?? graded vsp vs comodule of laurent polynomials ..... ????? ...... ?? just how equivalent are these ??

?? question of which formulation (?? ...) is more "fundamental" ... ??? graded module as graded _by_ abelian gorup, comodule as comodule _of_ hopf algebra .... ????? .... ???? abelian group as more fundamental than hopf algebra here ??? ..... ??? again, questions of "how equivalent ..." .... ???.....

???? .....

proj geom = dimensional analysis (2 or 3)

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 ??? ....

?? hmmm ... ?? so "classical scattering ..." dimensional theory example baez cooked up as ... ??? "classical" in "non-relativistic" sense ... ??? .... ??? so what _about_ relativistic version ??? ..... ?? is it problematic ?? ... ???? ...... .... ???? .... or suggestive ... ??? .... ?? or even more problematic ... ??? for dimensional theory philosophy in general ?? .... ???? ......

?? what sense of "cone" does dimitrov have in mind in saying:

"We will then see that toric varieties are constructed by “gluing” together affine
varieties which correspond to a specific kind of geometric objects — a type of cone."

?? pun ??

at first i thought that they did mean that the real or complex spectrum is a cone .... now though i think they probably meant the (P,+) spectrum .... ???? .....

?? relationship ... ??? ...

?? real algebraic variety "xy=0" (fe ...) as not giving pseudomanifold ?? .... ?? any interesting modification to "fix" this, or is it instead maybe an instance of "complex good, real bad" ?? .... ??? ...

?? trying to get something going relating distributions to differential forms via pluecker embedding ... ?? ....

?? problematic because embedding lands in projective space ?? .... ???....

?? alex ... ??? coordination between talks ?? ... prerequisite ... ??? .... ?? other talks ?? ...

?? hartog toric variety .... toric line bundle .... underlying ordinary line bundle .... ??? trying to understand its toric convolution comonoid structure .... ???? and/or other related structures .... ????? ....

?? intersection co/homology of singular child's drawing ??? .... ???? ..... ??? .... cartographic group aciton .... ???? ....

1d cartographic group action ..... orbifold ..... ???? ..... 1d pseudomanifold .....

"non-orbifold" cartographic group action .... ??? .... ?? "....exactly two..." .... ??

??? "pseudoorbifold" ??? ..... ???? .....

?? segal category of morse function .... geometric realization of that category ... ??? as "combinatorialization" of original space ... ??? ....

?? radial morse function on smoothed polytope ...

?? 2d cartographic group action ... ??? non-orbifold ?? .... ??? ....

??? real vs complex .... ??? ....

?? 1d stratified space .... ??? ....

?? "attaching map" ?? ....

?? geometric realization of functor on ... ???? poset of natural numbers ??? .... ???? ....

?? pseudomanifold .... ???? cartographic group action .... stratification .... baas-sullivan .... ?? .... baez-dolan .... ??? ..... ???? ..... .... woolf .... ??? ....

?? just (?? ...) noticed ... title "toposes of toric quasicoherent sheaves" .... ???? misleadingly natural to expect "sheaf" and "topos" to go together here .... ??? ....

?? .... "all toposes as honorary presheaf toposes, in about same way as all boolean algebras as honorary power-sets" ..... ???? ......

?? filteredly cocontinuous .... ??? ....

?? stuff allen knutson hinted at about extra dot buildings .... also such stuff gleaned from brown and from pressley and segal .... ???? "morse theory" .... ???? try to connect with stuff woolf talks about ... "lefschetz hyperplane theorem" .... and morse theory .... ???? generic and not-so-generic hyperplanes to given subvariety ..... ??? ..... ........ ????? ...... .....

??? cohomology of flag variety ....

??? cohomology and / or intersection co/homology of singular schubert variety .... ??? ....

???? ......

?? morse theory as generally indication of "cobordism" concept .... ??? ..... ?? case of "singular morse theory" ... ??? ...

?? octoberfest talk ?? ... ??? maybe on accidental topos again ??

??? injection/surjection factorization for geometric morphism .... ??? similarly for AG morphism ??? ....

?? confusion about various contrasting doctrine analogies here ??? ..... and various ..... ?? factorization systems ??? .... ??? relationship ... ???.....

?? reflection property of one adjoint vs what of other ???? ..... ???

?? "flatness of localization" ... ?? ... ?? generalize ... ???

?? grothendieck topology vs more general .... ????? (bit about generalized sheaves over canonical non-distributive or canonical non-modular lattice .... ??? ....)

?? snow-globe and exponentiation .... ?? "higher-order" .... ???? ...... ?? kleisli .... ??? ....

?? sequence of elements of [0,1] whose product is 0 ....


?? sequence of elements of [0,infinity] whose sum is infinity ... ???

?? "exp(-_)" ... ??? ....

??? convergence from divergence ..... ?????? ......

?? compactness of extended positives ... ???? .... ?? and/or reals ?? ....

?? gibbs-boltzmann rule and ... ???? .... ??? .... haar measure ..... ????? ...... ???? ... ??? ....

?? "annihilation operator as generator of annihilator ideal" .... ???? .... ?? "gaussian change of variable" .... ???? .....

??? .... ??? .... ....

?? toric convolution comodule of toric convolution coalgebra ?? ....

?? trivial action of arbitrary "AG group" on arbitrary AG theory ... ??? as good example to experiment with .... ??? kleisli ... ???? .... ???? .....

?? level slip ..... ?????? ...... ??? try to straighten out ... ??? ....

?? action of group on _model_ of theory .... duality flip .... diaconescu ..... ???? .....

?? if derived (infinity- ?? ...) category is more fundamental (?? ...) than original, then why even bother with the various cores ??? .... ???? .... ... rhetorical .... ??? .... answer ... ??? ...


?? toric convolution of chain complexes of quasicoherent sheaves .... ??? .... ?? whether perverse sheaves are closed under this ??? ....

?? quasicoherent sheaf of modules of initial sheaf of rings ... ??? .....

?? woolf's preface in kirwan and woolf ... talks about "rational" fan ... ??? ....

??? hmmm ... ?? relationship to some generalizations i thought about a bit ... ??? ....

"When the fan is rational there is a corresponding toric variety ..." .... ??? .....

?? (P,+) spectrum of "divisible" commutative monoid ??? .... ??? .... galois ... universal cover .... rational vs real ... ??? ....