the title of my talk is ... a lie! ... or at least an exaggeration ... but that's pretty typical when you're trying to set up one of these secret dictionaries, or "cryptomorphisms" as some people call them, translating between two branches of mathematics, or two _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 "part of the experience" / "feature rather than bug" ... stretching your conceptual view trying to make the picture fit together .... ???? .....
anyway, before focusing on the more nitpicking ways in which this equation is a lie, it makes sense to first focus on the ways in which it's true .... ?? ....
?? some people say that _every_ equation is a lie ... i'm not sure that i'd go that far, but certainly _this_ one is a lie. (?? where if anywhere to fit this in ? ,,, ?? pretty far above, maybe ?? ... or maybe just on reserve in case of audience anticipation of it ... ??? ...)
?? if the equation is a bit fuzzy it's perhaps ok because each of the two sides is already a bit fuzzy, meaning somewhat different things to different people. for example "projective geometry" comes in both a "synthetic" flavor and an "analytic" flavor, and as a matter of fact both of these aspects are important in the theory of group representations (which is a main theme of this conference) ... to a first approximation, you can assume that i'm talking about the analytic rather than the synthetic flavor of projective geometry in this talk, although analytic and synthetic get tangled together in interesting ways, and perhaps "algebraic" is a more apt description than "analytic" here ...
?? as to how "crypto" (hidden ... ?? ...) the dictionary here really is, well, it really didi 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 ....... ?????
anyway, i'd like to start with an example of dimensional analysis here ... this is an example that i asked john baez to cook up for me, as a semi-realistic toy example of how physicists, for example, think in terms of dimensional analysis ,,,
?? name baez called it .... ???? "classical elastic scattering problem for two particles in one space dimension" ...
(?? rant about multiple confusingly related usages of word "dimension" here .... ????? .......)
two basic (?? primitive ... ?? ...) "dimensions" : "mass" and "velocity" ...
??? "unit analysis" ..... ?????????????? ......
?? draw 2d (??? ....) grid depicting this .... ???? ....
?? then we have some basic quantities ... ...?? ...
??? then "axioms" / "constraints" / "equations" ....
??? "possible state of affairs" .... "free parameter" .... ???? .....
lawvere ... beck .... theory ... doctrine ....
moduli stack ... tannakian ....
?? tannakian program as unification of study of rep cats and quasicoherent sheaf cats .... ??? ...
and/or subsumption ... ??? ....
theory as category .... object as stuff ... moduli stack .... predicate theory vs propostional theory ... "logic" .... "doctrine" ....
???? doctrine ... 2-topos .... ?? ....
?? sources of lawvere's concept of "theory" ... ??? physics ... logic ...
vague memories of stuff lawvere said ....
fund thm of dim theories .... ???? ..... as very simple so hard to track down ??? ....
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