?? so ... given Z-filtered (nice ... ??? ...) commutative monoid x, trying to picture "rees construction" applied to it ... ??? dot (x1,j) in x X Z as answer to "is x1 born by the jth stage ... ????? ..... ??possible pos/neg confusion here ??? ... ideal power filtration .... ????? .... ?? hmm, this description makes "rees construction" sound somewhat tautological ... ??? .....

?? "convex" toric filtration ??? ..... ???? ..... ????? ..... ?? somehow stuff here reminds me of some of arnold's ideas ... ????? ..... ??? singularity ... "ray" .... "caustic" .... ??? duality ... ??? .... toric ... ???....

?? any possibility of generalizing convexness of filtration to non-toric case ??? .... ?? geometric interpretation ??? .... duality ... ??? ....

??? "toric hilbert basis theorem" .... ????? .....

??? toric case of "hilbert scheme" ..... ???? ......

?? feeling of entitlement to get nice toric variety / blow-up (... ??? ...) from poisson operad situation .... ??? so can it be pushed through somehow, or can we understand what goes wrong ??? .... ??? bit about "toric spectrum vs ordinary spectrum of toric variety" .... ???? .... ??? maybe distinction tends to vanish in "1d" case ????? ...... 1-by-1 matrix multiplication path integral as sharp (=?= toric ??? ....) .... ??????? ....... relationship to "toric" flavor of abelian class field theory ????? ..... ???? ..... transition from 1 to 2 (??...) as threshold for "non-commutativeness" and also "non-sharpness of path integral" =?= "non-toricness ??? .... ???? ..... ????? .....

(added later: ?? ...hmmm .... but there was that bit about ... ??? invertible module of comm ring r, not corresponding to any invertible action of gl(1,r) ... ????? ..... ??? ....)

??? "dynkin (vs coxeter ?? ...) kaleidoscope" (??? =?= "root system" ?? ...) as not much more than "sufficiently homogeneous (??? wrt discrete symmetry group ??? ....) toric variety" ????? ...... ????? "affine" (?? and beyond ?? ...) case ??????? ..... ???? still get perfectly good toric variety from such ????? .....

??? most obvious toric variety coming form kaleidoscope as conspicuously non-singular ??? ..... ???? vs ... ??? vague ideas about trying to relate kaleidoscopic toric variety to schubert singularities ... ???? ......

???when working on schubert singularities did we in fact venture away from the parabolic (or whatever .... ??? ....) lie subalgebra towards the rest of the weyl co-chamber ??????? ..... ??????? .......

????? so consider "N-fan" .... ??????? ...... ???? fan with extra structure ??? ..... lattice under Z, with all cones containing N+ ??? .... ?????? ...... ?? [Z = line-of-sight] perspective ...... ????? ...... wait a minute, bit of cone / cocone confusion there ... all cocones containing N+ ; all cones fitting inside a half-space .... ????? .....

?? by the way is there room for those more general "cone"s (more general than just official cones and cocones ... ??? ....) to show up here ??? ...

??? more generally, fan "respecting" another fan ??? .... respect as more general than subdivide here ??? ?? maybe still not quite general enough though ??? .... f fan .... ?? "f-fan" .... ???living on lattice mapping to lattice of f .... ????....

??? "toric etale fundamental group" ??? .....

?? so is there a classifying topos for "an object for which all restriction maps are invertible" ?? ..... ????? .......

?? "locally has enough global sections" ... ????dependence on action structure ???? ...

?? nice simple example of non-quasicoherent action .... ???? ..... ?? ?? non-universal N-equivariant map from N-set to Z-set .... ????? ..... ?? for example from N to terminal Z-set, or from initial N-set to Z .... ????? .....

?? is it clear whether concept of "quasi-coherent" is at least "site-independent" ??? .... ??? if so then does this maybe automatically imply that it can be expressed in some sort of "topos language" ... ?? .... ????? ...... ??? without necessarily being "geometric" ... ???? .... ???? .....

??? does "quasicoherent" maybe make sense only over _local_ comm ring ?? ... ??? ....

??? consider for example quasicoherent modules in ringed topos given by ... ??? space of real numbers, with ring object "constant" Z ... ???? .... ???? .....

??? identity homomorphism as localization ... ???? .....

?? hmmm .... ??? ....

?? sheaf of lattices vs of semilattices ... ????? ......

?? hmmm.... "comm ring equipped with complement to subobject given by invertibles" ... ???? conservative homomorphisms ?? .... ????? local homomorphisms as special case ???? ..... ??? anti-conservative as maybe not so easy .... ??????? .....

??? sets and surjections ?????? ..... ??? filtered colimits .... ????? ....

??? objects vs morphisms here ???? ...... ?????

?? arbitrary comm ring as "believing itself local" ?? .... ????? .... ???? ... ....... ??? local homomorphisms ....... ????? ......

??? vs .... formal colimits built out of localizations, but with more general morphisms between them .... ????? .......

??? "local localization" ....???? ..... ???? "(??which??) 2 of 3 ..." .... ??? ....

?? having (?? boolean ... ???) complement to invertibles as seeming rather drastic ??? .... ??? anything a bit less drastic ??? ... ??? ....

??? so consider ... ?? affine plane with origin blown up .... ???? ....

?? affine plane ... k[x,y]

k[x,y:0; a,b:1]/ay=bx .... ?????....

(?? presentation as "computer program"-like ??? ...)

?? exact sequence .... ??

-> /ay=bx -> Z ... ???

?? morally qualify as "short exact" ... ??????? ..... ???? .....

?? fiber over j ... ????....

??? wait a minute .... for negative j the fiber is empty, right ???? .....

... hmmmm .....

?? need to draw a picture ..... ?????? ........



???? .....

??? short exact sequence with base N instead of Z ??? .....

??? so ... ??? consider .... X ... = .... ???? with grading given by ... ??? ??? _is_ this exactly "rees" construction ??? .... ????.... filtration .... to grading ... ????.... ??? .....

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

?? b2 kaleidoscope toric variety as projective line squared with the 4 special (...) points blown up ??? ....

(?? "torus with cross-caps at 4 special points" ....? ????

?? relate this to the flag geometry .... ??? ....

?? kaleidoscope as fan .... ???? ?? cones and cocones here vs the more general closed subsets wrt the galois connection ... ??? parabolic subalgebras vs more general galois-schubert subalgebras .... ?????? .....

maximal torus .... lie algebra vs lie group .... ??? lie group as torus but lie algebra as affine toric variety ..... ???? how to glue copies together .... ???? .....

??? exponential as transcendental ... vs algebraic .... ??? confusion ??? ....

??? "double life of young diagrams" .... ????? .......

?? inclusion of quasicoherent actions among potentially quasicoherent ... ?? as geometric morphism ... ??? ...

?? given "TAG topos", consider unit object for the TAG structure .... ???? relationship to "torus object" (???) we used to think about ??? ..... ??? ....

??? non-toric analog ..... ????? .....

?? ... hmmm .... ???? .... ??? "core" ??? ..... ????? ..... ??? "n-affine ..." ... ???? ..... ????? .....

??? doctrine morphism ... ???? TAG doctrine to ringed topos doctrine .... ???? forcing objects in syntactic category of TAG theory to be quasicoherent .... ???? .....

??? ....

???? "abelian category equipped with commutative ring object" .... ?????? not evidently parsing yet .... ?????? ...... ?????? .....

[???actually even original toric case not parsing now ... ????? .... ??? confusion ???? ..... TAG unit in TAG topos as not topos commutative monoid .... ???? .... ????? ..... ???? .....

?? comm monoided topos as TAG topos with property .... ?????? .....

?? vs (?? ...) ... ??? maybe somehow systematically deriving TAG / topos compatibility structure (?? ...) from example of comm monoided topos ?? ....

?? blowing up point p of space x vs taking projectivized tangent bundle of x and then ..... ???? ..... ??? ....

?? collapsing each fiber except that over p ??? .... ???? .....

?? geometric morphisms between accidental topos and toric zariski locale .... ??? ....

?? retraction as not preserving filtered colimits (?? so in particular not left adjoint ... ???? ....), but section does ????? .... ????

??? whether invertible module is automatically quasicoherent .... ????? ......

?? comm monoided topos vs "TAG topos" .... ????? ......... relationships ..... ??????? ......

?? blow-up (??? ...) relationship between a2 and g2 ?? ..... ?????? ......

?? where a maximal parabolic subalgebra (for example ...) lives ... ??? "co-weight space" ??? .... cocone therein ... ???? ..... ??? dual to cone where corresponding extremal weight lies .... ???? ..... ?? more generally, higher dimensional wall of weyl chamber as cone dual to non-maximal parabolic subalgebra .... kaleidoscope-as-fan toric variety ... ??? .... conceptual / geometric interpretation ... ??? ....

notes from discussion with alex today

... we wondered whether, for example, rees algebra (??? ...) associated with ideal power filtration of certain (?? ...) ideal in cocone (?? corresponding to some infinitesimal but perhaps "irregular" neighborhood of the "basepoint" of the corresponding affine toric variety ... ??? ...) might relate to subdivision of corresponding cone in certain corresponding way ..... ????? ....... ????? .....

?? also tried to relate something like "standard blow-up of basepoint" to something like "barycentric subdivision" .... ??? at one point i thought that syzygy might get involved, but then thought that maybe it was just a relation ..... ???? ......

?? point in cone giving grading of cocone .... ???? .......

?? classifying monoided topos for quasicoherent action .... ???? ... ???? ?? explicit description ... ??? ... non-toric version as well .... ???? ....

x comm monoid ...

y comm x-monoid ... ??? or Z-graded comm x-monoid, for example ... ??? ...

?? for example x = ... y = /ad=bc .... ??? ....

?? x-fan ... ???? ....

??? 3-place chain complex here ??? ....

?? comm monoid or Z-graded comm monoid in accidental topos... wrt TAG structure ... ???? ...... ???? ....

?? some of this stuff in paper ??? ....

notes for discussion with alex tomorrow

?? galois-schubert correspondence and toric geometry ....

?? stratification ... ??? "toric perverse sheaf" .... ????? .....

?? spirograph-flower picture of co-fan ... ???....

!!! work on paper .... ???? .......!!!! .....

?? a-series : more general dynkin (?? ...) diagrams :: certain grassmanian variety is toric : partial flag variety contains interesting toric variety .... ?????? .....

??stratification of toric variety ??? .... ??? understanding woolf's stuff ??? .....

?? "apartment structure" (?? ...) on partial flag variety / schubert variety ... ??? .... ... toric structure on toric variety ... ??? ..... "stratification" ...... ?????? ......

?? "perverse sheaf" ....

???? "toric perverse sheaf" ???????? ..... ????? .....

?? galois-schubert correspondence galois connection .... ???? toric interpretation ... ???? ..... roots .... figures .... ??? but ... ?? how did that go, interpreting the figures as weights or something .... ??? "highest weight" ... ????? .... ???? ??? annoying sign choice ... ????? ....

?? kaleidoscope as _co-fan_ ... ??? ....

?? hmmm ... ?? vague memory of ... ??? having some reasonable visual intuition about co-fan while talking to baez on his last visit here ... ?? in car and hotel room ... ?? at least, visual intuition about co-face .... ???? .... ?? actually more tactile than visual ?? .... feeling edge (?? ...) of 3d object .... ????? .....

??? given face of cocone, corresponding coface of it

??? vague intuition about "infinitesimal neighborhood of face" .... ??? or "localization" ????????? ..... ?? "the only constraints that you keep are the ones that "touch" the given face" .... ????.... ?? "that touch face f" = "for which f lies completely on border hyperplane" here ... ??? ...

?? hmmm, sort of "flower" picture of 2d (maybe higher too ???) co-fan ??? ... ?? "spirograph" ??? ... ??? dual fan ???

?? hmm, maybe "flower" (with nicely separated (??...) petals (?? = cocones ?? ...), no petal over half a circle, circle of petals closes up after just one revolution ... ????...) is more apt than "spirograph" here .... ??? hmm, or maybe "spirograph flower" is better ... hmm, actually i'm still a bit confused ... ??? ....

?? "daisy" ??? ... ??? maybe other fancy names that people who think about certain sorts of curves (??) use .... ??? ....

??spirograph flower given by "modulus as function of argument" ???.... ??? zeros of such function ... ???? ..... ?? "fourier ..." ... ???? ....

??? trying to understand toric quasicoherent sheaf in terms of spirograph flower ... ???? ....

?? so ... when you blow up a point (??) on a toric variety / subdivide co-dim 0 (??) cone in fan ..... ???? ..... the new subvariety as ... ??? maybe not precisely a toric variety .... ???? .....???? .... ???? .... ?? bit about various sorts of ideals of interest .... ???? ..... generated by toric functions vs by equations between such ... ??? .....

?? might the subarieties which don't especially seem like they "should" (?? corresponding to ideals generated by single toric functions ... ??) be nevertheless "accidentally" toric after all ?? .... ??? or "generalized toric" ... ??? ....

??? "generalized fan" ... ??? as maybe somewhat clear from discussion about "non-linear fan" ... ???? ....

??? various ways of subdividing cone, vs ... ?? ess just one way to blow up point .. ??? .... ????? .....

?? blowing up a higher dim (??? good ???) subvariety on a toric variety ... ???? .... ?? subdividing higher-dim cocone in cofan ... ????.... ??? ....

subdivision, stackiness, poisson blow-up ... ?????.....

??? "subdivision" ... "blow-up" .... "explosion" ... ???? ....

??? comparison between two ideas about "toric morphism" ... 1 : comm monoid hom and / or TAG theory morphism ... ??? .... vs 2 : "torue-equivariant map riding torus morphism" ... ??? trying to compare in detail ... ?? nice simple examples ?? ....

???functor from _comm monoid_ to _dimensional algebra_, assigning to commutative monoid x the dimensional algebra [monoid-alg(x) graded by fraction-group(x)] ... ????

?? extending this in nice way to functor from _TAG theory_ to _AG theory_ ??? .... ???? ..... ??? extra objects, corresponding to .... ??? .... fraction-group of "center" ??? .... ???? ..... ???? .....

?? comparison to leaving out the grading ... ???

?? toric (?? ...) aspect of "blowing up "comm alg" sub-stack (?? ...) of "assoc alg" stack" ... ???? .... ??????????????? ..... poisson alg .... ???????...... ??? level slips and strange loops ... ???? .....

????level slips and strange loops ??????????? .......

?? dim 1 face (?? "ray" ??) of cone dual to cod 1 face of cocone ... ??? visualization ... ???... ?? "nail down the points on the cod 1 face; in fact mod out by them, and map the residual ray to the standard ray ..." ???

?? dim 2 face of cone dual to cod 2 face of cocone ... ??? visualization ... ??
?? "nail down the points on the cod 2 face; in fact mod out by them, and map the residual dim 2 "silhouette" (?? ...) wedge to the standard ray ..." ??? ....

?? cocone co-face _not_ as just "omitting corresponding cocone face" ... ?? do we really still believe that cocone co-face has corresponding cocone face ??? ... ?? well, maybe yes ... ??? ..... ?? some confusion ... ???....

?? can't just take an extreme ray of a cone and then take the convex hull of the other extreme rays ..." ... ??? at least, not without ... ??? expecting silly things to happen ?? ... ???? .....

?? corresponding to cocone face is cocone co-face ??? ...

?? dual to cone face also corresponding to cocone face ???

?? tensoring cocone with truth-value comm monoid t ... ?? then homming it into t also, and comparing that to combinatorial face structure of cone ... ???

?? ["subdivision morphism" between toric varieties] approached this way ... ???...

??? "dual to cocone face is cone face where the inequality it represents is the pullback of the fundamental inequality" ??? .....

"fundamental inequality" 0 <= p ???? ....

?? ... "fenchel ... " ??? ....

?? discussion with todd ... me at first thinking that cone / cocone duality was just about "N as schizophrenic between comm monoid and comm monoid" (???? ....) ... ?? but then todd pointing out what (?? ...) was wrong with that ... so then i suggested "0 <= f(x) galois connection between x and f" idea .... ?? possibility of distinction between these ideas as relating to some sort of "cohomology for comm monoids" ?? ... ????? ......

?? "folded schizophrenia" ... ??? torn between two identical copies of self ... ??? ....

??? but now ... "dual to cocone face is cone face where the inequality it represents is the pullback of the fundamental inequality" vs ... ?? contrasting idea ?? ... ... truth-values ... ????? ..... ??? or maybe not contrasting ?? ?? but ... i want to say / understand here about ... ?? how truth-values are fitting in here ????? .... ?? "zero on cocone face and "tribal infinity" off it" ?? ...

?? so what _was_ the anomaly (?? ...) that todd pointed out ??? ... ?? corresponding cone as not always precisely hom[cocone,N] ?? .... ?? example ?? ...

??? "boundary" fan of cone ??? .... ???? ..... ?? "child's drawing" and "through the looking-glass" ??? ..... "polytope" vs "combinatorial manifold" ... ?? ... "morse theory" ... ??? "removing stacky point to obtain projective space" ... ???? ...

?? through the looking-glass ... adjoining cones in fan, glued together along mutual face ... ???? ....

?? anti-fundamental inequality .... ????? ..... subdivision morphism ... ??? ....

?? case where adjoining cones have same "affine hull" ??? .... vs other case ... ??? .....

?? adjoining highest-dimension cones ... ??? ....

?? "toric morphism" ... ??? riding torus morphism ...?? ... equivariance ... ??? stackiness .... ????? ..... "sleeping with one eye open" ... ?????......

?? toric aspect of apartment structure (?? only ??) in a-series case as related to associative-rather-than-merely-lie aspect ??? ... ??? ...

?? fan as special case of "non-linear fan" ?? ...

?? laurent polynomial vs rational function ... ??? ....

?? inclusions from N^2 into ZxN ... ??? ....

?? non-snug inclusion from cocone x to cocone y as ... ???? certain faces of spec(x) not in image of spec(y) ... ??? those in which ... boundary points of x exhibit genuine "borderline behavior" that can't happen at an interior point .... ???? ...

?? faces (?? vs co-faces ???...) of cocones ... ??? .... ???? ....

?? blowup of origin of affine plane ... ?? ...

?? some sort of "barycentric subdivision" of cone ?? ....

?? non-comparable co-/cones ... vs comparable but non-snug ... ??? .... ?? "co-snug" ??? .... ??? ....

?? so poset of toric varieties on a lattice as not co-/complete ?? ... ???...

?? does it have some particular kinds of limits and colimits, though ??? ...

?? is there an infimum of fans ??? .... ??? ...

?? different approaches to fans ... "downward-closed" vs "closed under (non-nullary ...) intersection" ... ???? ..... ?? any relationship to "measure vs random variable" ??? ... ?? ....

?? so ... ??? consider the geometric homotopy-colimit of the TAG theories corresponding to all the cocones in the upward-closed collection generated by ... ?? well, try various flavors of arrangements here ... ???? ....

?? 2d lattice ...

?? overlapping cones ... non-overlapping ... ?? ...

... ?? ...

??intersection of delocalizations as not a delocalization in general ... ???? .....

still confused ... locale where dominance relatonships are localizations, vs .... ??? ....

?? two dominance relations ??? .... ??? ....

?? ?? "basic (toric) open" ... ???? .....

?? ... "toric opens" of a toric variety ... ??? ?? presumably we had some simple description of how this relates to fan ... ???? ... ?? if it forms a finite distributive lattice / heyting algebra then what's the corresponding poset ??? .... ???? ....

?? vague memory of ... pictures of cubes ... ?? barycentrically subdivided ?? ... ??? ....

?? so given an upward-closed collection of cocones, consider those cocones in it which are herditary co-faces of minimal such ... ???

?? hmm, does the upward-closed collection of _all_ cocones (?? for example on 2d lattice ... ???? ....) cause a big problem here ??? ..... ???? .....

?? try "defining order on natural numbers in terms of multiplication" (after in terms of addition ...) on kenji ??

?? toric analog of cremona group ....

?? toric variety .... projective variety .... ????

?? dimensional algebra ... ?? ....

?? .....

??? toric analog of topology on field ?? .... ????? ......

?? so consider inclusion morphism between cocones that's not a localization .... ????geometric interpretation ???? ... ???? ..... vs that of localization as "toric zariski-open (reverse ...) inclusion" .... ????....

?? for example x and xy as generators of subring of k[x,y] ?? .... ????....

??? formal colimits wrt localizations vs formal colimits wrt more general morphisms here (??? ...) .... ???? .... ??? some semi-systematic non-trivial way of thinking of former as special case of latter .... ???? ....... ???? ..... "planar graph theorem" ... ??? ..... ??? maybe ... ???property of diagram, less restrictive than requiring all arrows to be realized as localizations, but still sufficient to yield theory of "combined doctrine" ... ???? .....

?? delocalizations of the rational numbers, for example .... ???? .....

?? pathological for one cone in fan to be contained within another without being a face of it ?? ....

?? maybe prohibiting this kind of pathology still allows the fans on a given lattice to nicely form a lattice (!! different meaning !! ...) ... ??? ... with the dominance relations realized as nice (?? but generally not geometrically monic ??? ...) morphisms .... ???? .....

?? how this (...??...) might relate to "borrowing geometric (?? ... homotopy-)colimits from TAG theories" ?? .... ???? ...

?? "torus" definition of "toric variety" ... ?? as telling us anything about whether to exclude this sort of alleged pathology ?? .... ???? ....

?? non-toric analog of fan for certain reasonably nice sort of variety ... ??? pasted together from affine spectrums of subrings of field of meromorphic functions on the variety .... ????? ...... ???? "closure of generic point as entire variety" ... ???? .....

?? maybe trying to use this to approach those certain questions about non-toric analogs of theorems about how un-general TAG theories of toric quasicoherent hseaves (?? ...) are ... ????....

?? "holomorphic structure on meromorphic line bundle / vector bundle" ... ???? ....

?? "divisor" ... ???? .... ?? ....

?? maximal variety based on given field .... ????

?? analog of "no two cones in fan overlap in cone of nonzero volume" ??? .... ????....

?? blow-up as giving (?? toric or non-toric ... ?? ...) birational equivalence ... gluing variety and blow-up of it together ... ??? whether blow-up "supersedes" original ... ???? .... ?? confusion about fan ... ??? ... ??? "geometric colimit" ... hmmm .... ???? .... ?? original as superseding blow-up then ??????? .... ???? .....

??? geometric monic ... ????? .... ?? blow-up .... ???? .... ?? localization as automatically geometrically monic ... ??? .... ??? but ... ??? ... ??? .... ??? ...

?? dimensional category where all nonzero quantities are invertible ??? ... ??? toric analog as tame stable 2-type ... ??? .....

?? _is_ toric variety birationally equivalent (in ordinary sense ...) to its torus ?? .... ???? .....

?? getting toric dimensional category from fan ???? .....

?? wpa says:

The toric variety of a fan is given by taking the affine toric varieties of its cones and glueing them together by identifying Uσ with an open subvariety of Uτ whenever σ is a face of τ.

?? ... ??that seems to leave possibility of one cone in fan being contained in another without any face relationship ... ???? ?? is that really what they want ??? ?? does it give "separated" thing ? ... ???? ....

?? "delocalizations" of given field / torus (in toric case ...?? ...) ?? .... ???category of such delocalizations ??? .... ??? as forming lattice ??? ... ??? .... ??? .... ?? morphisms between elocalizations as ... ???? ....

?? thought for a moment that it might make sense for toric variety to be "[truth-value]-valued sheaf" (?? ...) here ... ??? ..... ???? ..... ??? ...

?? hmm, maybe it _does_ ?? ... ??? ... ?? encoding downward-closed subcollection as something like "anti-chain" ... only include irredundant generators ... ??? ....

?? kind of equivalence (??? ...) of diagrams for purposes of taking co-/limits ... ??vaguely reminds me of kind of equivalence between graphs (?? ...) involved in theorem about planarness .... ?????? ..... "subdivision ... " .... ??? ...

?? "locale of localizations" vs ... ??? "delocale of delocalizations" ??? ... ???? ....

?? localizations of x as part of delocalizations of x' = "spectrum of field of meromorphic functions on x" ??? .... ???? ....

?? but delocalizations of x' as forming poset that's some sort of union of posets of "poset of localizations" kind but not of that kind itself ??? .... ??? .... ?? maybe need some sort of (??co-)completion process ?? ... ??? .... ?? maybe some sort of hopefully obvious "canonical" (?? ...) grothendieck toplogy on category of delocalizations ... ??? ...

?? don't forget ... ??? previous attempts of ours to push analogy of ordinary varieties to toric varieties, emphasizing delocalizations (i think ... though not under that name ... ??? ....) ... "frankenstein doctrine" ideas ??? .... ???? ....

?? idea of morphism between toric varieties as ... ?? riding morphism of toruses ??? ..... ?? how well does this work in the _affine_ toric case ???.....

?? functoriality of (toric ... ??? ...) "birationalization" ?? ... ???....

cofan picture ... ??? ....

?? so consider ac=bb .... ?? ....

a = x+y

c = x-y

xx - yy = bb

bb + yy = xx

?? inverting a leaves out x+y = 0 ...

?? inverting c leaves out x = y ... ????....

?? inverting both should leave a torus ??? .... ??? ..... ??? hmmm, so perhaps it does ... ?? 4 half-cones in a double cone ... ?????? ..... ??? blowing up apex ... ???? ....

?? for b = 0, ac = bb as cross ...

?? "forgetting b" ... ?? "ramification" at bars of cross ?? ... ??? ... ?? real vs complex ... ???? ....

?? "basis of periodic functions of all periodicities, all of same "shape" ..." vs "basis of functions all of same "shape" but at all "scales" ..." ?? .... ????

?? so what about this idea that gunnarsen seems to attribute to weyl, of treating basic irrationality proofs (for example square root of 2) as galois-theoretic ? ...

?? "morphism of toric varieties" preserving some sort of internal hom of toric opens ?? ... ???? .... ???? ....

?? quantale ... ??? ....

notes from discussion with alex today

??"maximal" toric variety ... ??? asf os...

???? .......

?? need extra relations for blow-up in singular case ???? .... ??? ...


... check paper notes .... ???? .....

notes for discussion with alex today

paper ...

1 try to make progress on acutally writing it !! ...

2 singularities of toric varieties ... and blowing up .... ?? ... application of latter to resolution of former ...

2.1 singularities of toric varieties ...

2.1.1 simple example ... "checkerboard" ...

2.1.2 ?? other examples ... ??? ?? tangent cone ?? ...

2.2 ?? "toric galois theory" ?? ..... ?? and so forth ....

2.2 toric blowing up ...

2.2.1 example... blowing up point in projective plane .... ?? developing general idea of _toric_ blowing up ... ??? .....

d=ax, e=ay, f=az, g=by, h=bz .... dg=ee, dh=ef, eh=fg ... ??with d,e,f,g,h all grade 1 ?? ...

?? stuff you can invert .... d,f,g,h,df,dg,fh,gh,dfgh ... ?? ....

?? so .... inverting dfgh, the degree 0 stuff is lattice generated by x = d/f and y = e/f ... ?? so we should maybe try expressing the other localizations as cocones inside of this .... ?? ....

?? so consider inverting just d, for example .... ?? then we seem to get cocone freely generated by e/d = y/x and f/d = 1/x; is that correct ?? ....

?? inverting just f ...

??maybe it would help to look first at inverting the pairs ... ?? ....

?? inverting df, we get one side of the d/f = x line ... ????....

?? inverting dg, we get one side of the d/g = dd/ee = xx/yy line ... ???.... ??hmmm, so is that really the x/y line ??? ....

?? inverting gh, we get one side of ... ???

?? morphism of affine toric varieties corresponding to inclusion of checkerboard lattice into chessboard lattice ... ?? ... ??? geometric interpretation .... ??? "covering" ???? .... ??? ..... ????? .....

??? relationship to kummer's chemistry analogy ?? ...????.... ??? ....

?? actually nice (... ?? ...) maps going both ways ... actually lots of maps both ways; would be nice to understand them all ... ???? .....

?? relationship between .... geometric fiber size and algebraic cokernel size .... ?? ... ??? seems like pretty reasonable guess ... ?? .... ?? kummer ... ????? .....

??? hmmm, so then what about ... "galois group of covering" here ... ?????? ...... and / or "galois correspondent" .... ????? ........ ???? abelianness here ??? ..... ???? ....... .... torus of toric variety .... stable 2-group of toric dimensional theory .... ??? .....

??? "radical extension" of toric varieties ... ??? shortage of other kinds ???? ..... ?? "snugness" and "preservation of face structure" ??? .... ?? snugness in algebraic number theory .... ??? .....

?? galois-theoretic aspect of accidental topos ... ??? ..... .... kummer ... ???? ...

??? toric ramification .... face ?? .... ?? "toric local ..." .... ??? .... "toric birational ..." (?? stable 2-group ... ??? ...) .... "toric algebraic group" .... ?? "toric langlands ..." .... ??? .....

?? "toric [closure of geometric image ...]" ??? ....

?? "sophisticated fiber" ... (?? flatness ... ??? ...) ... dimension calculation therefor, vs for "zariski tangent space at singular point" ... ?? in toric case .... ???? .... ... spectrum of cohomology ring ... ???? .... ??? ... ???? ...

?? so ... ?? if blow-up systematically gets along with toric structure in some nice way (still not completely convinced of it yet, but ... ?? ...) then ... ??? ought to look for (and be able to find ... ?? ...) some nice conceptual explanation of why ... ???? ... ?? perhaps at TAG theory level .... ?? .... ??well, perhaps the "free commutative monoidal action on ideal, with Z-grading ..." idea goes through in toric context, but ... ?? not clear how conceptual this is, yet ... ??? ....

so does lurie (and/or ... ?? ...) use "locally presentable (infinity,1)-categories" as examples (in certain way ... ?? ...) of (infinity,1)-toposes ???? .... ???? ..... ??? .....

??? possibility that that stuff that [hazewinkel and arnold and [that dutch guy who wrote a master's thesis on this stuff... can't remember their name offhand ...] and so forth ...] talk about relates to .... kaleidoscope relationship between toric singularities and schubert singularities ... ???? ......

??? ... kleinian singularity .... polynomial algebra .... ???? anything toric going on here ??? ..... .... moduli stack of elliptic curves .... ??? ..... .... discrete group .... ???? ...... ???? ......

d=ax, e=ay, f=az, g=by, h=bz .... dg=ee, dh=ef, eh=fg ... ?? specializing to d=e=g=0 .... ????as giving a projective line, which is supposed to be the blown-up point .... specializing (?? ...) to d invertible .... ????? ??? both a and x being invertible .... ???? ...... ???well, maybe a isn't too important here ... ??? .....
?? well, when d is invertible, then g and h can be eliminated, co-eliminating the first two relations, and making the third one tautological .... ???? ..... ??? so that seems to help make it plausible that this really is the projective plane with a point blown-up .... ???? ......

?? so.... is this toric ?? ..... ?? .... ?? coming from fan ???? .... ??? ...

?? first try getting torus from it ... ??? .... ?? so, tame stable 2-group given by 3d lattice mapping to 1d .... ????...... ??? then we're supposed to take the kernel or something??? .....

?? just as a wild guess, maybe this will be something like ... ???? 2d lattice divided into 3 cones, one of them checkerboard ... ????? ....... ??hmmm, but ... ??? non-singular toric variety can't be made out of singular pieces, can it ??? ....

?? well, so for a perhaps somewhat different wild guess, maybe there's more cones in the fan here .... say maybe 5 ... ???...... ?? hmm, what dimension (?? ...) cones am i talking about here ??? ...

?? well, so what are the "localizations" (?? ...) here .... ??? also, what are "the localizations of a fan" ??? ..... ?? ....

?? stuff you can invert .... d,f,g,h,df,dg,fh,gh,dfgh ... ???? hmmm, sounds sort of like face-lattice of square ... ????? dfhg ... ???? ..... ???so are we getting some sort of square-ish fan here ??? ...

?? so maybe what's happening, roughly, is that we start with the fan for the projective plane ... and one of the three sectors corresponds to the point that's being blown up .... and that sector is getting "bisected", which somehow corresponds to the point getting blown-up, aka "(functorially ... ??? ...) replaced by a projective line" .... ???????? ......

??idea of resolving singularities by "triangulation" of cones in fan ???? ..... ???? .... .... ?? but what about functoriality of blow-up ... ?? ... ?? ....

?? cone / cocone confusion in trying to understand blow-up / "bisection" ... functoriality thereof ... ????? .....

???hmmm..... ??? so consider projective plane with all 3 apartment points blown up .... ??? is this ess just the a2 kaleidoscope toric variety ??? ..... ??? if so, then what's going on here ???? ......

?? hmmm ... ?? maybe more than one way to interpret kaleidoscope as fan ... ???? idea that for certain (maybe hopefully somewhat obvious ?? ...) way, resulting toric variety is _non_-singular ??? .... ???? weyl chamber as ... "orthant-ish" ... ???? .... ??? weight lattice vs root lattice ??? ..... ????? ....

?? try to tell kenji needs to be more outloud about when we're not communicating .... ?? .....

??? comparing line bundles over projective plane with point blown up to those over original ....

?? projective plane with point blown up as toric ??????? ......

?? vague memories of stuff about ... terry bisson ... categorified fourier duality ... smc vs cartesian smc .... ??? .... ?????? ..... ?? resonating now with "toric stack" .... ???????

2d affine toric varieties arising as cones of toric projective embeddings .... ???? .... ??relatively few because of shortage of 1d projective toric varieties?? .... ???? ....

?? hmmm, but lots of other stuff to play around with here ..... ?? for example graded commutative monoid /ac=b^[2n] .... ??? geometric interpretation ... ??? toric stack .... ??? "1d toric orbifold" ..... ????? ......

??hmm, well perhaps _any_ affine toric variety can be seen as "cone of toric multi-projective embedding of toric stack" ... ??? in multiple ways ??? ..... ??? .... just choosing homomorphisms to lattices .... ??? .....

?? but hmm, any interesting interaction between "1d toric orbifold" (???such as now you mention it traditional "moduli stack of elliptic curves" ... ??? ....) and .... ?? more vanilla stuff about "curves" ??? .... ????? .....

r comm ring .... j ideal in r ..... as r-module ... take symmetric r-alg of it ... left adjoint process so preserves presentation ... ?? ....
>>
for example r = k[x,y] ... j = ... j presented as ....

?? so .... ?? k[x,y,a,b]/ay=bx ?? .... ??? or ... k[x,y:0; a,b:1]/ay=bx ... ???....

?? so suppose that we specialize to some point (x,y) not (0,0) ... ????? then we get a linear dependency between a and b ... ???? but not at (0,0) ..... proj of k[a,b:1] as projective line ... ????.....

r graded comm ring .... j graded ideal in r .... as graded r-module .... take symmetric graded r-alg of it .... preserves presentation ... ??? ....

r = k[x,y,z:1] .... j = ... j presented as ... ?? k[x,y,z:(1,0); a,b:(1,1)]/ay=bx

?? specialize to (x,y,z) = (0,0,1) .... ???? ???z=1 then trivializes line object (1,0) .... ?? in effect getting k[a,b:1] .... ????? ...

?? so ... should (1,1) (???or something) then be ample here ???? ..... ??or maybe (2,1) ??? ......

ax,ay,az,bx,by,bz .... ay=bx .... ???? ....ax*by=ay*ay ... ax*bz=ay*az ... ay*bz=az*by ... az*by=bz*ay .... hmmm, those last two seem to be the same ... ?? and "close to a consequence of" the two before that?? ?? which might help the dimension arithmetic to work out ... ????.....

d=ax, e=ay, f=az, g=by, h=bz .... dg=ee, dh=ef, eh=fg ... ?? specializing to d=e=g=0 .... ????as giving a projective line, which is supposed to be the blown-up point .... specializing (?? ...) to d invertible .... ????? ??? both a and x being invertible .... ???? ...... ???well, maybe a isn't too important here ... ??? .....
?? well, when d is invertible, then g and h can be eliminated, co-eliminating the first two relations, and making the third one tautological .... ???? ..... ??? so that seems to help make it plausible that this really is the projective plane with a point blown-up .... ???? ......

?? so what about the picard group of this variety ... ????? ...... ?? well, so what about pulling line objects back along the projection ??? ..... ????? ..... possibility of non-ampleness here ... ??? ....

?? whether for example spec(k[x,y]/y^2=x^3) should (...) be (...) a toric variety according to conventional "torus action ..." philosophy ... ???? ....

?? toric variety weakly modded out by some part of its own torus ... ??? accidental topos thereof ...????.....

?? also case where accidental topos is sequences of sets, or of objects from simpler accidental topos ... ??? ....

??? f : N^d -> Z ... ???? where ... image of f perhaps "doesn't cleanly split off" ??? .... ?? anything like this ??? .....

??? .....

?? try blowing up point on projective plane ... ???....

?? so consider .... ???? ideal power filtration of the ideal generated by x and y in k[x,y]/x^3-y^2 .... ????.....

???? ...... ???? .... ??? ... hmmmm .... ideal power filtration of non-principal ideal .... ????? ......

??? confusion .... ????? ...... 0,23,45,67,... ???? .... ??? idempotent ideal ?? .... ?? exotic (?? ...) instances of associated graded alg of ideal power filtration .... ???? ...

?? associated graded ... ????.... ??? vs zariski tangent space .... ????? ...... ???? .....

??? have we thought about this before and forgot, or ... ??? ...

??? certainly we've thought about "double point" singularity ... ???? ?? this as degeneration of that ???? ....... hmmmm ....

??? "co-unit ideal" of monoid alg of comm monoid .... ???? .....

?? "filtered commutative monoid" .... ???? ...... ???? any relationship to what ben-zvi mentioned about carlos simpson ?? .... ????? .......

???? associated graded comm monoid of filtered comm monoid ... ??? .... ?? hmm, evidently the associated graded alg of a filtered comm monoid algebra isn't a monoid algebra in general ... ????...... ?? not that far off though ??? ..... ?? for some reason vaguely reminds me of ... ??how did that go .... ???"zero as point vs as ... ?? operator ..." ???? ..... or something ??? ....

?? "conical singularity" ... ??? is this actually making sense now ?? .... ??? related ideas ... ??? .... ??? criteria for ... various stuff ... ??? non-singularness, for example ... ???? ...... ?? ... local vs global .... ???? ..... ??? "local ring" ... ???? .... ???? maybe this _is_ making sense ??? .... ???? polynomials vs power series here ???? ...... ???? "completion" ??? .... ???? ....

?? blowing up co-dimension 0 subvariety .... ????? ......

?? for paper ... ??? ... ??? "combined doctrine" stuff ??? .....

?? classification of affine toric varieties in 2 dimensions ... ??? relationship to ... ?? cokernel of 2-by-2 matrix of integers ... ???.... ??? ...... ?? certain kind of morphism of torus and/or toric variety ?? .... ??? ....

?? some sort of morphism of affine toric variety corresponding to inclusion of cones .... ???? ..... ???? ....

?? so... given endomorphism f of Z^2, consider [... ?? inverse image under inverse (presuming it exists) of g := "f tensored with R" ...]] of N^2 (as "quadrant-lattice" in source of g = target of g^[-1]) ... ????..... ?? any particular nicer way of saying this ?? ....] ??? as affine coordinat monoid of affine toric variety ... ?? ...

?? vague feeling ... stuff about invariant distributions on flag varieties and so forth .... somethng we ran into there ... ??as somehow similar/related to ... ?? how a "face" of a toric variety adheres to it .... ???? ..... ???? singularity of schubert variety .... ????? ..... ?????? ..... .... hmmmmm ..... ?????? .... ??? are we maybe finally figuring out something about singularities of affine toric varieties ?????? ...... ??????? ........

??am i claiming that cone of "quadratic" (...) projective embedding of projective line is singular ??? .... ????hmmm, maybe that would make perfect sense .... ????... .... hmmmm .... ???classic conical singularity as having "toric" structure ???? ....... ??????? ....... ??? in some sense should have already been obvious, maybe ???? .... ... ?? ... (?? ... hmmm, still some confusion between ... remembering vs forgetting grading .... ???? ...... ?????.....) .....

??? toric world as to some extent then good playground for toy examples of singularities .... ????? ...... ????? ......

?? lot of puns on cone / conic / conical here .... ????? ..... ?? mapping cone .... ???? .....

??? blowing-up of toric singularity ..... ?????? ........ ... grading ... ???? ....

??? kaleidoscope as fan here (... ???....) ?????? ......

??? kummer's chemistry analogy .... ??????? .......

?? vague memories of other weird ideas relating to ways lattice can relate to hyperplane in vector space .... ?? .... ??? 2d case ??? ..... ???? .....

??relationship between ["cohomological" aspect of affine coordinate monoid of singular affine toric variety ... ??? or is this aspect maybe somewhat special to 2d case ????? .... ??? maybe misleading to try to generalize from that case ??? .... ????? .....] and [complex of ideas relating to "deformation cohomology" and blowing up of singularities ... ??? .... and so forth ... ???] ??? ....

??? .... "singular morse theory" .... cartographic group action ... ??? ..... mapping cone ..... ????? ...... .... ???? .....

ideal .... 2-squares theorem ... finally ... ??? .....

ideal ... lagrange .... ???what's going on here .... ???....

alex ....

todd ....

corfield ,,,

?? somewhere recently i read something about ... ?? way of defining adeles that seemed somewhat less annoying than usuql way ... ??? ... ??? maybe in wpa ??? ....

?? ...

k = 1^(1/20)+1^(19/20) ....

k^2 = 2 + 1^(1/10) + 1^(9/10) ...

?? ramification here ... ????

?? martin ... ?? alex mennen .... mike stay .... ?? emilio faro ....

?? ... cleaning out office ... ???

?? complex numbers as schizophrenic object between _set_ and _comm ring_ .... unit of resulting monad on _comm ring_ .... ?? specialized to case of number ring ... ????.....

??? some confusion here between "hom" and "tensor" ?????? ...... ???????.......

???? something about _algebraic_ commutative ring here ??? .... and/or "analytic" ????? ???? .....

??? underlying algebraic / analytic additive group .... cokernel of "unit" .... ????? ...... .... ???? "abelian variety with complex multiplication" .... ?????? ......

discussion with mike stay ....

"trace" of endo-profunctor ....

??relationship to stuff emilio faro talked about ??? .... ????

compact closed bicategory .....

??? "strong monad ..." .... ?????

haskell ... ???

recursively enumerable sets and computable partial functions ... ????

??? "abstract data type" .... ???? .....

?? swallowtail .... ???? .....

??? confusion about ..... ???question of extent to which arbitrary abelian extensions are simply radical extensions ... ???? well, perhaps simply question of whether roots of unity are already available .... ?? but still... ?? for example, are we claiming that .... ???without assuming all roots of unity already available.... abelian extensions are still radical extensions ... ???? ......

??actually, is that really true??? ..... having roots of unity as needed (??) for radical to imply abelian, but ...... ????? is it maybe also needed for abelian to imply radical ???? .......

???idea that it's disappointing if moduli field of elliptic curve with complex multiplication just amounts to some boring subextension of maximal radical extension .... ???? ..... ???????.....

???hmmm, or was i getting pretty mixed up here .... ??? are radical extensions maybe not all that boring, even after all roots of unity are available ??? .... or maybe they are ?????? ...... ????non-/abelianness of iterated radical extensions ??? .... well, of course we know about "solvable galois group" ..... ????? ......

???"compounded but not iterated radical extensions" ..... ?????? ......

?? "adjoining all radicals of all quantities already in the field" ..... ??????? ...... ??????? ...... ??????? ...........

??? possibility of worse confusion here than already noticing ... ??? .....

???? does "maximal cyclotomic extension" qualify as "boring subextension of maximal non-iterated radical extension" ???? ..... ??? well, perhaps yes ... ??well, certainly seems to be subextension, though maybe could argue about boringness .... roots _of unity_ .... ?????? .....

"p-torsion" ...... ????? roots of ..... ??????? ........

??????maximal abelian extension of cyclotomic extension .... ??? or of maximal cyclotomic extension ???? .... ??????? finding maximal abelian extension of sub-cyclotomic field inside maximal abelian extension of maximal cyclotomic field .... ?????? .......

??? "jacobi reciprocity" ...?? or am i misremembering name here ... ??? something like that as maybe close to encompassing full artin reciprocity ... ?????? .... ????? .......

??? maximal abelian extension of Q inside maximal abelian extension of imaginary quadratuc field .... ??? how it fits inside, in terms of jugendtraum ideas and so forth ... ???? ??? "idomeneal" ??? ..... "genus vs class" ... ???? ......

??? trying to interpret "normal cyclic" as "radical" in _"geometric"_ case .... ???? ....... ???? ..... ??? branch point .... ????? ......

?? so consider for example a real quadratic extension of the rationals, say q(sqrt(6)) or something .... ???? .....

??? these galois representations we seem to be imagining at the moment .... ??expressed as functors ........ ??????...... from algebraic closures of given number field k, to ... ??? modules of .... ??? .... ?????

k number field ....

j ideal in k .... ?? ....

?? assign to each algebraic closure x of k the int(k)/j-module given by homming int(x) into int(k)/j .... ???? over .... ????? or.... as .... ???? module of something ????.....

?? for todd ....

1 doctrine and 2-topos .... ??? lurie and (2,1) vs (2,2) ... ??? problematicness of dimensional doctrine and (2,2) ??? ....

??? ...

summer now ....

??? "algebraic closure as analog of universal unwrapping", so to speak ... ???.... ????vs .... ?????? "algebraic closure with particular subfield nailed down as analog of partial unwrapping" .... ??????

...piglet .... ?????...... ??? fiber ... decoration .... ??? algebra vs geometry ... ???? "algebra fiber-wise, geometry base-wise" ... ????.....

"discriminant" .... ?????? ...... symmetric rational functions of x,y vs general rational functions of x,y ... ???? spectrum pictures here .... ?????..... ?????? ?? permutation vs braid ... ????? .....

????? mysteriousness of "clock multiplication" ..... ????? ..... piglet and clock addition .... decoration .... monotony ... ???? ...... periodicity .... ????? ....


????? for thursday ....

galois's principle and 20th cyclotomic field .... ??? also and field of rational functions in n variables ... ??????

???? maybe start with gaussian integers example .... ????? ......

??? can you define _the_ twentieth root of 1 in terms of _the_ square root of -5 and _the_ 5th root of 1 ....?? ... ????? ..... hmmmm ... ????

??? mass hyperboloid picture of binary quadratic forms .... ???? ......

todd....


??? consider the fourier duals (maybe later also "algebraic fourier duals" ????) of the additive groups of the finitely presented modules of a number ring ... ???... ???do these naturally have a conformal structure ????????..... (??? maybe later also structures associated with non-archimedean primes .... ????? ....)

??? maybe didn't quite say that right yet, or at least not in good generality with good terminology ... ??? ..... ???? imaginary quadratic case to begin with ????...

??? "algebraic fourier dual" of finitely presented abelian group as algebraic group .... ???? "spectrum of group algebra" ?????

?????? game here of .... ????? studying underlying (???...) real affine variety of projective complex variety ?????? ....... ?????? but ..... ????? relationship to examples like ... ??? real projective line vs real circle .... ????????.....

??? taking seriously module of number ring r obtained by modding out fractions(r) by submodule given by fractional ideal ?????....... ??? for example rationals mod integers, or gaussian rationals mod gaussian integers ???? ..... ?????..... ???fourier dual of such quotient module ???? ....... ???? .....

?? htpy fiber of dimensional functor from invertible a-modules to invertible a/j-modules .... ???? acting somehow on ... ???? unit a/j-module ... ??? ..... ?????

?? variety y^2 = f(x) with f cubic with coefficients in z/p .... ???? .....

??? a given imaginary quadratic number field, seen mod p ..... ????? ......

?? "elliptic curve with complex multiplication" as "elliptic curve on which it makes good sense to draw pictures of quotient rings of the ring of integers in the corresponding imaginary quadratic number field" ..... ?????

?? "makes sense to draw pictures of quotient rings of Z in/on gl(1)" ???? .....

??? "orders" in number fields ????

?? for an elliptic curve with complex multiplication we have the option of getting from its p-torsion points a representation of the absolute abelian galois group of the corresponding imaginary quadratic number field, or a representation of the non-abelian extension including also complex conjugation ... ??? .... ?? while for one without complex multiplication, don't seem to have such option; no number field in plain view so only latter alternative seems available ... ??? ..... in the complex multiplication case, choosing the "abelian" alternative, seems like two perspectives on .... ???? 1d-ness of rep .... ??? first, as just consequence (??...) of abelianness, vs second, as linked with rep living over the imaginary quadratic field, or over _some_ thing connected with that field ... ??? adeles or something ???? ....


?? golden elliptic curve ... ??????

?? elliptic curve one hecke operator away from having complex multiplication .... ???? ......

???? frobenius .... ???? relationship between this idea that we seem to be groping towards now of "making quotient field (?? ... or more generally ring ?? .... ) of integer ring of field k into representation of absolute abelian galois group of k ... " and .... "frobenius ..." idea .... ????? ...... ??????? .......... ????? .....

??? "frobenius ... " in connection with groupoid cardinality interpretation of zeta functions ....

??? idea about ... ??? interaction between source and target galois interactions on homomorphisms from number rings to finite fields .... .... ??? "k-fields" ???? .... schematic / intuitive picture in terms of "map between fundamental groups induced by map between spaces ... " .... ?? "ramification" .... ????

??? "frobenius ..." in "non-abelian" case .... ?????.....

??? any interesting overlap between "complex multiplication" and "being a jacobian variety" (??? ...) in higher genus case ??? ... ???....

??? are we claiming that you should think of points on elliptic curve with complex multiplication over finite field as having ring structure ???? .... ????? .... ??? but .... ??? non-square numbers here ... ??? ... ??? maybe ok ?? ....

?? gl(1,Z/m) vs ... ?? Z/n .... ???? .... ????? ....

?? functor from abelian closures of imaginary quadratic number field k to [z/p]-vector spaces .... assigning to closure c the vector space of ("strict" ...) p-torsion points on elliptic curve x (with complex multiplication wrt k ...) ...

??? maybe getting from this a representation of gl(2,j-adeles(k)) ... where j is ... amount of ramification needed for getting the p-torsion points .... ????? ..... ?????? .......

??? integer ring of k as acting by endomorphisms on x .... ??so maybe acting by endomorphisms on strict p-torsion of x ??? .....

???maybe .... this as some really obvious module of int(k) .... ????? ??? and this statement as more or less just jugendtraum here .... ????? .... ????.......

??? trying to identify p-torsion points on elliptic curve with complex multiplication as being almost just like corresponding imaginary quadratic number ring ... ???? ..... ???? maybe with the identification map being ambiguous up to something ???? ...... ????? .....

????? cyclotomic case ????? .......

???? elliptic curve with complex multiplication as some sort of direct limit of stuff whose inverse limit is profinite completion of corresponding int(k) ... ?????? ....... ???????? ....... ???????? .......

???? fourier duality between elliptic curve and corresponding imaginary quadratic number ring .... ???? ..... ... complex numbers .... ??? archimedean vs non-archimedean ... ???????? ........ ?????? ......

?? ... values of elliptic function at elements of corresponding imaginary quadratic number ring .... ???? ...... or of fractional ideal therein ???? ..... ?????? .......

??? level slip here ???????? ..... ????? ......

????? frobenius .... ????? ......

????? combined action of int(k) and absolute abelian galois group of k on strict p-torsion points of corresponding x .... ????? ..... ??????? ......

2-step solvable galois group here ... ?????? .....

????hmmmm ....... i wrote:

"??? trying to identify p-torsion points on elliptic curve with complex multiplication as being almost just like corresponding imaginary quadratic number ring ... ???? ..... ???? maybe with the identification map being ambiguous up to something ???? ...... ????? ....."

... ??? maybe the something that it's ambiguous up to is ess just the (second ... ?? ...) galois group here ????? ...... ??? .....

??? quotient ring (?field?) of int(k) as rep of certain partial galois group ... ???? ...... ??? l-function of such rep ??? .... ????? ....

quotient ring int(k)/j as representation of j-absolute abelian galois group of k ???? ..... ????? ..... ??????? .....

?? "for any galois representation of the absolute galois group of (for example ...) the rationals over (for example ...) the 3-adics, there's a corresponding automorphic representaiton .... " ???? .....

?? for a (nice ??) homomorphism from the absolute galois group of the rationals to gl(1,3-adics), there's a corresponding "automorphic representation" of gl(1,adeles(the rationals)) ...." ???? ..... ??? and "gl(1,adeles(the rationals))"
is the ideles of the rationals ??? ..... ????? .....

??? for example consider ... ??? 3-torsion of gl(1) over the abelian closure of the rationals ...??? ?? functor from abelian closures of the rationals to vector spaces over the 3-adics ... ???? ??assigning to an abelian closure the vector space given by ... ????

??? "adeal" .... ??????.....

?? so ... consider endomorphism of projective variety "y^2 = x^3-x" coming from ...
?? x |-> 1+x / 1-x ... ????

ax+b / cx+d

a/c = -1

b/d = +1

a+b / c+d = infinity

x+1 / -x+1



cubes in f_2 =

since alex asked about possibly getting notes from the summer course maybe i'll make a slight attempt here to write some down ....

6/26 galois's principle ...

6/27 ....

6/28 piglet's 2-lake island ...

6/29 ?? salmon approach to solution by radicals ... ??structures on 4-element set

6/30 (at (other) alex's house...) .... more .... symmetry as problem-solving tool .... ??? .....

7/1 ?? (at kevin's house) defining simultaneity in terms of light-signalling and inertial motion ...

7/2 ?? .... fooling around with snubbed tetrahedron ....

??? so .... elliptic curve x with complex multiplication ... ??? getting galois representation from its p-torsion ... ??? but then .... ??trying to connect this with .... ?? points of x over finite fields ... ??? more specifically over finite k-fields where k is the imaginary quadratic number field associated to x .... ????....

??? "class field" .... ??? ??? how (??k-abelian) algebraic integer j transforms wrt absolute abelian galois group of k as equivalent to how it manifests in finite k-fields ... ?????? ......

??? counting solutions of y^2 = x^3-x (or something like that ...) over finite fields .... tying things in here somehow ... ????

?? points of x over finite k-field f as forming abelian group .... ??? how p-torsion of this abelian group varies as q varies, taking f = f_q .... ???? ... ??? ....

???? ......