?? 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 ..." .... ???.....
???? .....
Friday, September 30, 2011
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 ??? ....
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 ... ??? ...
"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 ... ??? ...
Thursday, September 29, 2011
Wednesday, September 28, 2011
?? 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 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 .... ??? ....
Tuesday, September 27, 2011
?? 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 .... ??? ....
?? .... "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" ... ??? ...
??? 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" ... ??? ...
Monday, September 26, 2011
??? 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 .... ??? ....
?? 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" .... ???? .....
??? .... ??? .... ....
?? 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" .... ???? .....
??? .... ??? .... ....
Sunday, September 25, 2011
?? 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 ..... ???? .....
?? 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 ... ??? .....
?? 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 ... ??? ....
??? 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 ... ??? ....
Saturday, September 24, 2011
?? tensoring (?? ... ?? syntactic coproduct ?? ....) AG theory with AG theory of g-torsor .... ???? ..... ?? change of enrichment-base ?? .... ???? ...
?? wrong idea maybe worth noting : ?? given category x and group g acting on it, and objects x1 and x2 in x, ask whether there's a sensible way for g to act on [x1,x2] ... and seems pretty clear there isn't ... ?? part of point being that x becoming enriched over _g-action_ (?? ...) (?? in some variant of situation described here ??? .... ?? ....) likely to involve original (?? ...) hom-space appearing as "invariants" of new one, rather than "total" .... ??? ....
?? wrong idea maybe worth noting : ?? given category x and group g acting on it, and objects x1 and x2 in x, ask whether there's a sensible way for g to act on [x1,x2] ... and seems pretty clear there isn't ... ?? part of point being that x becoming enriched over _g-action_ (?? ...) (?? in some variant of situation described here ??? .... ?? ....) likely to involve original (?? ...) hom-space appearing as "invariants" of new one, rather than "total" .... ??? ....
?? extension of AG theory of g-torsor x ...
?? g-equivariant morphism from x to y .... ??? where y is some particular g-space (??? .....) ....
for example g = gl(1) ... y = tautological g-space .... ???? ...
?? laurent polynomials .... ????? z-graded .... ????? ......
?? co-module of laurent polynomials ??? ....
??? toric ... ??? ....
??? degree 0 alg homomorphism from polynomials to laurent polynomials .... ???? = laurent polynomial of degree 1 ??? ....
??? "k1-based module of k1-algebra k2 as simply module of underlying ring of k2" ??? .... ??? sort of analog of this that fails, where base is purely stacky instead of purely spacey ??? ..... ??? ..... ...
?? affine algebraic geometry over purely stacky base .... ???? ....
??? "g-equivariant algebraic geometry" ... ??? .... ??? dorette pronk ... ??? ... ?? "morita" equivalence vs stricter equivalence
?? "g-torsor x tw g-equivariant x-parameterized family of [h_?]-torsors" .... ??? .... categorified version of ... ?? ....
?? g-equivariant morphism from x to y .... ??? where y is some particular g-space (??? .....) ....
for example g = gl(1) ... y = tautological g-space .... ???? ...
?? laurent polynomials .... ????? z-graded .... ????? ......
?? co-module of laurent polynomials ??? ....
??? toric ... ??? ....
??? degree 0 alg homomorphism from polynomials to laurent polynomials .... ???? = laurent polynomial of degree 1 ??? ....
??? "k1-based module of k1-algebra k2 as simply module of underlying ring of k2" ??? .... ??? sort of analog of this that fails, where base is purely stacky instead of purely spacey ??? ..... ??? ..... ...
?? affine algebraic geometry over purely stacky base .... ???? ....
??? "g-equivariant algebraic geometry" ... ??? .... ??? dorette pronk ... ??? ... ?? "morita" equivalence vs stricter equivalence
?? "g-torsor x tw g-equivariant x-parameterized family of [h_?]-torsors" .... ??? .... categorified version of ... ?? ....
Friday, September 23, 2011
Thursday, September 22, 2011
?? signature 1+1 (?? or more generally n+n ??) ... ??? isometry switching time and space .... ??? ....
??? usual (?? ) level slip confusion here ??? .... isometry from time to space vs auto-isometry of space-time ... ??? ....
?? tachyon ... two places at once .... all points along line at once ... ???? "space-like" .... ???? ... ?? ordinary particle appears to have speed zero in some frame, while tachyon appears to have infinite speed in some frame .... ???? ....
??? usual (?? ) level slip confusion here ??? .... isometry from time to space vs auto-isometry of space-time ... ??? ....
?? tachyon ... two places at once .... all points along line at once ... ???? "space-like" .... ???? ... ?? ordinary particle appears to have speed zero in some frame, while tachyon appears to have infinite speed in some frame .... ???? ....
?? general phenomenon of ... ??? "extension theory given by introducing new stuff, at first completely unconnected to original stuff, but then tied to it by also-introduced new structure ..." ....
?? "types, predicates, axioms" as lining up with "property, structure, stuff", but in "filtered" way vs "graded" .... ?? ....
(?? kontsevitch ?? .... ?? .... ... vasilieff invariants ... ??? ...)
?? showing up with for example "dimensional theory enriched over _g-rep_ " .... ??? with specified grading group ... ??? ...
?? ... annoying/intriguing "locally internal category" appendix in baby elephant ... ?? ...
?? "types, predicates, axioms" as lining up with "property, structure, stuff", but in "filtered" way vs "graded" .... ?? ....
(?? kontsevitch ?? .... ?? .... ... vasilieff invariants ... ??? ...)
?? showing up with for example "dimensional theory enriched over _g-rep_ " .... ??? with specified grading group ... ??? ...
?? ... annoying/intriguing "locally internal category" appendix in baby elephant ... ?? ...
??? gabriel-ulmer duality as .... ??? "entire spectrum determined (?? in particular straightforward way) by global models" ..... ?????? .....
?? maybe just side-effect .... ???? since there _are_ more general theories for which pretty much same way works .... ????? ...... ?????? ..... hmmm, model category closed under directed colimits (?? and in nice way ... ??? ...), vs under more general colimits (and in nice way ... ??? .....) .... ?? and / or "alexandroff theory" ... ??? and / or "generalized alexandroff" .... ???? ....presheaf as sheaf .... ????? ..... ???? ....
???? "global suffices" .... vs ..... ????? ...... ???? .... presheaf (vs sheaf) as aspect of this vs as inimical to it ??? ... maybe not inimical; maybe it's just sheaf that's inimical ..... ????? ..... but sheaf as presheaf .... ???? .....
?? maybe just side-effect .... ???? since there _are_ more general theories for which pretty much same way works .... ????? ...... ?????? ..... hmmm, model category closed under directed colimits (?? and in nice way ... ??? ...), vs under more general colimits (and in nice way ... ??? .....) .... ?? and / or "alexandroff theory" ... ??? and / or "generalized alexandroff" .... ???? ....presheaf as sheaf .... ????? ..... ???? ....
???? "global suffices" .... vs ..... ????? ...... ???? .... presheaf (vs sheaf) as aspect of this vs as inimical to it ??? ... maybe not inimical; maybe it's just sheaf that's inimical ..... ????? ..... but sheaf as presheaf .... ???? .....
?? "extension of AG theory of g-torsor" .... ?? which we're trying to construe as "g-rep-enriched AG theory" ... ?? ....
?? underlying just plain AG theory being .... ?? usual de-enrichment ??? .... ?? namely taking invariants subspace ... ??? ....
?? then also equipping model with frame of its underlying g-torsor ... ??? .... syntactic weak coproduct with g-rep-enriched theory of ... "a frame". i guess .... ??? which is what, syntactically ??? .... ??well, i guess that it is after all just something like ... "introducing the non-invariant elements in the enriched hom spaces as new morphisms" ... ?? "after all" because i had some vague guesses along those lines ... which, because of the way in which the new morphisms give rise as well to new objects via coequalizers ... leads me to think about ... ??? annoying / intriguing "locally internal category" appendix in baby elephant, and ideas connected in some way (in my mind ...) with that .... ????.... snow-globe / potemkin model ...
??? alternative (perhaps lax?) de-enrichment process here ??? ..... adjoint ... ??? .... "total vs fiber" ?? ... ?? .... not "preserving cocompleteness" ?? ... ???? .....
??? still bugged by level slip here ????? ..... ???? g acting on theory t, vs on t-model .... ??? and/or both, with "riding" ?? .... ??? "slipping/sliding along long exact homotopy sequence" ... ??? ...
?? simply "g-torsor (bundle/object)" after all ?? .... ??? .... x # [??cayley picture of g-torsor] -> y as "new/generalized,non-invariant morphism" .... ????? simply kleisli category ????? ...... ???? kleisli category and this "locally internal" / "snow-globe" / "potemkin" stuff ?????? ...... ?? kleisli, (and /) or co-kleisli ?? ... ??? ... ?? adjoining coequalizers of new morphisms sounds like kleisli rather than co-kleisli ??? ... i mean .... ??? .... ??? co-/kleisli vs co-/eilenberg-moore .... ??? ....
???? work out example of .... gl(1) acting on affine line .... ????? ....
k[x]-module ...
??? hmmm, maybe there really was a kleisli/co-kleisli duality flip just above, combined in some way with a long exact sequence level slip .... ??? "old vs new" confusion .... ??? affine line doesn't have interesting bundle of gl(1)-torsors, but orbit stack of tautological action of gl(1) on it does .... ???? ..... ???? ....
!! work this out !! ..... ??? ....
??? typical (?? ...) vector space with linear operator as ... ??? not turnable-into graded module of k[x]x .... ???
..... ???? .... ..... ???? .....
?? underlying just plain AG theory being .... ?? usual de-enrichment ??? .... ?? namely taking invariants subspace ... ??? ....
?? then also equipping model with frame of its underlying g-torsor ... ??? .... syntactic weak coproduct with g-rep-enriched theory of ... "a frame". i guess .... ??? which is what, syntactically ??? .... ??well, i guess that it is after all just something like ... "introducing the non-invariant elements in the enriched hom spaces as new morphisms" ... ?? "after all" because i had some vague guesses along those lines ... which, because of the way in which the new morphisms give rise as well to new objects via coequalizers ... leads me to think about ... ??? annoying / intriguing "locally internal category" appendix in baby elephant, and ideas connected in some way (in my mind ...) with that .... ????.... snow-globe / potemkin model ...
??? alternative (perhaps lax?) de-enrichment process here ??? ..... adjoint ... ??? .... "total vs fiber" ?? ... ?? .... not "preserving cocompleteness" ?? ... ???? .....
??? still bugged by level slip here ????? ..... ???? g acting on theory t, vs on t-model .... ??? and/or both, with "riding" ?? .... ??? "slipping/sliding along long exact homotopy sequence" ... ??? ...
?? simply "g-torsor (bundle/object)" after all ?? .... ??? .... x # [??cayley picture of g-torsor] -> y as "new/generalized,non-invariant morphism" .... ????? simply kleisli category ????? ...... ???? kleisli category and this "locally internal" / "snow-globe" / "potemkin" stuff ?????? ...... ?? kleisli, (and /) or co-kleisli ?? ... ??? ... ?? adjoining coequalizers of new morphisms sounds like kleisli rather than co-kleisli ??? ... i mean .... ??? .... ??? co-/kleisli vs co-/eilenberg-moore .... ??? ....
???? work out example of .... gl(1) acting on affine line .... ????? ....
k[x]-module ...
??? hmmm, maybe there really was a kleisli/co-kleisli duality flip just above, combined in some way with a long exact sequence level slip .... ??? "old vs new" confusion .... ??? affine line doesn't have interesting bundle of gl(1)-torsors, but orbit stack of tautological action of gl(1) on it does .... ???? ..... ???? ....
!! work this out !! ..... ??? ....
??? typical (?? ...) vector space with linear operator as ... ??? not turnable-into graded module of k[x]x .... ???
..... ???? .... ..... ???? .....
AG theory with syntactic category = k[x]-modules ... vs Z-graded k[x]-modules .... ???? ...
semantic :
g-torsor x
g-torsor x equipped with point y in f(x)
framed g-torsor x equipped with point y in f(x)
syntactic :
g-rep ...
?? g-action on (r,m) extending given g-action on r .... ???? .....
r-module m ... ?? ....
????? ...... ????? ....
semantic :
g-torsor x
g-torsor x equipped with point y in f(x)
framed g-torsor x equipped with point y in f(x)
syntactic :
g-rep ...
?? g-action on (r,m) extending given g-action on r .... ???? .....
r-module m ... ?? ....
????? ...... ????? ....
Wednesday, September 21, 2011
?? given comm monoid in AG environment of g-reps, consider "semi-direct product" ..... ???
?? given dimensional category enriched over g-reps ..... ?????? .....
?? given AG theory enriched over g-reps ....... ????? ......
?? x comm monoid in _g-rep_ ..... ??? then consider x-modules ...... ??????? .....
?? consider z/2 acting on k[x] by x |-> -x ... ??? ....
?? "k[x]-module equipped with ..." .... ????? .....
?????? ....... ...... ??? .....
?? re-express idea of "group acting on ring-module pair" in terms of module category autonomously, without relying on ring ...." ..... ???? .....
?? given dimensional category enriched over g-reps ..... ?????? .....
?? given AG theory enriched over g-reps ....... ????? ......
?? x comm monoid in _g-rep_ ..... ??? then consider x-modules ...... ??????? .....
?? consider z/2 acting on k[x] by x |-> -x ... ??? ....
?? "k[x]-module equipped with ..." .... ????? .....
?????? ....... ...... ??? .....
?? re-express idea of "group acting on ring-module pair" in terms of module category autonomously, without relying on ring ...." ..... ???? .....
Tuesday, September 20, 2011
?? so what _about_ "torically flat quasicoherent preseheaf over toric variety" ???? ....
??? property/structure confusion here ?????? ....... ?? hmmm .... cartesian monoidalness of functor as property, vs monoidalness of it as structure ??? ..... ???? ..... ... ??? .... ?? toric convolution as only "formerly cartesian" ??? .... ????? ......
??? vs toric quasicoherent sheaf ???? ...... ???? ..... ....
?? TAG morphism as generally _not_ flat (... ???? ... "cartesian-product-preserving" .... ???? .... ???? .... ?? equalizers .... ???? ....) ..... ???? .....so ..... ??? more suspicion about extent to which "toric convolution of hallmark of toric tannakian program" makes any sense .... ???? ..... ??? how (if at all ... ??? ...) preservation of toric convolution relates to "toric spectrum" .... ????? vs "flat spectrum" .... ????? ...... ?????? ......
??? still lots of confusion here .... ????? ......
???? .... ??? bialgebra ??? ...... ????? ...... ....
??? lax monoidalness ???? ...... ...
?? extremely restrictive for morphism from walking object AG theory to preserve toric convolution ????? ....... ????? ...... ????? ...... ?? "flat" .... ???? ... ???? ...
?? hmmm.... ??? f(x # y) vs f(x) # f(y) .... f(x # y -> x) .... ???? arbitrary functor between cartesian smc's as op-lax monoidal ??? .... ??? .... ??? not applicable here (?? ...) ??? ..... ???? ..... ??? ..... .... ??? ...
??? property/structure confusion here ?????? ....... ?? hmmm .... cartesian monoidalness of functor as property, vs monoidalness of it as structure ??? ..... ???? ..... ... ??? .... ?? toric convolution as only "formerly cartesian" ??? .... ????? ......
??? vs toric quasicoherent sheaf ???? ...... ???? ..... ....
?? TAG morphism as generally _not_ flat (... ???? ... "cartesian-product-preserving" .... ???? .... ???? .... ?? equalizers .... ???? ....) ..... ???? .....so ..... ??? more suspicion about extent to which "toric convolution of hallmark of toric tannakian program" makes any sense .... ???? ..... ??? how (if at all ... ??? ...) preservation of toric convolution relates to "toric spectrum" .... ????? vs "flat spectrum" .... ????? ...... ?????? ......
??? still lots of confusion here .... ????? ......
???? .... ??? bialgebra ??? ...... ????? ...... ....
??? lax monoidalness ???? ...... ...
?? extremely restrictive for morphism from walking object AG theory to preserve toric convolution ????? ....... ????? ...... ????? ...... ?? "flat" .... ???? ... ???? ...
?? hmmm.... ??? f(x # y) vs f(x) # f(y) .... f(x # y -> x) .... ???? arbitrary functor between cartesian smc's as op-lax monoidal ??? .... ??? .... ??? not applicable here (?? ...) ??? ..... ???? ..... ??? ..... .... ??? ...
?? past vs present attempts to express flag-geometry-flavor theories in various doctrines ...
dimensional doctrine ...
AG doctrine ...
geometric doctrine ...
?? somewhat confused about how far we got with some of this ... and what approaches we may have used .... ??? ....
??? "quasicoherent artin-wraith glueing" difficulties .... ???? .....
?? reps of galois-schubert lie subalgebra .... ???? .... ?? informally (?? ...) equating with nice conceptually/syntactically presented/axiomatized AG theory .... ??? and informally trying to understand reduction or near/partial reduction to dimensional doctrine .... ???? some confusion here ???? ....
"hecke" .... ???? .....
gl vs pgl ...
pgl(2) vs pgl(3) .... ??? .....
?? certain informal calculations that we did ... with alex and/or baez .... ??? .... confusion about what these were / attempted .... ??? ....
?? generalized pluecker relations .... ???? showing up in more than one way, maybe ??? .... various gradings ... treated in various ways .... projective, toric .... ??? .....
??? resorting to "subtraction of structure" (william burke ... gremlin ... "orbit stack" ... "gauge-fixing" ... "structure gambit" ... ??? ....) .... ???? extent to which we did so resort .... and / or succeed at it .... ????? ....
?? hmmm ... "g-torsor x equipped with point p of f(x)" for functor f ... ??? as not especially "gauge-fixing" / "structure gambit" ??? ..... ???? ..... ....
dimensional doctrine ...
AG doctrine ...
geometric doctrine ...
?? somewhat confused about how far we got with some of this ... and what approaches we may have used .... ??? ....
??? "quasicoherent artin-wraith glueing" difficulties .... ???? .....
?? reps of galois-schubert lie subalgebra .... ???? .... ?? informally (?? ...) equating with nice conceptually/syntactically presented/axiomatized AG theory .... ??? and informally trying to understand reduction or near/partial reduction to dimensional doctrine .... ???? some confusion here ???? ....
"hecke" .... ???? .....
gl vs pgl ...
pgl(2) vs pgl(3) .... ??? .....
?? certain informal calculations that we did ... with alex and/or baez .... ??? .... confusion about what these were / attempted .... ??? ....
?? generalized pluecker relations .... ???? showing up in more than one way, maybe ??? .... various gradings ... treated in various ways .... projective, toric .... ??? .....
??? resorting to "subtraction of structure" (william burke ... gremlin ... "orbit stack" ... "gauge-fixing" ... "structure gambit" ... ??? ....) .... ???? extent to which we did so resort .... and / or succeed at it .... ????? ....
?? hmmm ... "g-torsor x equipped with point p of f(x)" for functor f ... ??? as not especially "gauge-fixing" / "structure gambit" ??? ..... ???? ..... ....
toposes of toric quasicoherent sheaves
this talk is part of my "algebraic geometry for category-theorists" program, which is also what my talk at the conference that we just went to was about ... so that talk that i gave is a useful prerequisite for this talk ...
a big part of "algebraic geometry for category-theorists" is to convince category-theorists that, probably without realizing it, they've been engaging in certain prototypically algebraic-geometric activities for their whole category-theoretic lives ...
when i say "probably without realizing it", i'm extrapolating from my own experience ... my general method as to assume that everyone else made the same stupid mistakes that i did, and then try to offer gentle correction of those mistakes (?? bit about impulse to teach arising from constantly walking around in a daze muttering to yourself "if only i'd known _that_ when i was just starting out" ...) .... which can really confuse people if they never made those mistakes and/or never even had the chance to make them ... ??? ....
and of course it's good in a way when that (?? ...) sort of thing happens, that you find out that you already know something that you didn't know you knew .... because then you don't have to go through the bother of learning something really new; instead you just start taking credit for something you already knew .... ??? ....
so when is this alleged time in your category-theoretic lives that you've been engaging in a prototypically algebraico-geometric activity without realizing it? it's not when you might naively have guessed- that's why you probably didn't realize it ... instead, the time that i'm talking about is whenever you find yourself studying some kind of "structured categories" and thinking of them as "theories" of some kind of "logic" ... (?? and so forth .... thinking of the homomorphisms between such structured categories as "interpretations" or "models of the domain theory in the environment provided by the codomain theory" ...)
?? examples, with audience participation ..... ??? ....
?? bit of imagined history .... lawvere and theories, beck and doctrines .... ?? relatively contemporary ideas about some sort of 2-toposes .... ??? ..... where "my" doctrines fit in there ... "categorified gabriel-ulmer duality" ... categorification of "lex theories as special toposes" .... ???? ....
anyway that's a very general preamble, but today i'm hoping to actually get into interesting specifics ... describing in some detail a little part of this program .... so what i'm actually hoping to talk about today is how this (?? ...) big general tannakian philosophy / program specializes to a particular part of / topic in algebraic geometry known as _"toric varieties"_ ...
?? status of toric geometry as "easy toy example" in algebraic geometry ... ??? ....
?? actual (?? ...) toruses involved, vs "based on sets instead of vector spaces / abelian groups" .... latter approach as leading, perhaps not _too_ surprisingly, to involvement of toposes ....
???? so in fact, in addition to the whole big conceptual justification that i've just tried to give here, i have a secret entirely (?? ...) different justification for what i'm planning to talk about .... this alternative justification is that the examples of toposes that are going to turn up here happen to be what i consider somewhat amusing (?? ...) and educational examples of toposes .... they're what i informally (?? ...) might consider to be "the simplest examples of grothendieck toposes that aren't presheaf toposes" .....
?? so perhaps i should ask ... more audience participation .... your opinions about what sort of topos that might be ...
??? lots of different ways of thinking about toposes .... ?? might be interesting to hear about alternative ways ... ?? .....
?? filteredly cocontinuous .... ????? ....
??? various contrasting analogies (?? ...) from "topos" to ["abelian category" or some related concept ...] .... ??? ....
??? trying to formulate compatibility conditions ... ??? "combined doctrine" ... ??? exploit audience expertise ... ??? ....
?? bicommutative bialgebra .... ???? ......
this talk is part of my "algebraic geometry for category-theorists" program, which is also what my talk at the conference that we just went to was about ... so that talk that i gave is a useful prerequisite for this talk ...
a big part of "algebraic geometry for category-theorists" is to convince category-theorists that, probably without realizing it, they've been engaging in certain prototypically algebraic-geometric activities for their whole category-theoretic lives ...
when i say "probably without realizing it", i'm extrapolating from my own experience ... my general method as to assume that everyone else made the same stupid mistakes that i did, and then try to offer gentle correction of those mistakes (?? bit about impulse to teach arising from constantly walking around in a daze muttering to yourself "if only i'd known _that_ when i was just starting out" ...) .... which can really confuse people if they never made those mistakes and/or never even had the chance to make them ... ??? ....
and of course it's good in a way when that (?? ...) sort of thing happens, that you find out that you already know something that you didn't know you knew .... because then you don't have to go through the bother of learning something really new; instead you just start taking credit for something you already knew .... ??? ....
so when is this alleged time in your category-theoretic lives that you've been engaging in a prototypically algebraico-geometric activity without realizing it? it's not when you might naively have guessed- that's why you probably didn't realize it ... instead, the time that i'm talking about is whenever you find yourself studying some kind of "structured categories" and thinking of them as "theories" of some kind of "logic" ... (?? and so forth .... thinking of the homomorphisms between such structured categories as "interpretations" or "models of the domain theory in the environment provided by the codomain theory" ...)
?? examples, with audience participation ..... ??? ....
?? bit of imagined history .... lawvere and theories, beck and doctrines .... ?? relatively contemporary ideas about some sort of 2-toposes .... ??? ..... where "my" doctrines fit in there ... "categorified gabriel-ulmer duality" ... categorification of "lex theories as special toposes" .... ???? ....
anyway that's a very general preamble, but today i'm hoping to actually get into interesting specifics ... describing in some detail a little part of this program .... so what i'm actually hoping to talk about today is how this (?? ...) big general tannakian philosophy / program specializes to a particular part of / topic in algebraic geometry known as _"toric varieties"_ ...
?? status of toric geometry as "easy toy example" in algebraic geometry ... ??? ....
?? actual (?? ...) toruses involved, vs "based on sets instead of vector spaces / abelian groups" .... latter approach as leading, perhaps not _too_ surprisingly, to involvement of toposes ....
???? so in fact, in addition to the whole big conceptual justification that i've just tried to give here, i have a secret entirely (?? ...) different justification for what i'm planning to talk about .... this alternative justification is that the examples of toposes that are going to turn up here happen to be what i consider somewhat amusing (?? ...) and educational examples of toposes .... they're what i informally (?? ...) might consider to be "the simplest examples of grothendieck toposes that aren't presheaf toposes" .....
?? so perhaps i should ask ... more audience participation .... your opinions about what sort of topos that might be ...
??? lots of different ways of thinking about toposes .... ?? might be interesting to hear about alternative ways ... ?? .....
?? filteredly cocontinuous .... ????? ....
??? various contrasting analogies (?? ...) from "topos" to ["abelian category" or some related concept ...] .... ??? ....
??? trying to formulate compatibility conditions ... ??? "combined doctrine" ... ??? exploit audience expertise ... ??? ....
?? bicommutative bialgebra .... ???? ......
?? "toric convolution" on AG theory coming from TAG theory .... fe theory of an object ... "place-wise" tensor product of reps of symmetric groups .... ???? .... terry bisson ... ??? ..... "toric flatness" .... ??? ..... young diagram calculus .... ???? ....
?? preservation of toric convolution, and "ordinary spectrum vs toric spectrum of toric variety" .... ???? .... ?? whole big "spectrum" / "globalization" / "kan extension" bit here ... ??? .... ...
(?? vs "k1 spectrum vs k2 spectrum of k2 AG theory, with k1 -> k2" ??? ..... ????? ..... ...)
??? toric convolution of formulas, vs partially-defined toric product of models .... ????? .....
?? idea that progression from "algebras" to their (structured ...) "module" (?? ...) categories tends to promote good geometric colimits .... ??? funny additional echo of this in toric case ??? .... ????? ..... applied to extra toric convolution tensor product ...
?? preservation of toric convolution, and "ordinary spectrum vs toric spectrum of toric variety" .... ???? .... ?? whole big "spectrum" / "globalization" / "kan extension" bit here ... ??? .... ...
(?? vs "k1 spectrum vs k2 spectrum of k2 AG theory, with k1 -> k2" ??? ..... ????? ..... ...)
??? toric convolution of formulas, vs partially-defined toric product of models .... ????? .....
?? idea that progression from "algebras" to their (structured ...) "module" (?? ...) categories tends to promote good geometric colimits .... ??? funny additional echo of this in toric case ??? .... ????? ..... applied to extra toric convolution tensor product ...
Monday, September 19, 2011
?? hmmm .... idea that toric case of tannakian philosophy is about extra toric convolution tensor product ... ??? perhaps valid to some extent, but ... ?? don't forget _un_combined doctrine .... pure TAG doctrine .... just symmetric monoidal cocomplete category .... cartesian product not in TAG doctrine, so .... ??? no toric convolution .... ??? .... ??? so ... ???? really does seem to be true that TAG + niceness associated with "flatness" and "accidental topos" leads towards some genuine (?? ...) sort of "toric"-ness ??? .... "toric convolution" ?? .... ??? pushing envelope here with stackiness ??? ... ????
??? trying to adapt thoughts here to apply in other cases .... ???? AG instead of TAG .... ?????
??bit about tensor product on accidental topos as .... ???? how did it go ???? ..... ??? extra left adjoint of essential geometric morphism .... ???? adapting to AG case vs adapting to "TAG case from toric convolution viewpoint" ... ??? ....
????? accidental quasitopos ????? ..... ???? still analog of "toric convolution" .... ?????? ...... ???? ..... ....
??? recovering quasitopos from coalgebras in it ???? ...... ???? ....
?? hmmm .... ?? back when we were first (?? ...) noticing the existence of nice (?? ...) non-abelian cocomplete algebroids .... ???? and considering free monoidal and free symmetric monoidal enhancements of such .... ?? had feeling about .... seemingly "minor" variation on "hopf alg" / "bialg" / "algebraic group / monoid" that nevertheless seemed also impressively weird ... ?? ...
??? "filtered ..." .... ???? ....
... walking epi .... ???? .....
??? trying to adapt thoughts here to apply in other cases .... ???? AG instead of TAG .... ?????
??bit about tensor product on accidental topos as .... ???? how did it go ???? ..... ??? extra left adjoint of essential geometric morphism .... ???? adapting to AG case vs adapting to "TAG case from toric convolution viewpoint" ... ??? ....
????? accidental quasitopos ????? ..... ???? still analog of "toric convolution" .... ?????? ...... ???? ..... ....
??? recovering quasitopos from coalgebras in it ???? ...... ???? ....
?? hmmm .... ?? back when we were first (?? ...) noticing the existence of nice (?? ...) non-abelian cocomplete algebroids .... ???? and considering free monoidal and free symmetric monoidal enhancements of such .... ?? had feeling about .... seemingly "minor" variation on "hopf alg" / "bialg" / "algebraic group / monoid" that nevertheless seemed also impressively weird ... ?? ...
??? "filtered ..." .... ???? ....
... walking epi .... ???? .....
Sunday, September 18, 2011
?? theory of x-building ew triple of flags with first two generically oriented ... generalized pluecker algebra bi-multigraded ...
?? theory of x-building ew triple of flags with each two generically oriented ... ??? 3! symmetry ... ?? localize previous example ...
?? theory of x-building ew apartment .... kaleidoscope group and maximal torus ...
?? x-building vs ... ??? x-building with extra stuff .... torsor of simply-connected form .... ??? ....
?? hmmm, still didn't straighten out .... ???? irrelevant models here .... ????? ....
?? stupid guess about a2 toric variety and "anti-flag" ... ??? doesn't even seem to generalize to a3 ... ??? ...
ad + be + cf = 0 ....
... but that's the flag constraint; we wanted anti-flag ... ??? ....
?? theory of x-building ew triple of flags with each two generically oriented ... ??? 3! symmetry ... ?? localize previous example ...
?? theory of x-building ew apartment .... kaleidoscope group and maximal torus ...
?? x-building vs ... ??? x-building with extra stuff .... torsor of simply-connected form .... ??? ....
?? hmmm, still didn't straighten out .... ???? irrelevant models here .... ????? ....
?? stupid guess about a2 toric variety and "anti-flag" ... ??? doesn't even seem to generalize to a3 ... ??? ...
ad + be + cf = 0 ....
... but that's the flag constraint; we wanted anti-flag ... ??? ....
generalized toric quasicoherent sheaves .... concretely looking at the genuinely generalized examples .... ????....
compatibility between ordinary tensor product and toric convolution from bicommutative bialgebra viewpoint .... ???? relationship to bit about geometric morphism and generalized day convolution .... ???? .... ??? coalgebra and geometric morphism ... ???? .... ...
compatibility between ordinary tensor product and toric convolution from bicommutative bialgebra viewpoint .... ???? relationship to bit about geometric morphism and generalized day convolution .... ???? .... ??? coalgebra and geometric morphism ... ???? .... ...
Friday, September 16, 2011
?? what happened with getting toric varieties (not necessarily canonical kaleidoscopic) from a2 flag variety ?? ... did we give up on it for some sensible reason ??? .... ?? whereas ... ??? kept trying to work out b2 example .... ???? ...
?? well, i might have decided that it was too trivial, or already worked out, or something ... ?? but ... ?? maybe actually at least slightly interesting ... ???.... ?? projective line's worth of flag-flag-flag orientations where they're generic 2 at a time ?? .... ???.... ??? schubert calculus structure constant ??? ... categorified .... ??? .... ?? well, looks at least superficially like a projective line .... ??? .... ?? compactness paradox here ??? .... ???? ..... ....
?? "motive" here ?? .... ?? categorified term in structure constant .... ???? .....
?? "anti-flag" ... ??? "point as far off line as possible" ....center of triangle and line at infinity ... ??? .... ??? degeneration ??? ....
?? more generally ...a_n flag variety's worth of a_[n+1] flag-flag-flag orientations when they're generic 2 at a time ??? ..... ???? .... ?? .... ??? maybe not correct ... ??? .... ???? ....
?? well, i might have decided that it was too trivial, or already worked out, or something ... ?? but ... ?? maybe actually at least slightly interesting ... ???.... ?? projective line's worth of flag-flag-flag orientations where they're generic 2 at a time ?? .... ???.... ??? schubert calculus structure constant ??? ... categorified .... ??? .... ?? well, looks at least superficially like a projective line .... ??? .... ?? compactness paradox here ??? .... ???? ..... ....
?? "motive" here ?? .... ?? categorified term in structure constant .... ???? .....
?? "anti-flag" ... ??? "point as far off line as possible" ....center of triangle and line at infinity ... ??? .... ??? degeneration ??? ....
?? more generally ...a_n flag variety's worth of a_[n+1] flag-flag-flag orientations when they're generic 2 at a time ??? ..... ???? .... ?? .... ??? maybe not correct ... ??? .... ???? ....
projective geometry = dimensional analysis (50 minutes)
?? lies to be admitted ....
1 omission of stacky points .... ???? ...
2 projective vs multi-projective ... ??? ...
3 ??? .... realness, positiveness ....... ???? ..... ???? ....
dictionary ...
dimension = line bundle
quantity of given dimension = section of line bundle
?? theory = ... ??? theory
?? "choice of units" = "affine open neighborhood" ????? ......
(?????? hmmmmm ..... ?? confusion here about inverting vs setting to 1 ... ??? .... maybe sort of ok ???.... no 1 to set to in non-endo hom-space ..... ???? ..... ??? in affine case, relationship between closed subspace obtained by setting various quantities to 1, and open subspace obtained by inverting them ??? .... any sort of "homotopy equivalence" here ???? .... ??? ..... "neighborhood" .... ???? ..... wait a minute, "localizing at a point" ....??? possibly a somewhat "generic" point .... ????? ...... ??? conservative/localization factorization .... ???? localization as flat ...... ???? anti-flat ???? ..... ????? ...... ??? "invert all the quantities inverted by f" .... =?= "invert all the quantities set to 1 by f" ????? ..... (?? hmmm .... ?? no ??? .... ???? pure localization as not setting much to 1, but inverting stuff .... ??? .... ??? geometric interpretation ... ?? "remove the zero-points of all functions that are 1 on the closed subvariety" .... ????? doesn't that sound like localization ????? ..... ?? hmmm, maybe if you do "remove the zeros of ones ..." to an inclusion of a _closed subvariety_, then it _is_ localization ??? ..... more generally maybe localization of zariski closure ????? ...... ???? semi-recent bit about .... ??? reflecting isomorphisms vs reflecting limits or colimits or something ???? ...... non-basic opens .... ????? .....) ..... ??? localization as "substitute" for non-flat morphism .... ????? ......... ????? ..... doctrine ..... combined doctrine ...... classification of flat .... ??? ..... snow-globe / potemkin .....)
?? .....
?? algebraic/analytic vs synthetic .... ???? ... ?? rep th .... ???? ....
?? example baez cooked up ...
?? lawvere's usage of "theory" ...
?? speed of light as section of line bundle "distance tensor time^*" .... ??? ....
?? timing of "response" ??? ...
?? broad (?? and / or somewhat heterodox ?? ...) interpretation of "tannakian philosophy" ... ??? ...
brandenburg ...
??? "while i was planning this talk, i was reminded that this word "scattering" is one of those obscurantist words that physicists use to obscure the real nature of their ideas from outsiders and probably from themselves as well, and that it might be interesting to try sometime to translate this word into plain english ... but that's _not_ what this talk is about ... for purposes of this talk you're probably just supposed to think of "scattering" as some sort of allegedly cool physics jargon ..."
i'd like to start with an example of dimensional analysis .... asked baez to cook up .... scattering (digression above? ...) .... ?? dimensional analysis as member of loose family of "type discipline" / "structured thinking" ideas .... ???
(?? confusion (on my part ....) about ungraded vs graded, vs graded vs multi-graded ....... ????? ..... 0 vs 1 vs 1 vs more .... ???? ..... ... ??? "logic" ... ??? predicate vs propostional .... vs mono-typed vs typed ..... ????? ..... ?? lawvere ... theory as category ... object as type .... ???? ....)
?? lies to be admitted ....
1 omission of stacky points .... ???? ...
2 projective vs multi-projective ... ??? ...
3 ??? .... realness, positiveness ....... ???? ..... ???? ....
dictionary ...
dimension = line bundle
quantity of given dimension = section of line bundle
?? theory = ... ??? theory
?? "choice of units" = "affine open neighborhood" ????? ......
(?????? hmmmmm ..... ?? confusion here about inverting vs setting to 1 ... ??? .... maybe sort of ok ???.... no 1 to set to in non-endo hom-space ..... ???? ..... ??? in affine case, relationship between closed subspace obtained by setting various quantities to 1, and open subspace obtained by inverting them ??? .... any sort of "homotopy equivalence" here ???? .... ??? ..... "neighborhood" .... ???? ..... wait a minute, "localizing at a point" ....??? possibly a somewhat "generic" point .... ????? ...... ??? conservative/localization factorization .... ???? localization as flat ...... ???? anti-flat ???? ..... ????? ...... ??? "invert all the quantities inverted by f" .... =?= "invert all the quantities set to 1 by f" ????? ..... (?? hmmm .... ?? no ??? .... ???? pure localization as not setting much to 1, but inverting stuff .... ??? .... ??? geometric interpretation ... ?? "remove the zero-points of all functions that are 1 on the closed subvariety" .... ????? doesn't that sound like localization ????? ..... ?? hmmm, maybe if you do "remove the zeros of ones ..." to an inclusion of a _closed subvariety_, then it _is_ localization ??? ..... more generally maybe localization of zariski closure ????? ...... ???? semi-recent bit about .... ??? reflecting isomorphisms vs reflecting limits or colimits or something ???? ...... non-basic opens .... ????? .....) ..... ??? localization as "substitute" for non-flat morphism .... ????? ......... ????? ..... doctrine ..... combined doctrine ...... classification of flat .... ??? ..... snow-globe / potemkin .....)
?? .....
?? algebraic/analytic vs synthetic .... ???? ... ?? rep th .... ???? ....
?? example baez cooked up ...
?? lawvere's usage of "theory" ...
?? speed of light as section of line bundle "distance tensor time^*" .... ??? ....
?? timing of "response" ??? ...
?? broad (?? and / or somewhat heterodox ?? ...) interpretation of "tannakian philosophy" ... ??? ...
brandenburg ...
??? "while i was planning this talk, i was reminded that this word "scattering" is one of those obscurantist words that physicists use to obscure the real nature of their ideas from outsiders and probably from themselves as well, and that it might be interesting to try sometime to translate this word into plain english ... but that's _not_ what this talk is about ... for purposes of this talk you're probably just supposed to think of "scattering" as some sort of allegedly cool physics jargon ..."
i'd like to start with an example of dimensional analysis .... asked baez to cook up .... scattering (digression above? ...) .... ?? dimensional analysis as member of loose family of "type discipline" / "structured thinking" ideas .... ???
(?? confusion (on my part ....) about ungraded vs graded, vs graded vs multi-graded ....... ????? ..... 0 vs 1 vs 1 vs more .... ???? ..... ... ??? "logic" ... ??? predicate vs propostional .... vs mono-typed vs typed ..... ????? ..... ?? lawvere ... theory as category ... object as type .... ???? ....)
Thursday, September 15, 2011
?? possibility of allowing "toric convolution" to get only braided instead of strictly symmetric ??? ....
??problematic in a bunch of ways ... ?? association between toric convolution and cartesian product ... importance of antipode in quasitriangular hopf algebra ... ????? ..... ???? .... ?? involvement of quasitriangular structure with both multiplication and comultiplication ?? ..... ????? ...... ??? ...
??? ......
??problematic in a bunch of ways ... ?? association between toric convolution and cartesian product ... importance of antipode in quasitriangular hopf algebra ... ????? ..... ???? .... ?? involvement of quasitriangular structure with both multiplication and comultiplication ?? ..... ????? ...... ??? ...
??? ......
?? euclidean algorithm ... infinite loop ... loss of innocence ... henceforth algorithms would crash on a regular basis ... ??
?? symmetries made to be broken ... ??humans will try to apply them even when they're broken ... ??? sometimes it even works .... tic-tac-toe .... bugged me my inability to draw perfectly straight lines, perfectly symmetric pictures ... but the symmetry application worked anyway ... in the abstract context .... ????? ....
?? symmetries made to be broken ... ??humans will try to apply them even when they're broken ... ??? sometimes it even works .... tic-tac-toe .... bugged me my inability to draw perfectly straight lines, perfectly symmetric pictures ... but the symmetry application worked anyway ... in the abstract context .... ????? ....
?? what happens to idea of superman-flavor equivalence between _nice TAG theory_ (in form of accidental topos ... ??? ...) and _nice AG theory with extra toric convolution tensor product_ when ... ??? the first tensor product is omitted, or perhaps identified with the extra one ?? .....
?? encoding topos as special kind of AG theory, from which it can be recovered by taking the nice comonoids in the AG theory ??? .....
??construing above as special case of encoding ringed topos as AG theory ... ???? .... from which it can be recovered by taking the nice comonoids, and taking the special ring object to be the unit-object-as-coalgebra ... ?? afterwards can ask whether the ring object is "the integers", which is supposed to tell you that it's the original special case .... ???....
?? relative toric variety .... ??? ....
?? this original special case as distinguishing the toric convolution product from the "first" tensor product ??? .... ???? .... ????? ...... ??????? ........ .........
??? abstract generalization (?? ...) of "small zariski topos" here ????? ...... ??? .... ?? "spectrum" in tierney/johnstone sense ??? .... ???? ....
??? quasicoherence here (?? ...) ?? .....
?? nice coalgebras of a commutative ring ......for example nice coalgebras of k[x] .... ???? .... ?? given an object s in the small zariski topos of a commutative ring r .... try to get nice coalgebra of r .... ????? .... ?? take free r'-module on s where r' is the "structure sheaf" .... ?? _will_ this generally be quasicoherent ??? ..... i think no, but don't really remember at the moment how this works .... ??? .... hmmmm ......
?? "quasicoherentization" ... ??? ...
?? possibility we had the right idea above only up to the point where we tried generalizing from topos to ringed topos .... ???? ....
?? encoding topos as special kind of AG theory, from which it can be recovered by taking the nice comonoids in the AG theory ??? .....
??construing above as special case of encoding ringed topos as AG theory ... ???? .... from which it can be recovered by taking the nice comonoids, and taking the special ring object to be the unit-object-as-coalgebra ... ?? afterwards can ask whether the ring object is "the integers", which is supposed to tell you that it's the original special case .... ???....
?? relative toric variety .... ??? ....
?? this original special case as distinguishing the toric convolution product from the "first" tensor product ??? .... ???? .... ????? ...... ??????? ........ .........
??? abstract generalization (?? ...) of "small zariski topos" here ????? ...... ??? .... ?? "spectrum" in tierney/johnstone sense ??? .... ???? ....
??? quasicoherence here (?? ...) ?? .....
?? nice coalgebras of a commutative ring ......for example nice coalgebras of k[x] .... ???? .... ?? given an object s in the small zariski topos of a commutative ring r .... try to get nice coalgebra of r .... ????? .... ?? take free r'-module on s where r' is the "structure sheaf" .... ?? _will_ this generally be quasicoherent ??? ..... i think no, but don't really remember at the moment how this works .... ??? .... hmmmm ......
?? "quasicoherentization" ... ??? ...
?? possibility we had the right idea above only up to the point where we tried generalizing from topos to ringed topos .... ???? ....
?? "geometric plethysm" .... ??? picking out projectively embedded partial flag variety ... ???
"generalized clebsch-gordan decomposition as finer as more structure is given ...." ...
??? confusion between "invariant tensor" and "invariant subspace" ... ???? .... relationship ... ??? ....
?? relators for multi-homogeneous coordinate algebra of partial flag variety .... ???? .....
..... ???? ....
"generalized clebsch-gordan decomposition as finer as more structure is given ...." ...
??? confusion between "invariant tensor" and "invariant subspace" ... ???? .... relationship ... ??? ....
?? relators for multi-homogeneous coordinate algebra of partial flag variety .... ???? .....
..... ???? ....
Wednesday, September 14, 2011
?? powers of 7 mod 10 ...
1,7,9,3
?? powers of 10 mod 7 ...
1,3,2,6,4,5
?? ... so what's the relationship to 1,4,2,8,5,7 ??
hmm, at least i see why i tried the first one first ... because clearly 1,4,2,8,5,7 live mod 10 rather than mod 7 ... but so what's going on here ??? _is_ it some sort of "reciprocity" ?? .... ???? .....
obviously very "simple" stuff going on here, but never learned about it, as far as i can remember ...
7 goes into 10 once, with remainder 3 .... ??? hmm, maybe 3,2,6,4,5,1 are remainders corresponding to truncated quotients 1,14,142,1428,14285,142857 ???
10/7,100/7,1000/7,10000/7,100000/7,1000000/7
?? somehow you're done when you get to a remainder of 1 ... 999999/7 = 142857 ....
?? harmonic series ... ??? .....
?? remainders live modulo 7 .... ?? but then look at the truncated quotients mod 10 .... ??? remainders wrt 10 of truncated quotients wrt 7 .... ??? ..... ???? ..... ??? ....
?? analogy between polynomials over z/n (especially with n prime ?? ...) and integers .... ??? but .... actual map from former to latter .... ????? parody of evaluation at zero .... ????? ...... ?????? .... ??? map other way .... ???? ..... ???? ..... ?? mediated by .... polynomials over z .... ???? ......
1,7,9,3
?? powers of 10 mod 7 ...
1,3,2,6,4,5
?? ... so what's the relationship to 1,4,2,8,5,7 ??
hmm, at least i see why i tried the first one first ... because clearly 1,4,2,8,5,7 live mod 10 rather than mod 7 ... but so what's going on here ??? _is_ it some sort of "reciprocity" ?? .... ???? .....
obviously very "simple" stuff going on here, but never learned about it, as far as i can remember ...
7 goes into 10 once, with remainder 3 .... ??? hmm, maybe 3,2,6,4,5,1 are remainders corresponding to truncated quotients 1,14,142,1428,14285,142857 ???
10/7,100/7,1000/7,10000/7,100000/7,1000000/7
?? somehow you're done when you get to a remainder of 1 ... 999999/7 = 142857 ....
?? harmonic series ... ??? .....
?? remainders live modulo 7 .... ?? but then look at the truncated quotients mod 10 .... ??? remainders wrt 10 of truncated quotients wrt 7 .... ??? ..... ???? ..... ??? ....
?? analogy between polynomials over z/n (especially with n prime ?? ...) and integers .... ??? but .... actual map from former to latter .... ????? parody of evaluation at zero .... ????? ...... ?????? .... ??? map other way .... ???? ..... ???? ..... ?? mediated by .... polynomials over z .... ???? ......
"?? using zero quasicoherent sheaf as "undefined" succeeds in making global toric convolution tensor product well-defined ??? ...."
?? well, bit more subtle (but also straightforward ... in some sense ...) than that ... global toric convolution as colimit wrt fan of local such ... ?? ...
?? superman-flavor equivalence theorem ... ?? role in pushing / establishing boundary of definition ... ?? cases at hand ... nice accidental topos ... and so forth ... ?? ...
?? "toric convolution" ... "generalized day convolution" ..... ????? ...... ??? bialgebra duality .... ??? .... ??? "generalized toric convolution" .... ????? ....
?? transport bialgebra duality across tannakian bridge ?? ...
?? well, bit more subtle (but also straightforward ... in some sense ...) than that ... global toric convolution as colimit wrt fan of local such ... ?? ...
?? superman-flavor equivalence theorem ... ?? role in pushing / establishing boundary of definition ... ?? cases at hand ... nice accidental topos ... and so forth ... ?? ...
?? "toric convolution" ... "generalized day convolution" ..... ????? ...... ??? bialgebra duality .... ??? .... ??? "generalized toric convolution" .... ????? ....
?? transport bialgebra duality across tannakian bridge ?? ...
?? extra "toric convolution" tensor product ... modules of commutative bialgebra .... ??? "bialgebra dual" here ??? ... ??? what happens if you try to "switch roles of the two tensor products ("commutative" and "comultiplication" ...) without dualizing thre bialgebra" ???
??? superman-flavor equivalence theorem between "bialgebra" (?? ...) and "nice monoidal category" .... ???.... what happens to such in "globalized" context ?? .... ???? "generalized day convolution" ????? .....
??set-based vs vsp-based such (? ...) theorem .... ?? ... day convolution ... ?? "tannaka-krein" ??? ... .... ??? "set-based tannaka-krein / tannakian program / philosophy" ... ?? well, that _is_ sort of what this (...) toric stuff is about .... ??? ....
?? some sort of unification of "grothendieck topology" and "specification of filtered colimits to be preserved" ??? .... ???? ....
?? generalized day convolution and ... ??? superman-flavor equivalence theorem between tensor product operation on nice category and "virtual" such on corresponding "site" ??? ..... ??? .... ??? ...
??? superman-flavor equivalence theorem between "bialgebra" (?? ...) and "nice monoidal category" .... ???.... what happens to such in "globalized" context ?? .... ???? "generalized day convolution" ????? .....
??set-based vs vsp-based such (? ...) theorem .... ?? ... day convolution ... ?? "tannaka-krein" ??? ... .... ??? "set-based tannaka-krein / tannakian program / philosophy" ... ?? well, that _is_ sort of what this (...) toric stuff is about .... ??? ....
?? some sort of unification of "grothendieck topology" and "specification of filtered colimits to be preserved" ??? .... ???? ....
?? generalized day convolution and ... ??? superman-flavor equivalence theorem between tensor product operation on nice category and "virtual" such on corresponding "site" ??? ..... ??? .... ??? ...
?? non-toric analog of [ ?? ... loop periodicity and / or tail length constraints having effect of "trivializing global aspect ..." ... ??? ...] ... ???? ... ..... ???? ...
?? weil conjectures .... ???? ..... ??? ....
?? points of toric varieties over finite fields .... again ... ??? .... ?? also archimedean ..... ????? .....
?? weil conjectures .... ???? ..... ??? ....
?? points of toric varieties over finite fields .... again ... ??? .... ?? also archimedean ..... ????? .....
?? using zero quasicoherent sheaf as "undefined" succeeds in making global toric convolution tensor product well-defined ??? ....
?? equivalence of doctrines between ... "nice AG theory + toric convolution" and "nice TAG theory" ?? ... ?? .... ..... ?? recovering toric convolution tensor product on abelian group objects in accidental topos by ..... ????? .... ?? hmmm, might this be the ordinary tensor product of abelian group objects in a topos, about which we've remarked before ... i mean in the accidental case .... ??? .... ???? ....
?? generalized toric quasicoherent sheaves here ??? .....
?? equivalence of doctrines between ... "nice AG theory + toric convolution" and "nice TAG theory" ?? ... ?? .... ..... ?? recovering toric convolution tensor product on abelian group objects in accidental topos by ..... ????? .... ?? hmmm, might this be the ordinary tensor product of abelian group objects in a topos, about which we've remarked before ... i mean in the accidental case .... ??? .... ???? ....
?? generalized toric quasicoherent sheaves here ??? .....
Tuesday, September 13, 2011
?? "tangent space" of AG theory t at model m .... ?? as group object in _vsp_ ??? ... object as model over d with iso to m at basepoint .... ?? addition of objects ?? ... ?? addition of morphisms ?? ...
?? "deformation cohomology" ... ???? ...
?? "quasicoherent artin-wraith glueing" ....
?? "serre subcategory / quotient category" .... ??? ......
?? "ext as higher hom" ... ??? ....
?? representing extension by cocycle ( ?? ...) .... ???? representing morphism between such extensions by (?? higher? ...) cochain (?? ....) .... ??? ....
kernel _and_ cokernel .... ??? "double spectrum" ??? .... ??? ....
?? "deformation cohomology" ... ???? ...
?? "quasicoherent artin-wraith glueing" ....
?? "serre subcategory / quotient category" .... ??? ......
?? "ext as higher hom" ... ??? ....
?? representing extension by cocycle ( ?? ...) .... ???? representing morphism between such extensions by (?? higher? ...) cochain (?? ....) .... ??? ....
kernel _and_ cokernel .... ??? "double spectrum" ??? .... ??? ....
remove ....
??? "spectrum" .... "cayley-hamilton" .... ??? ..... "zeros of characteristic polynomial" .... ??? infinite-dimensional case ... purely algebraic .... determinant ..... ????? ...... bootstrap ..... ???? ...... tempting false proof of .... ?? cayley-hamilton ?? .... ?? "spectral theorem" ??? ..... ???? ..... ???? ....
pair of commuting operators ....
triple of commuting operators, and grading .... ??? .... hmmmm .... category .... diagram .... "non-commutative geometry" .... ???? ...... category ring ... heisenberg / connes .... ?? "non-commutative" in sense of ... ?? context where "commute" doesn't even parse ... ??? .... ??? algebraic projective geometry / "global algebraic geometry" as maybe aspect of "non-commutative geometry" ??? .... ???? .....
?? anyway, trying to link "removal of point from spectrum of commutative ring" with some definition of "spectrum of operator" involving ... ????? non-invertibility of determinant .... ????? ....
?? characteristic polynomial of operator f as ... ???? polynomial in variable t .... ???? determinant of f - t*[identity operator] ??? ... ??? but how to generalize to arbitrary-dim case ???? ...... ?????? consider kernel and/or cokernel of f - t*[identity operator] ???????? ........ ???????? ..... ???? ....
???? resultant ...... ????? ..... hmmmmm ..... ???? .....
?? coherent sheaf as living over localization iff certain fiber (?? ...) is "trivial" ??? .... .... ????? .....
?? mutual kernel of commuting operators .... ???? co-mutual cokernel ?????? ..... ????....
kernel of homomorphism vs of operator .... confusion ... ??? .... ideal vs fiber .... ?????? ...... ????? ....... ?? hmm, relationship to mystical stuff about "ideal" and "divisor" .... ????? ..... underlying module of (?? good ... "divisorish" ... ??) ideal as "twisted over corresponding codimension 1 subvariety" .... ???? ..... ??? ..... ??? more general codimension ???? ...... ???? .....
?? "kernel and / or cokernel" here as reminding me of "thick subcategory" .... ???? and i was also going to say "quasicoherent artin-wraith glueing", but .... ????? ....
?? "fringe functor" given by some sort of "projective cover" ??? .... ????? .... ??? .... ?? benson ... ??? ....
???? ... ??? also ... ?? "index theorem" ... ???? ....
??? "spectrum" .... "cayley-hamilton" .... ??? ..... "zeros of characteristic polynomial" .... ??? infinite-dimensional case ... purely algebraic .... determinant ..... ????? ...... bootstrap ..... ???? ...... tempting false proof of .... ?? cayley-hamilton ?? .... ?? "spectral theorem" ??? ..... ???? ..... ???? ....
pair of commuting operators ....
triple of commuting operators, and grading .... ??? .... hmmmm .... category .... diagram .... "non-commutative geometry" .... ???? ...... category ring ... heisenberg / connes .... ?? "non-commutative" in sense of ... ?? context where "commute" doesn't even parse ... ??? .... ??? algebraic projective geometry / "global algebraic geometry" as maybe aspect of "non-commutative geometry" ??? .... ???? .....
?? anyway, trying to link "removal of point from spectrum of commutative ring" with some definition of "spectrum of operator" involving ... ????? non-invertibility of determinant .... ????? ....
?? characteristic polynomial of operator f as ... ???? polynomial in variable t .... ???? determinant of f - t*[identity operator] ??? ... ??? but how to generalize to arbitrary-dim case ???? ...... ?????? consider kernel and/or cokernel of f - t*[identity operator] ???????? ........ ???????? ..... ???? ....
???? resultant ...... ????? ..... hmmmmm ..... ???? .....
?? coherent sheaf as living over localization iff certain fiber (?? ...) is "trivial" ??? .... .... ????? .....
?? mutual kernel of commuting operators .... ???? co-mutual cokernel ?????? ..... ????....
kernel of homomorphism vs of operator .... confusion ... ??? .... ideal vs fiber .... ?????? ...... ????? ....... ?? hmm, relationship to mystical stuff about "ideal" and "divisor" .... ????? ..... underlying module of (?? good ... "divisorish" ... ??) ideal as "twisted over corresponding codimension 1 subvariety" .... ???? ..... ??? ..... ??? more general codimension ???? ...... ???? .....
?? "kernel and / or cokernel" here as reminding me of "thick subcategory" .... ???? and i was also going to say "quasicoherent artin-wraith glueing", but .... ????? ....
?? "fringe functor" given by some sort of "projective cover" ??? .... ????? .... ??? .... ?? benson ... ??? ....
???? ... ??? also ... ?? "index theorem" ... ???? ....
?? quasicoherent sheaf over toric variety that's (at least locally ... ?? ...) toric convolution comonoidal, and also ordinary pointwise tensor monoidal, or comonidal, or whatever ... ???? ....
case where convolution comonoidal structure is/n't "discrete" .... ???? ....
??? invertibility wrt toric convolution tensor product .... ????
?? relationship to generalized ("translational" ... ?? ...) morphism of toric varieties .... ??? ....
?? seeing via "ordinary AG interpretation of underlying ordinary quasicoherent sheaf of toric quasicoherent sheaf" how concept of "toricness" of quasicoherent sheaf depends on "basepoint(s?)" of toric variety .... ??? .... ?? but then idea of "allowing translates of toric quasicoherent sheaves" (?? ...), and how that might relate to other (more "TAG-intrinsic" ... ???...) ideas about what happens to accidental topos when TAG is "mad emore translation-independent" by using generalized morphisms of toric varieties ...
??? toric convolution of torically equivariant stuff .... ??? decategorifies to "convolution of constant functions" ???? ..... which sounds weirdly degenerate, but .... ??? maybe more interesting categorified ?? .... ??? ....
???? _are_ (?? some ?? ...) line bundles on toric varieties maybe simultaneously toric-convolution-comonoidal and toric-equivariant ???? ..... ??????? ..... ????? ....... ?? hmmm .... ??? magic of invertibility ... ???? ....
case where convolution comonoidal structure is/n't "discrete" .... ???? ....
??? invertibility wrt toric convolution tensor product .... ????
?? relationship to generalized ("translational" ... ?? ...) morphism of toric varieties .... ??? ....
?? seeing via "ordinary AG interpretation of underlying ordinary quasicoherent sheaf of toric quasicoherent sheaf" how concept of "toricness" of quasicoherent sheaf depends on "basepoint(s?)" of toric variety .... ??? .... ?? but then idea of "allowing translates of toric quasicoherent sheaves" (?? ...), and how that might relate to other (more "TAG-intrinsic" ... ???...) ideas about what happens to accidental topos when TAG is "mad emore translation-independent" by using generalized morphisms of toric varieties ...
??? toric convolution of torically equivariant stuff .... ??? decategorifies to "convolution of constant functions" ???? ..... which sounds weirdly degenerate, but .... ??? maybe more interesting categorified ?? .... ??? ....
???? _are_ (?? some ?? ...) line bundles on toric varieties maybe simultaneously toric-convolution-comonoidal and toric-equivariant ???? ..... ??????? ..... ????? ....... ?? hmmm .... ??? magic of invertibility ... ???? ....
"toric ag as fourier duality generalized from abelian groups to commutative monoids" ....
(?? but ... ??? "global" aspect ??? .... ?? ... ?? categorification ?? ... ... ??? ...)
??the / a here ??? ..... ??? dualizing object/s ??? ....
incorporating semilattice duality ....
doctrines .... AG + extra (?? secretly "convolution" ?? ...) tensor product ....
vs ... "flat AG" ... "abelian + extra tensor product" .... ???? .... ?? .... ??? .....
??? "keeping biproduct as coproduct but taking new product" .... ???? .....
??? relationship between toric quasicoherent sheaf and corresponding ordinary quasicoherent sheaf as something like "fourier tranform" ?????? ...... ????? .......
??? "convolution tensor product of Z-reps" .... ??? tensor product of underlying vector spaces ... that's the "usual" tensor product of group reps .... becomes addition/convolution on fourier dual (??algebraic vs analytic ???? ....); hence name "convolution product" ..... ?????..... ????? other ("pointwise") tensor product as seeming culturally weirder here .... "intersection with multiplicity" ..... ???? distributivity here ??????
(a pointwise b) convolve c =?= (a convolve c) pointwise (b convolve c) ??????
wasn't expecting anything like this because of ..... "fourier _duality_" .... ?? and apparent lack of decategorification motivation .... ??? ....
??? semilattice duality ????? .....
??? "newton polytope" ... ????? .....
??? bi-distributivity =?=> bi-idempotence and bi-absorption ????? ..... ????....
x*x = (x*x)+0 = (x+0)*(x+0) .... ?? not too promising ?? ...
??? thing c st (a pointwise b) convolve c = (a convolve c) pointwise (b convolve c) for all a,b ... ???? ....
?? case b = "constantly 1" ... ???... ??? so c convolve constantly 1 must be constantly 1 ?? ..... using all a ??? .....
?? which maybe has flavor of ... requiring c to be "pointlike" ???? ..... ... ??? depending ... ???? ....
(a pointwise b) convolve (c + c') = (a pointwise b) convolve c + (a pointwise b) convolve c' = (if c and c' are "good") (a convolve c) pointwise (b convolve c) + (a convolve c') pointwise (b convolve c') =?= (a convolve (c+c')) pointwise (b convolve (c+c')) =
((a convolve c) + (a convolve c')) pointwise ((b convolve c) + (b convolve c')) = (a convolve c) pointwise (b convolve c) + (a convolve c') pointwise (b convolve c) + (a convolve c) pointwise (b convolve c') + (a convolve c') pointwise (b convolve c') .... ?? ...
?? so looks like sum of good tends to be bad ???? ..... matrix vs its main diagonal ... ??? ....
??? distributivity vs modularity ... ?????? .....
??? two kinds of tensor product of ab group reps, vs two kinds of hom .... ????? ..... ?? don't seem that related, since the tensors adjoint to the homs live in somewhat different places ... ?????? .....
?? toric quasicoherent sheaf vs generalized toric quasicoherent sheaf ... ???? "dirac measure" .... derivatives thereof .... ???? level slip ??? ..... ??? contrasting ideas of "measure" ???? .....
?? given theory of certain combined doctrine, taking just those objects obeying certain distributivity condition ... ???? ....
?? distributivity comparison morphism here ???? ..... ??? inclusion of main diagonal of matrix ???? ...
??? modularity ???? ..... ???? ..... _is_ there really a commonality here ???? ..... are modular lattices "laxly distributive" in particular ??? ..... (a sup b) inf c ... (a inf c) sup (b inf c) ... ???? .... ??? in the classic non-distributive lattice there's a comparison morphism from latter to former ??? is that correct ??? ..... ... ?? in classic non-modular lattice .... ???? .... ?? maybe still a comparison morphism ?? ... ????? .... obvious by universal property of latter as sup ??? .... ... ??? ..... but still .... ??? ......
?? "toric tannakian program/philosophy" .... ?? 2 versions ... toric quasicoherent sheaf vs [generalized toric quasicoherent sheaf ..... and/or ordinary quasicoherent sheaf with toric convolution tensor product .... at least locallly .... ???? .....] .... extra "toric convolution" tensor product" (??? at least locally ????? ....) vs"thorough-going set-based approach" .... ???? ....
analogy of cartesian product of toric quasicoherent sheaves to cartesian product of ordinary quasicoherent sheaves, vs to toric convolution tensor product of (at least locally ..... ????? .....) ordinary quasicoherent sheaves ... ??? relationship to "toric convolution tensor product as usurping biproduct's role as cartesian product though not as coproduct" ... ??? ....
?? relationship / vague similarity feeling of problematicness of tensor product on cocone category to problematicness of global toric convolution tensor product .... ???? .... ?? but problematicness dealt with by interesting new (??) "zero" (=?= "undefined") vs that dealt with by interesting new "one" .... ??? non-totalness of binary product vs of nullary .... ????? .....
??? locally a toric convolution comonoid .... ????? ....
??? "fourier analysis" ..... ??? analyzing vector space into comonoids vs vice versa .... ???? .... ??? chain complex vs filtration ??? ........ ????? .....
???? "tail tree structure" and toric convolution comonoidalness ....
???? toric convolution tensor product ..... ???? vague feeling of relationship to .... ???? ..... generalized day convolution as involving geometric morphism ... ??? ..... "adjoint to tensor product" .... ????? ......
??? toric orbit stack of toric variety, and "quantum double" .... ???? ....
?? more generally (?? ...), interaction between toric quasicoherent (and/or generalized such ...) structure and toric equivariance structure .... ???? ...... .... ?? ...
?? underlying real variety of elliptic curve .... ?? toric aspect ?? .... or higher abelian variety ??? ... ??? also higher genus curve .... ??? and relationship between last two .... ?? "jacobian" ... ??? ....
(?? but ... ??? "global" aspect ??? .... ?? ... ?? categorification ?? ... ... ??? ...)
??the / a here ??? ..... ??? dualizing object/s ??? ....
incorporating semilattice duality ....
doctrines .... AG + extra (?? secretly "convolution" ?? ...) tensor product ....
vs ... "flat AG" ... "abelian + extra tensor product" .... ???? .... ?? .... ??? .....
??? "keeping biproduct as coproduct but taking new product" .... ???? .....
??? relationship between toric quasicoherent sheaf and corresponding ordinary quasicoherent sheaf as something like "fourier tranform" ?????? ...... ????? .......
??? "convolution tensor product of Z-reps" .... ??? tensor product of underlying vector spaces ... that's the "usual" tensor product of group reps .... becomes addition/convolution on fourier dual (??algebraic vs analytic ???? ....); hence name "convolution product" ..... ?????..... ????? other ("pointwise") tensor product as seeming culturally weirder here .... "intersection with multiplicity" ..... ???? distributivity here ??????
(a pointwise b) convolve c =?= (a convolve c) pointwise (b convolve c) ??????
wasn't expecting anything like this because of ..... "fourier _duality_" .... ?? and apparent lack of decategorification motivation .... ??? ....
??? semilattice duality ????? .....
??? "newton polytope" ... ????? .....
??? bi-distributivity =?=> bi-idempotence and bi-absorption ????? ..... ????....
x*x = (x*x)+0 = (x+0)*(x+0) .... ?? not too promising ?? ...
??? thing c st (a pointwise b) convolve c = (a convolve c) pointwise (b convolve c) for all a,b ... ???? ....
?? case b = "constantly 1" ... ???... ??? so c convolve constantly 1 must be constantly 1 ?? ..... using all a ??? .....
?? which maybe has flavor of ... requiring c to be "pointlike" ???? ..... ... ??? depending ... ???? ....
(a pointwise b) convolve (c + c') = (a pointwise b) convolve c + (a pointwise b) convolve c' = (if c and c' are "good") (a convolve c) pointwise (b convolve c) + (a convolve c') pointwise (b convolve c') =?= (a convolve (c+c')) pointwise (b convolve (c+c')) =
((a convolve c) + (a convolve c')) pointwise ((b convolve c) + (b convolve c')) = (a convolve c) pointwise (b convolve c) + (a convolve c') pointwise (b convolve c) + (a convolve c) pointwise (b convolve c') + (a convolve c') pointwise (b convolve c') .... ?? ...
?? so looks like sum of good tends to be bad ???? ..... matrix vs its main diagonal ... ??? ....
??? distributivity vs modularity ... ?????? .....
??? two kinds of tensor product of ab group reps, vs two kinds of hom .... ????? ..... ?? don't seem that related, since the tensors adjoint to the homs live in somewhat different places ... ?????? .....
?? toric quasicoherent sheaf vs generalized toric quasicoherent sheaf ... ???? "dirac measure" .... derivatives thereof .... ???? level slip ??? ..... ??? contrasting ideas of "measure" ???? .....
?? given theory of certain combined doctrine, taking just those objects obeying certain distributivity condition ... ???? ....
?? distributivity comparison morphism here ???? ..... ??? inclusion of main diagonal of matrix ???? ...
??? modularity ???? ..... ???? ..... _is_ there really a commonality here ???? ..... are modular lattices "laxly distributive" in particular ??? ..... (a sup b) inf c ... (a inf c) sup (b inf c) ... ???? .... ??? in the classic non-distributive lattice there's a comparison morphism from latter to former ??? is that correct ??? ..... ... ?? in classic non-modular lattice .... ???? .... ?? maybe still a comparison morphism ?? ... ????? .... obvious by universal property of latter as sup ??? .... ... ??? ..... but still .... ??? ......
?? "toric tannakian program/philosophy" .... ?? 2 versions ... toric quasicoherent sheaf vs [generalized toric quasicoherent sheaf ..... and/or ordinary quasicoherent sheaf with toric convolution tensor product .... at least locallly .... ???? .....] .... extra "toric convolution" tensor product" (??? at least locally ????? ....) vs"thorough-going set-based approach" .... ???? ....
analogy of cartesian product of toric quasicoherent sheaves to cartesian product of ordinary quasicoherent sheaves, vs to toric convolution tensor product of (at least locally ..... ????? .....) ordinary quasicoherent sheaves ... ??? relationship to "toric convolution tensor product as usurping biproduct's role as cartesian product though not as coproduct" ... ??? ....
?? relationship / vague similarity feeling of problematicness of tensor product on cocone category to problematicness of global toric convolution tensor product .... ???? .... ?? but problematicness dealt with by interesting new (??) "zero" (=?= "undefined") vs that dealt with by interesting new "one" .... ??? non-totalness of binary product vs of nullary .... ????? .....
??? locally a toric convolution comonoid .... ????? ....
??? "fourier analysis" ..... ??? analyzing vector space into comonoids vs vice versa .... ???? .... ??? chain complex vs filtration ??? ........ ????? .....
???? "tail tree structure" and toric convolution comonoidalness ....
???? toric convolution tensor product ..... ???? vague feeling of relationship to .... ???? ..... generalized day convolution as involving geometric morphism ... ??? ..... "adjoint to tensor product" .... ????? ......
??? toric orbit stack of toric variety, and "quantum double" .... ???? ....
?? more generally (?? ...), interaction between toric quasicoherent (and/or generalized such ...) structure and toric equivariance structure .... ???? ...... .... ?? ...
?? underlying real variety of elliptic curve .... ?? toric aspect ?? .... or higher abelian variety ??? ... ??? also higher genus curve .... ??? and relationship between last two .... ?? "jacobian" ... ??? ....
Monday, September 12, 2011
for jon woolf
hi ... i've been following to some extent your work developing some ideas which you attribute at least in part to john baez and myself ...
actually i haven't been following it as closely as i'd like ... which is part of why i'm writing to you ... trying to find out more ...
actually i haven't been following it as closely as i'd like ... which is part of why i'm writing to you ... trying to find out more ...
for alex
i think that i've mentioned that the question of how interesting the paper that we're trying to write turns out to be probably depends somewhat (perhaps obviously) on how well-studied the concept of "toric quasicoherent sheaf" already is; and that as usual for me, i'm not familiar enough with the literature to have much idea how well-studied it is...
but i've reached a point where i think i understand the topic well enough that probably even i should be able to find where it's studied in the literature...
this is mostly a result of what i mentioned in my e-mail the other day, about understanding the geometric interpretation of the underlying ordinary quasicoherent sheaf of a toric quasicoherent sheaf...
having thought about a bit, i'd say that the key idea about this geometric interpretation is that the underlying ordinary quasicoherent sheaf is (at least locally) cocommutative comonoidal wrt the "convolution" tensor product of quasicoherent sheaves over an affine toric variety....
??? blecchh, am i still / again horribly confused here ??? .... convolution wrt lattice vs wrt torus .... ???? .... ?? commutative monoid ring as bialgebra .... mult from mult, comult from diagonal ... ??? but in affine toric variety picture it's somewhat reversed ... ??? algebra/geometry duality for affine toric varieties as sort of extension of fourier duality (??? ....) from abelian group case to commutative monoid case ??? .... ???anyway, somehow it sort of makes sense to call the tensor product coming from the comultiplication here "the ocnvolution product" .... convolution on the affine toric variety ... comultiplication on the commutative monoid ring ... diagonal on the commutative monoid .... ??? .....
??? skycraper sheaves closed under convolution tensor product here ???? ..... giving actual monoid structure on affine toric variety ???? ....
?? for affine toric variety mutliplication is already totally defined, so no need to adjoin "undefined" element ... ????? ..... ?? confusion about adjoining "undefined" (??? =?= "zero" ???? ....) vs adjoining "1" ...... ???? .....
?? freyd trick using "undefined" to construe category as semigroup / monoid ... ??? ..... ???? .... cocone category .... ????? .....
??? _non_-cocommutative comonoid for convolution of quasicoherent sheaves over affine toric variety ??? .... ???? .....
?? "sheaf of convolution cocommutative coalgebras over structure sheaf of "monoidally" ringed topos" ..... ???? .....
but i've reached a point where i think i understand the topic well enough that probably even i should be able to find where it's studied in the literature...
this is mostly a result of what i mentioned in my e-mail the other day, about understanding the geometric interpretation of the underlying ordinary quasicoherent sheaf of a toric quasicoherent sheaf...
having thought about a bit, i'd say that the key idea about this geometric interpretation is that the underlying ordinary quasicoherent sheaf is (at least locally) cocommutative comonoidal wrt the "convolution" tensor product of quasicoherent sheaves over an affine toric variety....
??? blecchh, am i still / again horribly confused here ??? .... convolution wrt lattice vs wrt torus .... ???? .... ?? commutative monoid ring as bialgebra .... mult from mult, comult from diagonal ... ??? but in affine toric variety picture it's somewhat reversed ... ??? algebra/geometry duality for affine toric varieties as sort of extension of fourier duality (??? ....) from abelian group case to commutative monoid case ??? .... ???anyway, somehow it sort of makes sense to call the tensor product coming from the comultiplication here "the ocnvolution product" .... convolution on the affine toric variety ... comultiplication on the commutative monoid ring ... diagonal on the commutative monoid .... ??? .....
??? skycraper sheaves closed under convolution tensor product here ???? ..... giving actual monoid structure on affine toric variety ???? ....
?? for affine toric variety mutliplication is already totally defined, so no need to adjoin "undefined" element ... ????? ..... ?? confusion about adjoining "undefined" (??? =?= "zero" ???? ....) vs adjoining "1" ...... ???? .....
?? freyd trick using "undefined" to construe category as semigroup / monoid ... ??? ..... ???? .... cocone category .... ????? .....
??? _non_-cocommutative comonoid for convolution of quasicoherent sheaves over affine toric variety ??? .... ???? .....
?? "sheaf of convolution cocommutative coalgebras over structure sheaf of "monoidally" ringed topos" ..... ???? .....
?? linear logic .... ???? .... ?? in light of new ideas about how toric quasicoherent hseaves relate to convolution tensor product of ordinary quasicoherent sheaves ... ?? revisiting vague idea of relationship of TAG and/or combined doctrine to linear logic .... ???? .....
standard ag approach to toric varieties ... "toric line bundle" .... ??? do they use convolution-comonoidalness here ??
?? this concept of "generalized toric quasicoherent sheaf" ... ??? what sort of category these form ?? ... cartesian closed ... plus other tensor product ....
?? parallel :
tensor and cartesian in accidental topos ...
tensor and cartesian in extension of accidental topos ... ???? coalgebras in accidental topos ??? .... ???? ....
ordinary ("pointwise" ...) tensor and convolution tensor of quasicoherent sheaves on (?? affine ??) toric variety ... ??? ....
???? ....
??? do coalgebras form a quasitopos ???? ..... ?? seems unlikely, but ... ??? ... ?? what was that weird fact about coalgebras and directed colimits (?? ...) that todd mentioned ... ?? heard in turn from ezra getzler ?? ....
coalgebra for monad vs for monoid here .... ???? ....
?? right adjoint property of "cocomm monoids in smc" ....
?? how much did we ever think about whether "monoids in mc" has left adjoint ??? ..... ???? ....
standard ag approach to toric varieties ... "toric line bundle" .... ??? do they use convolution-comonoidalness here ??
?? this concept of "generalized toric quasicoherent sheaf" ... ??? what sort of category these form ?? ... cartesian closed ... plus other tensor product ....
?? parallel :
tensor and cartesian in accidental topos ...
tensor and cartesian in extension of accidental topos ... ???? coalgebras in accidental topos ??? .... ???? ....
ordinary ("pointwise" ...) tensor and convolution tensor of quasicoherent sheaves on (?? affine ??) toric variety ... ??? ....
???? ....
??? do coalgebras form a quasitopos ???? ..... ?? seems unlikely, but ... ??? ... ?? what was that weird fact about coalgebras and directed colimits (?? ...) that todd mentioned ... ?? heard in turn from ezra getzler ?? ....
coalgebra for monad vs for monoid here .... ???? ....
?? right adjoint property of "cocomm monoids in smc" ....
?? how much did we ever think about whether "monoids in mc" has left adjoint ??? ..... ???? ....
??? projective plane with all points removed .... ????
?? describe explicitly ... quasicoherent sheaf ...
?? .... toric analog ... ?? .... "non-linear fan" .... ??? ....
?? weird variants of cremona group ... ???? ....
?? analog/variant of "partial fraction basis" ???? ..... ???? ..... ??? categorified ??? .....
?? describe explicitly ... quasicoherent sheaf ...
?? .... toric analog ... ?? .... "non-linear fan" .... ??? ....
?? weird variants of cremona group ... ???? ....
?? analog/variant of "partial fraction basis" ???? ..... ???? ..... ??? categorified ??? .....
?? toric line bundle ... ??? trying to formalize in nice "conventional algebraic-geometry" way, and then prove equivalent to ... even nicer way ... ??? .....
?? tendency for toric line bundle concept to be somewhat orthogonal to "loops and tails" idea .... ????? .....
?? module of "quantum double" ???? ..... ?????..... no, not correct yet .... but .... vague idea of combining being a module of the monoid ring with some extra structure associated with .... ????? comultiplication on monoid ring ... seems maybe more correct ... perhaps eventually obviously so ...
?? how coalgebra structure gets along with comultiplication here .... ????
?? hmmm .... i've been missing something a bit obvious here ??? .... modules of commutative monoid ring .... ?? as having two tensor products ???? ......
??but ... ??? special case of abelian group ring ... ????....
hmmm ... some confusion here .... probably not too bad ...
?? _invertible_ rep of abelian group ... ???? ....
?? line bundle over torus .... ?? ......
?? 1-dim fibers vs 1-dim total vsp and/or sections ... ??? ....
?? invertible line bundle over torus =?= invertible Z-rep ... ?? ...
?? hmmm, maybe there really are two different tensor products here ... ??? again seems amazing for me not to have really noticed (?? ...) it, if so .... ???? ....
?? must (?? ...) have noticed there are two _reasons_ for tensor products to exist here ... ?? mult and comult, or 2 mults (equal by eckmann-hilton ...) ...
?? p-tuples of vector spaces can be tensored pointwise, or convolved .... ??? ... and this is a pretty familiar idea .... with perhaps more familiar decategorification .... hmmmmm .....
?? wait a minute ... even on a non-abelian group (or monoid) you can both convolve and multiply pointwise .... hmmm, wait another minute ... that's using measure-function duality ..... ????? .....
????? haar measure ???? ..... ??? and what happens to it in trying to extend from group to monoid ??? ... ??? compactness and/or "moral compactness" issues ???? .... torus ..... ??? ....
"categorified haar measure" .... ??? ....
?? i was going to throw in something about "serre duality" here ...
but ... ?? actually some of what's going on here turns out to be very simple ... (not that it mightn't be interesting to try to straighten out some of the other stuff sometime ...) ...
?? commutative monoid acting on cocommutative coalgebra ... as equivalent to cocommutative monoid wrt convolution tensor product (that is, arising from comultiplication ...) of quasicoherent sheaves over affine toric variety ... ?? ....
?? what happens to convolution tensor product in passage from affine to more general toric variety ??? ....
?? well for one thing it becomes the cartesian product of toric quasicoherent sheaves, right ?? ...
??terry bisson ??? ..... ?? categorified fourier transform ... ??? ....
?? but still, "global funny stuff" here ??? .... relationship to "generalized day convolution" and so forth ??? .... toric variety as semi-group object, or almost so .... ???? .....
?? tendency for toric line bundle concept to be somewhat orthogonal to "loops and tails" idea .... ????? .....
?? module of "quantum double" ???? ..... ?????..... no, not correct yet .... but .... vague idea of combining being a module of the monoid ring with some extra structure associated with .... ????? comultiplication on monoid ring ... seems maybe more correct ... perhaps eventually obviously so ...
?? how coalgebra structure gets along with comultiplication here .... ????
?? hmmm .... i've been missing something a bit obvious here ??? .... modules of commutative monoid ring .... ?? as having two tensor products ???? ......
??but ... ??? special case of abelian group ring ... ????....
hmmm ... some confusion here .... probably not too bad ...
?? _invertible_ rep of abelian group ... ???? ....
?? line bundle over torus .... ?? ......
?? 1-dim fibers vs 1-dim total vsp and/or sections ... ??? ....
?? invertible line bundle over torus =?= invertible Z-rep ... ?? ...
?? hmmm, maybe there really are two different tensor products here ... ??? again seems amazing for me not to have really noticed (?? ...) it, if so .... ???? ....
?? must (?? ...) have noticed there are two _reasons_ for tensor products to exist here ... ?? mult and comult, or 2 mults (equal by eckmann-hilton ...) ...
?? p-tuples of vector spaces can be tensored pointwise, or convolved .... ??? ... and this is a pretty familiar idea .... with perhaps more familiar decategorification .... hmmmmm .....
?? wait a minute ... even on a non-abelian group (or monoid) you can both convolve and multiply pointwise .... hmmm, wait another minute ... that's using measure-function duality ..... ????? .....
????? haar measure ???? ..... ??? and what happens to it in trying to extend from group to monoid ??? ... ??? compactness and/or "moral compactness" issues ???? .... torus ..... ??? ....
"categorified haar measure" .... ??? ....
?? i was going to throw in something about "serre duality" here ...
but ... ?? actually some of what's going on here turns out to be very simple ... (not that it mightn't be interesting to try to straighten out some of the other stuff sometime ...) ...
?? commutative monoid acting on cocommutative coalgebra ... as equivalent to cocommutative monoid wrt convolution tensor product (that is, arising from comultiplication ...) of quasicoherent sheaves over affine toric variety ... ?? ....
?? what happens to convolution tensor product in passage from affine to more general toric variety ??? ....
?? well for one thing it becomes the cartesian product of toric quasicoherent sheaves, right ?? ...
??terry bisson ??? ..... ?? categorified fourier transform ... ??? ....
?? but still, "global funny stuff" here ??? .... relationship to "generalized day convolution" and so forth ??? .... toric variety as semi-group object, or almost so .... ???? .....
Sunday, September 11, 2011
?? approach to b2 flag variety complexified bi-multi-homogeneous coordinate algebra via ... replacing quaternions by 2-by-2 matrixes (since that's what they are in complexified case ...), and replacing span of u and v (2 imaginary quaternions ...) by ???, and imaginary quaternions by .... ??? sl(2,c) ??? .... ???? ...
"de-anchoring" ... ??? that is, i was thinking about "theory of projective plane", construed in various doctrines .... and thinking about how in some, projective planes over different fields all just live together, whereas in others seem "segregated" by base over which they live ... "anchor" ... then started drifting .... (??? ...) .... ??lax vs weak/strong "ringed geometric morphism"... ?? idea of "base" formed by some sort of "complex of coherently related models of all coxeter/dynkin diagram theories" .... ????? conceptual status of such complex ????? ...... trying to do de-anchoring in AG doctrine .... ????? .....
strong=weak ... lax vs strict ... flexible ... "robust vs delicate" ... ?? idea that "robust vs delicate" fits in well with whole idea of "topology as rubber sheet geometry" ???? .... ..... ????? "topology" vs "homotopy" .... progression from topology to homotopy as built into topology ... ??? analogy with category ??? .... ??? how far to push this analogy ??? ... hmmmm ... category of categories .... space of spaces .... ??? .... ?? discrete space of spaces vs interesting space of spaces .... ???? .....
visualization of rank 2 real algebraic buildings
?? disclaimer about ... not "serious" ??? ?? "this is a talk about visualization so it's intended to be a bit more on the fun side than the serious side ... besides i already did it so i might as well inflict some of the fun on you ..." ... ??? ....
1 concrete visual stuff ... a2, b2, g2 .... s^0, s^1 (gimbal ... penrose ..), s^2 X s^0 (rolling ball) .... concrete relationship to bi-multi-homogeneous coordinate algebra ... (??how to handle g2 "tweak map" issue ?? ...?? ...) .... warnings/jokes about tradeoff between real and complex alg geom ... for example naive real view thinks projective quaternions form a group .... ??? ....
?? idea even in real case quaternion zero hints at problem ... ??? .....
g2 crown grassmanian vs quincunx grassmanian ... (?? analogous names for b2 and even a2 ?? ... ?? triangle and line ?? too many other conceptual associations ?? ...) ... quincunx grassmanian as projective light cone and maybe worth explaining how to analyze into revolution and rotation state ... ?? crown grassmanian as traditionally somewhat hard to get at ?? .... ?? maybe not so bad, depending on ... ??? above "tradeoff(...)"-flavored issues again ... ???? ... trying to minimize dimension of projective embedding .... ???? .....
?? funny switch with b2 tradeoff issues ?? ... ??? it's some of the b2 quincunx (??? ...) variables that you're tempted to eliminate, whereas ... ??? morally it's the b2 square irrep that's "causing the problem" ... namely latent non-invertible complexifed quaternions (?? "biquaternions" ... ??) .... ?? anything like this in g2 case ??? ....
?? gimbal-axis as "skewer" ... independent rotations around it of globe and gimbal ... ??? globe as sun=hot .... roast gimbal on spit ... ???.... don't use this (...), but ...
gyre and gimbal ...gyroscope from nested gimbals ?? ....
?? from octonions to rolling ball via quaternions ??? ... ?? octonionic approach to bi-multi-homogeneous coordinate algebra here ... ?? cheating to use so(7) ?? ...
?? no "folding" stuff on this talk ??? .....
?? building as easier to visualize than it's symmetry group ??? ....??? except for maximal compact subgroup ??? .... ???? is that exactly correct in all 3 cases ???? ..... ?? trickiness of visualizing "boost" in b2 case ??? .... "mixing time and space = gimbal and 2-sphere" ... ??? .... ??? flat (2,3) vs conformal (1,2) here .... ???? ....
?? a2 animation ... point (with doppelgaenger ...) moving geodesically on 2-sphere ... ?? comment about geodesic structure (without parameterization) as much weaker than metric structure ?? ?? or maybe that's sufficiently obvious (when in right mood ...) not to mention ... ???? .... ??well, etymology (?? ...) of "geodesic" might justify comment .... ??
2 toric stuff ??? .... kaleidoscope as fan .... ??? ..... ?? schubert singularities ... ??? .... light cone, penrose ... 2,3,5 ... roll, spin, skid ... (root system picture here ...)
3 pretend to justify appropriateness for this conference ,,, ?? very short ...
theory .... propositional : predicate :: scheme/variety : stack .... ????? .....
(emphasize screwed-up-ness of alg geom (??) terminology here ?? ....)
?? slightly (?? ...) different theories here (...) depending on how may gradings you use ...
?? proj ... ????.... "tannakian" version ??
doctrines ...
theory of projective planes, for example .... live interestingly in more than one doctrine ....
tannakian program ...
!! ask alex about talk length ... ?? also mathematica at university ...
???? trying to limk together .... rolling ball and octonions .... rolling ball and quaternions .... quaternions and b2 flag variety bi-multi-homogeneous coordinate algebra ... b2 and b3 ... b3 and g2 ... g2 and octonions and rolling ball .... g2 fvbmhca and rolling ball ... ???? ... in some combination .... ??? ...
?? "steering wheel" ... ?? in trying to explain intutitie sense of flag concept (rank 2 case ...) and "generic flag pair" .... "6 degrees ..." ....
?? disclaimer about ... not "serious" ??? ?? "this is a talk about visualization so it's intended to be a bit more on the fun side than the serious side ... besides i already did it so i might as well inflict some of the fun on you ..." ... ??? ....
1 concrete visual stuff ... a2, b2, g2 .... s^0, s^1 (gimbal ... penrose ..), s^2 X s^0 (rolling ball) .... concrete relationship to bi-multi-homogeneous coordinate algebra ... (??how to handle g2 "tweak map" issue ?? ...?? ...) .... warnings/jokes about tradeoff between real and complex alg geom ... for example naive real view thinks projective quaternions form a group .... ??? ....
?? idea even in real case quaternion zero hints at problem ... ??? .....
g2 crown grassmanian vs quincunx grassmanian ... (?? analogous names for b2 and even a2 ?? ... ?? triangle and line ?? too many other conceptual associations ?? ...) ... quincunx grassmanian as projective light cone and maybe worth explaining how to analyze into revolution and rotation state ... ?? crown grassmanian as traditionally somewhat hard to get at ?? .... ?? maybe not so bad, depending on ... ??? above "tradeoff(...)"-flavored issues again ... ???? ... trying to minimize dimension of projective embedding .... ???? .....
?? funny switch with b2 tradeoff issues ?? ... ??? it's some of the b2 quincunx (??? ...) variables that you're tempted to eliminate, whereas ... ??? morally it's the b2 square irrep that's "causing the problem" ... namely latent non-invertible complexifed quaternions (?? "biquaternions" ... ??) .... ?? anything like this in g2 case ??? ....
?? gimbal-axis as "skewer" ... independent rotations around it of globe and gimbal ... ??? globe as sun=hot .... roast gimbal on spit ... ???.... don't use this (...), but ...
gyre and gimbal ...gyroscope from nested gimbals ?? ....
?? from octonions to rolling ball via quaternions ??? ... ?? octonionic approach to bi-multi-homogeneous coordinate algebra here ... ?? cheating to use so(7) ?? ...
?? no "folding" stuff on this talk ??? .....
?? building as easier to visualize than it's symmetry group ??? ....??? except for maximal compact subgroup ??? .... ???? is that exactly correct in all 3 cases ???? ..... ?? trickiness of visualizing "boost" in b2 case ??? .... "mixing time and space = gimbal and 2-sphere" ... ??? .... ??? flat (2,3) vs conformal (1,2) here .... ???? ....
?? a2 animation ... point (with doppelgaenger ...) moving geodesically on 2-sphere ... ?? comment about geodesic structure (without parameterization) as much weaker than metric structure ?? ?? or maybe that's sufficiently obvious (when in right mood ...) not to mention ... ???? .... ??well, etymology (?? ...) of "geodesic" might justify comment .... ??
2 toric stuff ??? .... kaleidoscope as fan .... ??? ..... ?? schubert singularities ... ??? .... light cone, penrose ... 2,3,5 ... roll, spin, skid ... (root system picture here ...)
3 pretend to justify appropriateness for this conference ,,, ?? very short ...
theory .... propositional : predicate :: scheme/variety : stack .... ????? .....
(emphasize screwed-up-ness of alg geom (??) terminology here ?? ....)
?? slightly (?? ...) different theories here (...) depending on how may gradings you use ...
?? proj ... ????.... "tannakian" version ??
doctrines ...
theory of projective planes, for example .... live interestingly in more than one doctrine ....
tannakian program ...
!! ask alex about talk length ... ?? also mathematica at university ...
???? trying to limk together .... rolling ball and octonions .... rolling ball and quaternions .... quaternions and b2 flag variety bi-multi-homogeneous coordinate algebra ... b2 and b3 ... b3 and g2 ... g2 and octonions and rolling ball .... g2 fvbmhca and rolling ball ... ???? ... in some combination .... ??? ...
?? "steering wheel" ... ?? in trying to explain intutitie sense of flag concept (rank 2 case ...) and "generic flag pair" .... "6 degrees ..." ....
Saturday, September 10, 2011
"theory of a linear isometry k^2 -> k^3 and an input-output pair for it" .... ???? why not reduce to just the isometry and the input ??? ..... recover the output by evaluation ..... ??????? ........
?? "evaluation" here as maybe tricky ??? .... ?? involving conjugation by and thus inversion of quaternion ... ??? ...
?? "evaluation" here as maybe tricky ??? .... ?? involving conjugation by and thus inversion of quaternion ... ??? ...
?? conventional-geometric interpretation of toric quasicoherent sheaf ...
?? what's special about spectrum of a permutation matrix ?? .... ?? "subgroup-like" ??? ....
?? relationship to constraining loop periodicity / tail length ?? .... ???? ...
?? fan geometry here ?? ....
?? coalgebra / coalgebra .... ????? ....
??toric divisor ... ??? ...
?? recovering arbitrary quasicoherent sheaf as ab gp toric quasicoherent sheaf .... ??? ....
?? what's special about spectrum of a permutation matrix ?? .... ?? "subgroup-like" ??? ....
?? relationship to constraining loop periodicity / tail length ?? .... ???? ...
?? fan geometry here ?? ....
?? coalgebra / coalgebra .... ????? ....
??toric divisor ... ??? ...
?? recovering arbitrary quasicoherent sheaf as ab gp toric quasicoherent sheaf .... ??? ....
d-isotropic grassmanian of k^[a+b] as mapping to d grassmanian of k^a .... ????? ...... ????? .....
g2 crown grassmanian .... "2d subspace of k^3, tw ... " ... ??? ....
?? 2d isotropic subspace of k^[3+4] ... rank 2 partial isometry from k^3 to k^4 ... ??? .... "rolling track" tw isometry ( ?? ... ) to "orientation" space ... ??? ...
?? paradox about maybe getting lower-dim than possible projective embedding of crown grassmanian ?? ....
g2 crown grassmanian .... "2d subspace of k^3, tw ... " ... ??? ....
?? 2d isotropic subspace of k^[3+4] ... rank 2 partial isometry from k^3 to k^4 ... ??? .... "rolling track" tw isometry ( ?? ... ) to "orientation" space ... ??? ...
?? paradox about maybe getting lower-dim than possible projective embedding of crown grassmanian ?? ....
Friday, September 9, 2011
?? form quaternion (e-g)j + (f-h)k .... and another (e+g)j + (f+h)k + jl .... ???? ....
??and then impose that the first one conjugated by a+bj+ck+dl is the second one .... ???? ....
??? maybe "grading problems" here ??? ....
???? very vague feeling about some sort of resemblance here to ... ??? dirac approach to clifford algebra ... ??? ....
?? j collision here ... ??? use u,v,w for basic imaginary quaternions for the moment ...
((e-g)u+(f-h)v)(a + bu + cv + dw) = (a + bu + cv + dw)((e+g)u + (f+h)v + jw) ??? .....
(gb-eb+hc-fc) + (ea-ga+fd-hd)u + (fa-ha+gd-ed)v + (ec-gc+hb-fb)w = (-be-bg-cf-ch-dj) + ()u + ()v + ()w
2bg+2ch + dj = 0 ????
??? ......
??and then impose that the first one conjugated by a+bj+ck+dl is the second one .... ???? ....
??? maybe "grading problems" here ??? ....
???? very vague feeling about some sort of resemblance here to ... ??? dirac approach to clifford algebra ... ??? ....
?? j collision here ... ??? use u,v,w for basic imaginary quaternions for the moment ...
((e-g)u+(f-h)v)(a + bu + cv + dw) = (a + bu + cv + dw)((e+g)u + (f+h)v + jw) ??? .....
(gb-eb+hc-fc) + (ea-ga+fd-hd)u + (fa-ha+gd-ed)v + (ec-gc+hb-fb)w = (-be-bg-cf-ch-dj) + ()u + ()v + ()w
2bg+2ch + dj = 0 ????
??? ......
?? b2 ....
k[a:(x,x),b:(x,y-x),c:(x,-x),d:(x,x-y),e:(y,y),f:(y,y-2x),g:(y,-y),h:(y,2x-y),j:(y,0)] / ef+gh+j^2, ????
so(m+1,n+1) acting on .... ??? conformal completion of flat signature (m,n) space .... ???? .....
projective light cone in flat (m+1,n+1)-space as ess that conformal completion ?? ....
?? _are/were_ we having some point/line confusion here again ??? .....
?? light cone lives _in_ point space, not in light-ray space ... ?? ....
?? gimbal configuration space = so(3) = rp^3 .... ???? = light-ray space ???
?? axis configuration space = conformal 2+1 space-time .... ???? ....
??? interpret a,b,c,d as giving isometry from s^1 to s^2 .... ????? .... and separate e,f,g,h,j into 2 parts, one giving point of s^1, other giving point of s^2 .... ???? .... evaluation .... ????.....
e-g and f-h ... ???
e+g, f+h, and j .... ?????.....
k l m
n p q
??? ....
?? quaternions .... ??? projective quaternion .... ????? .... ?? conjugation action on imaginary quaternions ... ??? 2d subspace in imaginary quaternions ... ??? ....
k[a:(x,x),b:(x,y-x),c:(x,-x),d:(x,x-y),e:(y,y),f:(y,y-2x),g:(y,-y),h:(y,2x-y),j:(y,0)] / ef+gh+j^2, ????
so(m+1,n+1) acting on .... ??? conformal completion of flat signature (m,n) space .... ???? .....
projective light cone in flat (m+1,n+1)-space as ess that conformal completion ?? ....
?? _are/were_ we having some point/line confusion here again ??? .....
?? light cone lives _in_ point space, not in light-ray space ... ?? ....
?? gimbal configuration space = so(3) = rp^3 .... ???? = light-ray space ???
?? axis configuration space = conformal 2+1 space-time .... ???? ....
??? interpret a,b,c,d as giving isometry from s^1 to s^2 .... ????? .... and separate e,f,g,h,j into 2 parts, one giving point of s^1, other giving point of s^2 .... ???? .... evaluation .... ????.....
e-g and f-h ... ???
e+g, f+h, and j .... ?????.....
k l m
n p q
??? ....
?? quaternions .... ??? projective quaternion .... ????? .... ?? conjugation action on imaginary quaternions ... ??? 2d subspace in imaginary quaternions ... ??? ....
Thursday, September 8, 2011
?? "generalized pluecker relations" (jargon term for this ?? ...) for g2 ... ?? presented in various hopefully nice conceptual ways .... octonion-ish ... rolling-ball-ish .... ?? use of long root subalgebra ??
1d isotropic subalgs of imaginary split octonions ....
7 basic imaginary octonions ...
?? confusion about whether i should think in terms of these or ["coefficients of" them] =?= dual basis vectors ... even though sort of doesn't matter in this case .... ????....
2d ....
?? start with grassmanian "7 choose 2" ... ??? which is 21-dimensional ...
?? then add isotropy constraint ????? .....
??? generalized pluecker relations for 2 vanilla end dots of so(7) ... 1d and 2d isotropics ??? plus extra relations saying that the 2d ones are isotropic in the stronger sense ... octonionic ...
?? well let's see ... as far as the duality confusion here, i mean ... should be able to straighten it out .... ?? anti-tautological line bundle has sections ... ???
?? 21-dimensional irrep of so(7) .... ?? adjoint irrep in fact ?? ... with ?-dim grassmanian in its 20-dim projective space ... ???..... 14-dim adjoint rep of g2 inside 21-dim .... 5-dim grassmanian inside that .... ???? the 13-dim and the ?-dim inside the 20-dim intersect "higher-dimensionally than expected" ??? ...
1d isotropic subalgs of imaginary split octonions ....
7 basic imaginary octonions ...
?? confusion about whether i should think in terms of these or ["coefficients of" them] =?= dual basis vectors ... even though sort of doesn't matter in this case .... ????....
2d ....
?? start with grassmanian "7 choose 2" ... ??? which is 21-dimensional ...
?? then add isotropy constraint ????? .....
??? generalized pluecker relations for 2 vanilla end dots of so(7) ... 1d and 2d isotropics ??? plus extra relations saying that the 2d ones are isotropic in the stronger sense ... octonionic ...
?? well let's see ... as far as the duality confusion here, i mean ... should be able to straighten it out .... ?? anti-tautological line bundle has sections ... ???
?? 21-dimensional irrep of so(7) .... ?? adjoint irrep in fact ?? ... with ?-dim grassmanian in its 20-dim projective space ... ???..... 14-dim adjoint rep of g2 inside 21-dim .... 5-dim grassmanian inside that .... ???? the 13-dim and the ?-dim inside the 20-dim intersect "higher-dimensionally than expected" ??? ...
Wednesday, September 7, 2011
?? functor taking "generalized toric variety" to ordinary toric variety ?? ... ???? ... analogous to functor taking affine vsp to its translation vector space .... ?????......
?? comm monoid under (k,mult) ... ???.... cokernel functor ??? ...
?? but ... what happens under such functor to morphism taking main torus to non-main ??? ..... ????
?? hmmm ... (k,mult) vs gl(1,k) here .... ???? ..... "killing off 0" (in sense of setting it to 1 .... ???!!??? ...) as "dangerous" .... ??? .....
?? comm monoid under (k,mult) ... ???.... cokernel functor ??? ...
?? but ... what happens under such functor to morphism taking main torus to non-main ??? ..... ????
?? hmmm ... (k,mult) vs gl(1,k) here .... ???? ..... "killing off 0" (in sense of setting it to 1 .... ???!!??? ...) as "dangerous" .... ??? .....
?? fixed subspace of tweak map .... ??? ...
?? zariski closure of strict [framed apartment]-flag orientation "their line contains our favorite point" .... ??? ....
?? or ... ??? "their line goes through at least one of our points" ... ????....
?? _do_ apartment-flag orientations give kaleidoscope-invariant toric varieties ? .... ??? ...
?? zariski closure of strict [framed apartment]-flag orientation "their line contains our favorite point" .... ??? ....
?? or ... ??? "their line goes through at least one of our points" ... ????....
?? _do_ apartment-flag orientations give kaleidoscope-invariant toric varieties ? .... ??? ...
Monday, September 5, 2011
?? .... k[a:(x,x),b:(y-x,x),c:(-y,x),d:(-x,y),e:(x-y,y),f:(y,y)]/ad+be+cf ...
?? ideal generated by ac-eg .... ??? ..... ?? case g = infinity .... ???? ..... ???? case g = 0 .... ???? ....
?? also df-bg .... ???? .....
?? hmm, maybe we really want ideals homogeneous for both gradings ...
?? so ... maybe simplify a bit for the moment by just working with points instead of full flags ... so just a,b,c coordinates ...
?? hmm, but then .... ?? seems clear that there's a dense orbit ... ??? ...
?? frame for projective n-space given by generic (flag,flag,point) ?? .... ?? point as precisely (??) "calibrating aspect ratio" ?? ....
?? so ... ??? toric variety here ??? ...... projective n-space .... ?? "simplicial" fan .... ???? .... ?? n-simplex and [n+1]-cube ... ???? ..... ?? or perhaps [n+1]-diamond ?? ... ?? long axises of [n+1]-diamond corresponding to (max dim) faces of n-simplex .... ?? glueing technique for real spectrum of toric variety here .... ?? some poincare duality confusion here, but probably not bad ... ?? .....
?? blowing up to get full kaleidoscope toric variety ??? ..... ???? .....
?? ideal generated by ac-eg .... ??? ..... ?? case g = infinity .... ???? ..... ???? case g = 0 .... ???? ....
?? also df-bg .... ???? .....
?? hmm, maybe we really want ideals homogeneous for both gradings ...
?? so ... maybe simplify a bit for the moment by just working with points instead of full flags ... so just a,b,c coordinates ...
?? hmm, but then .... ?? seems clear that there's a dense orbit ... ??? ...
?? frame for projective n-space given by generic (flag,flag,point) ?? .... ?? point as precisely (??) "calibrating aspect ratio" ?? ....
?? so ... ??? toric variety here ??? ...... projective n-space .... ?? "simplicial" fan .... ???? .... ?? n-simplex and [n+1]-cube ... ???? ..... ?? or perhaps [n+1]-diamond ?? ... ?? long axises of [n+1]-diamond corresponding to (max dim) faces of n-simplex .... ?? glueing technique for real spectrum of toric variety here .... ?? some poincare duality confusion here, but probably not bad ... ?? .....
?? blowing up to get full kaleidoscope toric variety ??? ..... ???? .....
??? toric variety from graded object ... ??? ....
?? augmentation ... ??? bialg counit ?? .... ??? ....
?? ....
?? process involving modding out by homogeneous ideal .... ??? why not just "mod out by it ahead of time" ?? .... ???? ..... ??? property .... ????? .....
???? ... "zariski closure of torus orbit" ... ??? .....
"homogeneous core" ....
?? so ... various (?? generalized ??) properties of dimensional algebra ... (?? or of AG theory under AG theory coming from discrete abelian group, maybe later ... ?? ....)
?? being monoid algebra, with grading group being its groupization ...
?? being bialgebra ... ??? .... ??? "augmentation" ... ???? ....
?? "spectrum having dense orbit" .... ????? .......
??? grades being small .... ??????....
??? vague idea about "superimposing multiple orbits" .... ?????.......
?? augmentation ... ??? bialg counit ?? .... ??? ....
?? ....
?? process involving modding out by homogeneous ideal .... ??? why not just "mod out by it ahead of time" ?? .... ???? ..... ??? property .... ????? .....
???? ... "zariski closure of torus orbit" ... ??? .....
"homogeneous core" ....
?? so ... various (?? generalized ??) properties of dimensional algebra ... (?? or of AG theory under AG theory coming from discrete abelian group, maybe later ... ?? ....)
?? being monoid algebra, with grading group being its groupization ...
?? being bialgebra ... ??? .... ??? "augmentation" ... ???? ....
?? "spectrum having dense orbit" .... ????? .......
??? grades being small .... ??????....
??? vague idea about "superimposing multiple orbits" .... ?????.......
Sunday, September 4, 2011
?? playing around with ..... ????.... toric varieties coming from models of dimensional theories .... ????
??affine toric varieties this way ... but then also .... ???? ....
??case of dimensional (???? ....) theory of flags .... ???? .... ?? or do i mean flagged n-d vector spaces, or bi-flagged .... ???? ....
?? the two gradings here .... ??? one for getting toric variety, other for having it be projective .... ???? ....
?? relate to a2 kaleidoscope toric variety and relatives of it ... ??? ....
?? lots of other examples to play around with here ....
?? cuboquadratic algebra ...
?? baez's scattering example ...
?? lie alg with sum decomposition example ...
?? random simple syntactic examples ... ?? ...
??affine toric varieties this way ... but then also .... ???? ....
??case of dimensional (???? ....) theory of flags .... ???? .... ?? or do i mean flagged n-d vector spaces, or bi-flagged .... ???? ....
?? the two gradings here .... ??? one for getting toric variety, other for having it be projective .... ???? ....
?? relate to a2 kaleidoscope toric variety and relatives of it ... ??? ....
?? lots of other examples to play around with here ....
?? cuboquadratic algebra ...
?? baez's scattering example ...
?? lie alg with sum decomposition example ...
?? random simple syntactic examples ... ?? ...
Saturday, September 3, 2011
??a1 real cartan torus .... ? as "genuine" aspect-ratio adjustment group ... ???? ....
?? generic flag ordered pair .... 2 points ... first as infinity, second as 0 ... so projective line becomes affine and then linear ... ??re-scaling of this linear line as adjusting aspect-ratio ??? .... ??? horizontal and vertical as dual ?? ... ??? symplectic structure ??? ....
?? so cartan torus of symplectic group here as "corresponding" linear group ... ??? ....
?? "cartan torus" ... ?? "maximal torus" .... ???? ....
?? generic flag ordered pair .... 2 points ... first as infinity, second as 0 ... so projective line becomes affine and then linear ... ??re-scaling of this linear line as adjusting aspect-ratio ??? .... ??? horizontal and vertical as dual ?? ... ??? symplectic structure ??? ....
?? so cartan torus of symplectic group here as "corresponding" linear group ... ??? ....
?? "cartan torus" ... ?? "maximal torus" .... ???? ....
?? was also thinking about .... ?? possibility of "one x-destabilization away" toric (??? ....) variety as being slightly-crypto-morphic to "one y-destabilization away" one via some sort of "x-thing as equivalet to y-thing in the presence of ..." phenomenon .... ????? .....
?? at the moment seems like it probably doesn't "work" ??? ...
?? at the moment seems like it probably doesn't "work" ??? ...
?? nice way for t-orbit in g/b to get kaleidoscope symmetry ... ??
?? kaleidoscope group = normalizer(b)/b .... ????? = automorphisms of g/b as g-space ???? ....
?? seems a bit screwed up ....
?? normalizer_g(h)/h as automorphisms of g/h as h-space, or equivalently as invertible h,h double cosets ??? .....
?? invertible apartment-apartment orientations ??? .... ?? = arbitrary flag-flag orientations ... ???? .....
?? straightening out toric varieties given by ... ??? various flag-flag orientations degenerately (??...) re-construed as flag-flag-flag ... ??? ..... ?? still some confusion here, but ... ???? ...
?? so normalizer(t)/t ... ??? ...
?? action by normalizer(t) on toric variety .... ????? ..... ????? .... ???? ....
?? kaleidoscope group = normalizer(b)/b .... ????? = automorphisms of g/b as g-space ???? ....
?? seems a bit screwed up ....
?? normalizer_g(h)/h as automorphisms of g/h as h-space, or equivalently as invertible h,h double cosets ??? .....
?? invertible apartment-apartment orientations ??? .... ?? = arbitrary flag-flag orientations ... ???? .....
?? straightening out toric varieties given by ... ??? various flag-flag orientations degenerately (??...) re-construed as flag-flag-flag ... ??? ..... ?? still some confusion here, but ... ???? ...
?? so normalizer(t)/t ... ??? ...
?? action by normalizer(t) on toric variety .... ????? ..... ????? .... ???? ....
?? trying to visualize a2 real cartan torus action on flag variety ... orbit decomposition ...
?? 3d vector space ... with usual visualization of standard coordinate axises ...
?? cartan torus as "3d aspect ratio adjustment" group ...
?? subjecting 2d subspace to particular aspect ratio adjustment while _not_ subjecting 1d subspace to same adjustment ??? .....
??? "grassmanian blow-down" .... ???? hmmm, so what _does_ for example g2 "one-roll-away" give on _flag variety" as opposed to grassmanian ??? .... ???? .....
?? "pick direction, then magnitude" ... ????
?? span of schubert varieties here ..... ????? ......
?? non-/toric blow-ups/downs as spans of a sort ... ??? (?? better than as co-spans ?? ... ?? ...) .... ??relationship to bit we tried working out somewhat long time ago ... birational geometry .... ??? non-/toric .... ??? ....
?? confusion between ... ??choice of which destabilizer comes first, and which grassmanian coordinate gets omitted to give the partial flag blowdown ... ??? ..... ?? natural to omit the first one ?? .... ???? .... omitting the second one as giving the fiber that gets blowndown ??? ????? ....
??? blowing up light cone apex .... ???? ....
???? incidence span, pulled back .... ????? ......
??confusion ....
cartan .... cartan ..... ????? ......
??? span from blowndown variety to fiber of blowup ????? ..... ??? sounds screwed up ... ??? ....
?? many projections ... ??? composite span .... ??? ..... ???? ....
?? 3d vector space ... with usual visualization of standard coordinate axises ...
?? cartan torus as "3d aspect ratio adjustment" group ...
?? subjecting 2d subspace to particular aspect ratio adjustment while _not_ subjecting 1d subspace to same adjustment ??? .....
??? "grassmanian blow-down" .... ???? hmmm, so what _does_ for example g2 "one-roll-away" give on _flag variety" as opposed to grassmanian ??? .... ???? .....
?? "pick direction, then magnitude" ... ????
?? span of schubert varieties here ..... ????? ......
?? non-/toric blow-ups/downs as spans of a sort ... ??? (?? better than as co-spans ?? ... ?? ...) .... ??relationship to bit we tried working out somewhat long time ago ... birational geometry .... ??? non-/toric .... ??? ....
?? confusion between ... ??choice of which destabilizer comes first, and which grassmanian coordinate gets omitted to give the partial flag blowdown ... ??? ..... ?? natural to omit the first one ?? .... ???? .... omitting the second one as giving the fiber that gets blowndown ??? ????? ....
??? blowing up light cone apex .... ???? ....
???? incidence span, pulled back .... ????? ......
??confusion ....
cartan .... cartan ..... ????? ......
??? span from blowndown variety to fiber of blowup ????? ..... ??? sounds screwed up ... ??? ....
?? many projections ... ??? composite span .... ??? ..... ???? ....
?? not sure what i mean by this, but ... ?? vague sense in which schubert variety coming form "composite" kaleidoscope group element can be thought of as arising by sort of "convolution" .... ???? ....
?? relationship to "convolution" description of hecke operators and/or their composition described for example by vaughan jones .... ????? .....
"hecke operator with truth value coefficients" ... ??? ....
?? relationship to "convolution" description of hecke operators and/or their composition described for example by vaughan jones .... ????? .....
"hecke operator with truth value coefficients" ... ??? ....
Friday, September 2, 2011
?? so .... ?? seems pretty clear that even if grassmanian blowdowns of kaleidoscope toric variety v have chance of being those "one-step-away" varieties, v itself can't be just those assembled together somehow; they just don't fit ...
?? also, we may have been talking recently as though t were stabilizer of apartment, rather than of generic ordered flag-pair ....
?? fans for _all_ different maximal torus orbits on partial flag varieties and ... ??? maybe some other interesting (?? ...) homogeneous (?? ...) spaces here .... ??? ...
?? which of these toric varieties have full kaleidoscope symmetry ??? ... ???? ....
?? whether arbitrary "flag^n-tuple coset" can be defined in incidence logic ... ???? ....
?? also, we may have been talking recently as though t were stabilizer of apartment, rather than of generic ordered flag-pair ....
?? fans for _all_ different maximal torus orbits on partial flag varieties and ... ??? maybe some other interesting (?? ...) homogeneous (?? ...) spaces here .... ??? ...
?? which of these toric varieties have full kaleidoscope symmetry ??? ... ???? ....
?? whether arbitrary "flag^n-tuple coset" can be defined in incidence logic ... ???? ....
??? .... toric picard (and/or "jacobian ..." ... ??? ...) of kaleidoscope toric variety .... ????? ...... ?? relationship to ordinary picard / jacobian / whatever of flag variety .... ??? push "analogy" here .... ??? bott-borel-weil .... ???? .....
?? complications here (?? ...) concerning divisor theory in singular case ?? ... ?? hmmm, don't forget apparent non-singularity of _full_ kaleidoscope toric variety .... ???? .....
?? complications here (?? ...) concerning divisor theory in singular case ?? ... ?? hmmm, don't forget apparent non-singularity of _full_ kaleidoscope toric variety .... ???? .....
?? maximal affine portion of kaleidoscope toric variety v ... ??? try to understand in terms of ... v embedded in g/b .... ?? nice maximal affine portions of g/b .... associated with apartment .... ?? cover of g/b formed by such maximal portions .... pulled back aling embedding ... ??? hopefully coincide with nice toric fan-nish cover ... ???....
?? relationship between cover and cw-complex structure here ... ??? .... .... schubert calculus ... ??? .... ???? ......... double role of schubert varieties .... giving nice special representatives of cohomology classes, vs cutting (??) variety into pieces of open cover ... ??? .... ??? confusion / paradox here ??? ... ?? "cutting" as requiring real co-dimension 1 .... ????? ......
?? relationship between cover and cw-complex structure here ... ??? .... .... schubert calculus ... ??? .... ???? ......... double role of schubert varieties .... giving nice special representatives of cohomology classes, vs cutting (??) variety into pieces of open cover ... ??? .... ??? confusion / paradox here ??? ... ?? "cutting" as requiring real co-dimension 1 .... ????? ......
?? orbit closure wrt t-action on g/b .... ?? choosing which t vs choosing which orbit ... ???? ....
?? actually choose not just orbit, but basepoint for it .... ???? ....
?? apartment-flag configurations .... ?? bit about shortage of "infinitesimal double cosets" above continuity/wildness threshold ?? ...
?? a1 kaleidoscope toric variety .... projective line .... apartment-flag orientations ... ???? hmm, still below continuity threshold ?? ... ?? cross-ratio still needs another point to kick in ... ???.....
?? generalized morphisms of toric varieties ... ?? kleisli morphisms for monad "direct sum with (k,mult)" ??? .... ?? geometric interpretation via inclusion into monad "monoid k-algebra" .... ??? amounting to "allowing translations" ????? .....
?? does sort of fit with certain past ideas about ... ?? using something like this (...) to unify "x=0" flavor ideals with "x=y" flavor .... ???....
?? possibility of u(1,k) instead of (k,mult) here ??? ....
??? but what about generalizing not just the morphisms but the objects too ?? ... not just kleisli category but eilenberg-moore category .... ?? ...
?? twisted monoid algebras for (?? how exhaustive ??) example ???
?? subvarieties of affine / projective plane ... ??? .....
??? nilpotence ... ??? ....
(?? progression eilenberg-moore / object, kleisli / morphism, ... ??? .... ??? ....)
?? generalized fans here ??? ..... ?? maybe similar to what we already sort of worked out with monad "direct sum with affine coordinate ring of affine toric variety" ?? .... ??? .... ??? some added trickiness / interestingness ??? ....
???? "flatness" issue here ????? ......
?? how this all relates to accidental topos theory ??? .... ??? complications with zero / pointedness ??? ....
?? test this "allowing translations" idea on torus .... ???? not at all clear to me yet whether it's correct .... ???? ....
?? actually choose not just orbit, but basepoint for it .... ???? ....
?? apartment-flag configurations .... ?? bit about shortage of "infinitesimal double cosets" above continuity/wildness threshold ?? ...
?? a1 kaleidoscope toric variety .... projective line .... apartment-flag orientations ... ???? hmm, still below continuity threshold ?? ... ?? cross-ratio still needs another point to kick in ... ???.....
?? generalized morphisms of toric varieties ... ?? kleisli morphisms for monad "direct sum with (k,mult)" ??? .... ?? geometric interpretation via inclusion into monad "monoid k-algebra" .... ??? amounting to "allowing translations" ????? .....
?? does sort of fit with certain past ideas about ... ?? using something like this (...) to unify "x=0" flavor ideals with "x=y" flavor .... ???....
?? possibility of u(1,k) instead of (k,mult) here ??? ....
??? but what about generalizing not just the morphisms but the objects too ?? ... not just kleisli category but eilenberg-moore category .... ?? ...
?? twisted monoid algebras for (?? how exhaustive ??) example ???
?? subvarieties of affine / projective plane ... ??? .....
??? nilpotence ... ??? ....
(?? progression eilenberg-moore / object, kleisli / morphism, ... ??? .... ??? ....)
?? generalized fans here ??? ..... ?? maybe similar to what we already sort of worked out with monad "direct sum with affine coordinate ring of affine toric variety" ?? .... ??? .... ??? some added trickiness / interestingness ??? ....
???? "flatness" issue here ????? ......
?? how this all relates to accidental topos theory ??? .... ??? complications with zero / pointedness ??? ....
?? test this "allowing translations" idea on torus .... ???? not at all clear to me yet whether it's correct .... ???? ....
?? so is a2 in b2 ???? ..... ?? via "axis configuration" variety being projective plane ?? .... ?? question sounds vaguely related to some root system idea we had ... ??? pattern involving rank 2 systems .... talking with john huerta ... idea didn't go far; i almost sort of remember what happened to it ... ??? .... might be able to dig up in notes .... ???? ...
??? a2 borel ... ??? .....
?? no wait, so a2 obviously _is_ in b2 ??? .... no no no .... ok, some silly confusions here ... so(3) vs sl(3) .... ??? .... b2 as so(2,3) so includes so(3) .... ???? ....
?? so in trying to line up a2, b2, g2 in some sort of attempted pattern where each involves something moving on a 2-sphere... ?? resp s^0, s^1, s^0 X s^2 ... ???? ..... ???? 2-sphere always has its so(3) metric symmetry .... ???? .... ??? but that makes it seem like it's not just straightforwardly corresponding to one of the two simple roots .... which would give so(2,1)'s, right ?? .... ??? ....
... hmmm ...
... maximal compact .... ???? ....
?? the thing moving on the 2-sphere as always having non-singular "one-move-away" schubert variety .... ??? making it seem like there _is_ some nice simple long root / short root / invariat distribution / root system pattern going on here .... ???....
?? 2-sphere versus thing moving on it, vs stopped configurations vs paths of motion ... ???? ....
?? would be nice to find notes about that (...) discussion with huerta ... at del taco ...
??? a2 borel ... ??? .....
?? no wait, so a2 obviously _is_ in b2 ??? .... no no no .... ok, some silly confusions here ... so(3) vs sl(3) .... ??? .... b2 as so(2,3) so includes so(3) .... ???? ....
?? so in trying to line up a2, b2, g2 in some sort of attempted pattern where each involves something moving on a 2-sphere... ?? resp s^0, s^1, s^0 X s^2 ... ???? ..... ???? 2-sphere always has its so(3) metric symmetry .... ???? .... ??? but that makes it seem like it's not just straightforwardly corresponding to one of the two simple roots .... which would give so(2,1)'s, right ?? .... ??? ....
... hmmm ...
... maximal compact .... ???? ....
?? the thing moving on the 2-sphere as always having non-singular "one-move-away" schubert variety .... ??? making it seem like there _is_ some nice simple long root / short root / invariat distribution / root system pattern going on here .... ???....
?? 2-sphere versus thing moving on it, vs stopped configurations vs paths of motion ... ???? ....
?? would be nice to find notes about that (...) discussion with huerta ... at del taco ...
?? how entire galois-schubert correspondence "fits inside" kaleidoscope toric variety .... ?????? ... well, entire apartment does indeed "fit inside" it, it seems ... ???? .....
?? for these "mechanical visualizations" of rank 2 simple lie algebra real flag geometries, easy to see the mechanical system, not always so easy to see the symmetries of the correspondence relation ...
?? for these "mechanical visualizations" of rank 2 simple lie algebra real flag geometries, easy to see the mechanical system, not always so easy to see the symmetries of the correspondence relation ...
?? how gimbal picture of b2 relates to penrose diagrams ... ?? ...
?? schubert singularity .... light cone .... one light-beam away ... ?? in gimbal picture one of the schubert varieties as hopefully obviously non-singular ??? .....
?? b2 apartment ... weyl group dihedral ... flipping gimbal over along either of 2 orthogonal axises ... ?? ...
?? how to understand "other" a2 in g2 ... other ball, i mean .... ?? maximal compact of split form .... ???? ..... root system pattern ... real forms ... ?? separating out the long root subalgebra ... ??? .....
?? schubert singularity .... light cone .... one light-beam away ... ?? in gimbal picture one of the schubert varieties as hopefully obviously non-singular ??? .....
?? b2 apartment ... weyl group dihedral ... flipping gimbal over along either of 2 orthogonal axises ... ?? ...
?? how to understand "other" a2 in g2 ... other ball, i mean .... ?? maximal compact of split form .... ???? ..... root system pattern ... real forms ... ?? separating out the long root subalgebra ... ??? .....
kaleidoscope toric variety ... ?? basepoints of the torus orbits ?? ...
?? also the semigroup (?? ...) structure .... ??????.....
?? points of "finite order" wrt the semigroup structure ?? ... in real or complex spectrum ... ?? "unitary" .... ???? ..... "compact order" ????? ...... ?? real forms .... ???? ....
?? also the semigroup (?? ...) structure .... ??????.....
?? points of "finite order" wrt the semigroup structure ?? ... in real or complex spectrum ... ?? "unitary" .... ???? ..... "compact order" ????? ...... ?? real forms .... ???? ....
Thursday, September 1, 2011
?? projective vs complete for toric varieties ... ??? .... structure vs property ?? .... ?? specific line bundle ... ??? ...."canonical" ... ??? .... ??? ..... ... no good ... ??? .... ???? ....
??? "off-center center" ... serre duality ... ???? generalizing some understanding from flag variety case .... ???? ....
?? cohomology .... ???? .....
?? so given dimensional theory, consider .... ???? ..... commutative monoid obtained as follows :
?? take ab gp of iso classes of dimensions ...
?? take subset of "occupied" dimensions ...
?? take submonoid generated by it ...
??? .....
??? conceptual / geometric interpretation supposed to be ... ?? affine toric variety (?? ...) given by .... ???? ..... ????? .....
??? well, this started out as just a straightforward attempt to extract an affine toric variety in the form of a commutative monoid from a [torus acting on an affine variety, equipped with a point of the variety ...] in the form of a [dimensional theory equipped with a model ... ??? ...] ... ?? but then the model started seeming a bit superfluous to me ... though lack of it may in fact be screwing up / complicating conceptual /geometric interpretation ... ??? ....
??? "picard stack" and toric variety .... ????? ......
??? "jacobian ..." .... ?? ... "albanese ..." .... ???? ....
??? elliptic curve .... ???? .....
?? underlying real variety of complex variety .... ????? .....
?? kaleidoscope case?? .... ??? hmmm, as what more or less prompted the idea in the first place ??? .... ?? and maybe it's in fact tying in in a nice way here .... ???? .... ???? ......
?? issue of how monoid algebras can sometimes be twisted .... ???? .....
?? take ab gp of iso classes of dimensions ...
?? take subset of "occupied" dimensions ...
?? take submonoid generated by it ...
??? .....
??? conceptual / geometric interpretation supposed to be ... ?? affine toric variety (?? ...) given by .... ???? ..... ????? .....
??? well, this started out as just a straightforward attempt to extract an affine toric variety in the form of a commutative monoid from a [torus acting on an affine variety, equipped with a point of the variety ...] in the form of a [dimensional theory equipped with a model ... ??? ...] ... ?? but then the model started seeming a bit superfluous to me ... though lack of it may in fact be screwing up / complicating conceptual /geometric interpretation ... ??? ....
??? "picard stack" and toric variety .... ????? ......
??? "jacobian ..." .... ?? ... "albanese ..." .... ???? ....
??? elliptic curve .... ???? .....
?? underlying real variety of complex variety .... ????? .....
?? kaleidoscope case?? .... ??? hmmm, as what more or less prompted the idea in the first place ??? .... ?? and maybe it's in fact tying in in a nice way here .... ???? .... ???? ......
?? issue of how monoid algebras can sometimes be twisted .... ???? .....
"special-relativistic space-time geometry is "round" in the sense that a combination of two "boosts" (the kind of shift in perspective that you experience when you step onto a fast-moving train) is itself not necessarily just a boost anymore."
?? so ... train riding train ... non-commutative ... ??? .... ???
?? so ... train riding train ... non-commutative ... ??? .... ???
?? besides blowdowns of kaleidoscope toric varieties there's also all sorts of sub-varieties (and sub-quotients ...) that should be good to understand ... ??? ...
?? in fact, idea of "bottom-up" approach here, vs top-down ?? ... ??? very "disjunctive" ... ?? "analytic" / "frankenstein" ... ???? maybe more conjunctive (????? "synthetic" ????) approach afterwards ???? ... ????? ......
?? in fact, idea of "bottom-up" approach here, vs top-down ?? ... ??? very "disjunctive" ... ?? "analytic" / "frankenstein" ... ???? maybe more conjunctive (????? "synthetic" ????) approach afterwards ???? ... ????? ......
?? g acts homogeneously on g/b, of course .... and t maps to g ..... ?????? orbit decomposition of g/b wrt t ???? ...... (g,t)-double cosets .... ?? flag-apartment orientations ... ??? but which one to pick ?????.......
?? maybe does seem somewhat promising though ?? ... ?? "all the flags whose orientation towards the standard apartment is ..." .... ??? ....
??? orientation towards standard apartment .... giving orientations towards parts of standard apartment ... ??? .... ??? .... logic ... disjunction, conjunction .... ???? ....
?? also ... _of_ parts, towards parts ... ???? .....
?? irreducibility of toric variety ?? .... disjunction vs conjunction ... ??? ....
?? blow-downs ... ?? same torus, so still orientations towards the standard apartment ... ?? but now you only look at part of the flag ...
?? some orbit closure of t -> g -> perm(g/h) with h general parabolic now instead of borel ???
?? flag-apartment orientations ... ?? classification of flag triples and/or flag [n+1]-tuples ... ??? .... ?? showing up in .... ?? hecke algebras and/or cohomology algebras of flag varieties ... ??? .... multiplication operations therein ... ???....
?? maybe does seem somewhat promising though ?? ... ?? "all the flags whose orientation towards the standard apartment is ..." .... ??? ....
??? orientation towards standard apartment .... giving orientations towards parts of standard apartment ... ??? .... ??? .... logic ... disjunction, conjunction .... ???? ....
?? also ... _of_ parts, towards parts ... ???? .....
?? irreducibility of toric variety ?? .... disjunction vs conjunction ... ??? ....
?? blow-downs ... ?? same torus, so still orientations towards the standard apartment ... ?? but now you only look at part of the flag ...
?? some orbit closure of t -> g -> perm(g/h) with h general parabolic now instead of borel ???
?? flag-apartment orientations ... ?? classification of flag triples and/or flag [n+1]-tuples ... ??? .... ?? showing up in .... ?? hecke algebras and/or cohomology algebras of flag varieties ... ??? .... multiplication operations therein ... ???....
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