make the picture fit together ....

in any case, before worrying about some of the more nitpicking ways in which this equation might be false, i'm going to start by explaining the ways in which it's _true_ ...

[??? emphasizing idea of category of graded vector spaces as category of group representations ..... ??? when we get to that ... ???? .....]

[?? as to how "crypto" (hidden ... ?? ...) the dictionary here really is, well, it really did take me by surprise ... though in retrospect does seem really obvious in some ways .... ?? both study of "homogeneous quantities", which can be taken to mean "quantities that transform in a particularly nice simple way under rescaling transformations" .... ??? which hints at relationship of this stuff to conference theme of representation theory .... ??? _abelian_ case ....... ????? ??? save this "retrospectively obvious" stuff for right after audience opinions ... ???? ... "i described this relationship as a "cryptomorphism", hidden, secret .... but in retrospect it seems really obvious to me now ..... homogenous quantities ....re-scaling .... line-bundle = one-dimensional object ..." .... ????? .....]

?? "so we have 2 key ingredients here:

1 _dimensions_, which are the boxes in the table here ...

2 _quantities_ that live in these dimensions, which are the variables that appear in those boxes ... (???? .....)

and my claim is that we can translate what we're doing here from the language of dimensional analysis into the language of projective algebraic geometry ....

so let me whether anyone in the audience has any opinions about how these 2 key ingredients should get translated .... ????? .....