Saturday, March 31, 2012

?? polynomials in x,y, constant on y-axis .... ??? .... ?? "term with y must also contain x" ??? ....

1

x

xy

xyy

xyyy

?? not finitely generated ??? .... ??? bizarre that / if i never noticed this before ?? ... ???? ....

k[a,b] -> k[x,y]

a |-> x

b |-> xy

?? a=x, b=xy, c=xyy

ac=bb .... ???? ....

?? cones approaching allegedly "too big" one ... ??? ...

?? cocones approaching allegedly "too small" .... ???? ... ??? ....

?? some confusion / mix-up ?? ... ??? not giving blow-down i thought i wanted ... ??? .... ??? .... ?? vague idea was ... ?? having just noticed that blow-down seems to be a sort of ideal limit, and having been in search of an ideal limit recently, wondering whether this might be it ... ??? .... ??? ....

?? "localize at ideal" vs "blow-up ideal" .... ????? ..... ?? vs blow-down too .... ??? ....
?? idea of fan picture as sort of "octopus's brain" .... ??? tangent space of unit point .... ???? or could think of it as translation space of torus bulk ... ?? anyway, i'm imagining this homunculus or octopunculus or whatever sitting at the nexus point ... sending out and receiving messages from the extremities ... and ... ?? each tangent vector points to (generally ...) one of the extreme points ... ???? ....

?? how this picture, if straigthened out, relates to ... ?? prescription for constructing real manifestation of toric variety from fan ??? .... ???? .... ?? hmmm, role of lattice here ???? ..... ????? ....

?? also, how it relates to ... ?? general idea of straightening out peculiar "closed / open reversal" aspect of relationship between "fan" picture and "ordinary geometric" picture of toric variety .... ??? .... ?? ambiguity in "which cone given point lies in" (??? ...) because subset whose definition involves relative complementation is ... ?? tricky .... ??? ... ??? ... ??? .... ?? ..... ??? distr lattice vs boolean alg ... ??? .... ??? ... ??? .... simplicial complex vs fan ... ??? ... ??? ... ??? .....

?? straightening out idea of "fan as filter in poset of cones" ... ?? .... ?? what's that word that i'm looking for that combinatoricists use ?? ... ?? "cut-set" ?? ... i don't think tha that's it, but something like it ... ?? .... ?? extreme limiting cases of fans, and certain such arising canonically from various things ... ?? ....

?? taking fan picture of toric variety seriously in various ways ... ?? trying to see toric line bundle "twisting" in this picture ... ??? ... ??? to what extent do total and/or fiber spaces of toric line bundle have toric structure ?? ... ??? ....
?? "auto-skin-graft" metaphor (for "ordinary" geometric interpretation of "non-snug cone inclusion" in fan picture ... ??? ...) as weird because ... ??? arguably there is donor and recipient, and they're not the same .... ???? ... ??? still, vague idea of "one location benefit at expense of other" seems to apply somehow ... ??? .... ?? hmm, maybe the single patient is supposed to be something like the torus ?? ... ??? ...

?? moment ago were toying with "blow-up" being canonical only for codimension one, but now back to thinking that it(?? ...)'s always canonical ... ??? "ideal power filtration" .... ???? .... ??? .... (?? vague memory of some uncertainties that we had about that though ... i mean some pretty minor ones, about intersection of ideals vs pullback of associated modules over 1, or something like that ????? ....... not the major ones about ... some attempted big picture ... ??? ...) ?? though is the toric structure canonical ????? ..... ???? .... not completey sure yet ... something's got to give .... ??? ......

?? when fan cone gets "too big" (?? meaning zero's in convex hull of rest ?? ...) doesn't so much "disappear" as "collapse to point" ... ???? ... ?? wait, does that make any sense ??? ..... ?? corresponding co-fan co-cone .... ???? .... ?? vague memory about "co-fan co-cone "loses" dimension by becoming invariant in it" ... ???? ?? yes, but now we're considering "non-admissible" co-cones (not generating the whole co-fan lattice), and the corresponding "non-admissible cone" does seem like it could be construed as quotient .... ??? actually i'm probably confused enough here that i shouldn't trust what i'm saying yet .... ??? .... ?? should straighten out ?? ....

?? toric singularity ... ?? .... cone ... checkerboard ... ??? ....

??? so i think that i do remember now thinking a bunch about "blow-down" and "ideal as sub-rng" ... on a particular day when a bunch of that stuff really seemed to make sense, or threaten to do so ... ?? what was that idea about "universal way of making a subvariety a cartier divisor", or something like that ??? ... ?? some adjoint functors here .... ??? some vivid mental pictures involving curve in 3-space .... ???? ..... hmmm .... ?? but i don't offhand recall thinking about how ... ?? ... ?? when you blow down a subvariety of an affine variety, the bit about "subring inclusion as roughly corresponding to quotient map, but often not quite surjective" may bite ... ??? so that when you blow-up again you may get something projective (?? ...) extending / completing (?? ...) the original affine variety ... ????? .....

?? knot invariant coming from invariant of its blow-up ??? .... ???? ....

?? hmmm, so ... ?? issue of compatibility of blow-_down_ with toric structure ... ??? .... ?? and ... ?? canonicalness of toric blow-down ... ??? ....

??"knocking out wall ..." ... ??? of cone, vs of co-cone ... ??? ... ?? alleged canonical result here as generally "too big" ??? .... ???? ....

?? again, sure we thought about this to _some_ extent before, but ... ?? how sharp is paradox here involving .... ???? ..... "non-singular" ways of subdividing standard lattice quadrant ... ??? ... ?? hmm, maybe quite dull rather than sharp ... ?? usual bit about invertibility of aut(n) -> aut(free comm monoid(n)) ?? ....

?? "toric (??prime ?? ...) ideal" as wall complement ... ??? .... ?? or face complement ........ ???? .... ???? .... ?? hmmm, is there a somewhat nice picture here of "specialness of co-dimension one" here ??? .... ?? wall complement as "_minimal_ prime" ideal ... ??? ....
?? to what extent did we ever work out and / or legitimately reject idea of "filter" approach to fan ??? ...

subdivision comparison morphism seems to actually exist ... from subdivided to original ...

?? -> ...

x |-> a

y |-> ab

?? ....

?? "given x,y, what are the solutions for a,b to x=a,y=ab?" ... ???? .....

?? aka "solutions for b to y=xb" .... ???? ....

?? "subring tend to correspond to quotient space" .... ??? .... ?? except that the exceptional case tends to become the usual case .... ???? .....

?? b = y/x .... ???? one solution, except none when x = 0 but y =/= 0 ... (?? when working over a field ??? ... ??? ... ?? ...) ...?? hmmm, but also _lots_ of solutions when both x = 0 and y = 0 .... ... ... ??? .... ??? so not "dense open subspace" ..... ??? ...... ???? has "blow-up" aspect to it ???? ..... ???? ..... hmmmm ....... ???? ......

?? image neither open nor closed ??? ....

?? ingredient (?? ...) in blow-up but not complete blow-up by itself .... ?? .... ?? sort of like "transplanting skin from elsewhere on your body ..." .... ???? ...... "skin graft" ... ??? "auto-skin-graft" ?? ....

?? so should be interesting trying to see actual blow-up as glued together from these things ..... ???? ...... ?? "two ribbons, each with its own third dimension, giving four dimensions total, glued together to form blow-up" .... ??? ....

??? was going to mention before contrast between "voodoo limit" and "blow-up limit" ..... ???? .....

?? so birational isomorphism as somewhat more general than dense zariski open inclusion .... well, hmmm, we really should have / secretly (?? from ourself ??) did know that, because blow-up comparisons qualify too ... but now we're seeing these blow-up / inclusion hybrids (?? simply composites, in some case ... ??? .... ?? hmmm, maybe yes ... thinking in "fan subdivision" picture ... ?? ... usual danger of over-generalizing too much from 2d pictorial intuition .... ??? ..... ?? inclusion of toric open into blow-up, followed by blow-down .... ??? ....) .... ??? ......

?? mental image of "ribbon twist" here .... ????? ..... ?? ribbon very wide though ??? ..... line-of-sight singularity .... ???? ....

?? "infinite subdivision" ... ??? ... ??? "cauchy completion of filter poset" ??? ... ???? ... ????? .... ??? ...

??? "taking affine wedge and not subdividing it, but supplementing it with maximally subdivided wedges to fill it whole fan ...." ... ??? ...

?? confusion about non-/canonicalness of blow-up here .... ??? role of co-dimension .... ??? .... ???? ..... ?? again, to what extent did we work this out before ..... ??? .....

?? "blowing up higher codimension subvariety by blowing up codimension one subvarieties intersecting in it and then combining (?? ...) together ?? ... ??? resemblance / relationship to "removing higher co-dimension subariety by removing codimension one subvarieties intersecting in it and then combining (?? ...) together" ??? .... ???? ..... ????? .....

?? confusionabout what sort of subvariety can get blown up in "toric" case ... ?? ...

?? "toric divisor" ... ??? .... ?? higher codimension analog .... ??? .... ??? ... intersection ... ??? ....
?? questions for baez ...

?? "euler summand" .... ??? ..... categorified .... ??? ....

?? piston equation ....

?? walking symmetric bimorphism object sym mon cat .... ??? .... ??? complement of image as ... ??? points with empty fiber .... ??? .....

Friday, March 30, 2012

?? toric projective line ...

"line object l tw good epi from 1+1" ... ?? straightening out what good means here ... ??? ....
?? legend about .... ???? soliton ??? .... not the horseback legend, the computer legend .... ??? .... ?? anomalously long mixing time ... ???? ..... ???? .....
?? comparison morphism of blow-up ... from blown-up to original .... toric case ... fan picture .... ??? "subdivision" .... comparison map from subdivided to original ...... ????? ......

?? problematicness (?? ...) of " ...... ?????? .....


?? hmmm, confusion here between .... "subdividing" and "adding on extra chambers" .... ??? ... ???? relationship(s ?? ...) ...

??? "dense zariski open inclusion" =?= "birational isomorphism" ???? .....

(thinking about "birational geometry as guide to glueing" here ... ??? ...)

??? "universal completion of toric variety" ... ??? ...(also non-toric case ...) ... ??? as maybe problematic in way relating to problematicness of "universal blow-up" ???? ..... ???? .....

?? still confused about which of these things really are problematic .... ??? .... ?? as long as there's a genuine comparison morphism, then doesn't that let you do stuff here .... ???? ....

?? "toric algebraic completion" .... "forgetting lattice" ??? .... ????.... ?? q vs r ??? ... ??? .... ?? confusion about whether there's galois symmetry here .... ???? ....... ???? .... ?? ... "cover" .... ??? ... vs "canonicalness / uniqueness of roots" .... ????? ..... ???? roots of unity .... ???? .... fan picture vs .... ??? ...... ????? .....

?? hmmm, maybe a lot of the confusion that i seem to be experiencing here (?? this post in general ?? ....) has to do with ... ??? having at least temporarily totally forgotten how the "poincare duality" aspect of fans works .... ?? open / closed reversal .... ???? ..... ???? ....

?? actually not sure we did completely forget that ... because whole (?? ...) point of question as ... ?? involving _other_ geometric picture, where open / closed is not reversed .... ???? ....

?? anyway, lots of stuff to be straightened out here ...
?? structure type convolution monoids and comonoids .... certain games with this ... ???

?? "composing and then decomposing as restoring original structure, but not ice versa" .... ???? .... ?? some confusion here ..... ??? "kind of structure that can be decomposed given partition, vs kind that reveals its own partition" .... ????? ..... ?? co-unit as pretty drastic here ????? ..... ?? co-associativity as about equally drastic ??? ..... ????? .... ??? ....

?? non-free structure-type convolution monoid ??? .... ???? ...

??? non-symmetric convolution monoid ..... ???? ..... free symmetric as non-free non-symmetric .... ????? ..... ?? but any non-free symmetric .... ???? ....

??? convolution-monoidal stuff type ???? ...... ????? .... ??? vague feeling of (?? level-slippery) relationship to .... ??? quandle-based knot polynomial ..... ???? ..... hmmmm ...... ?????? ..... ?? action of monoidal structure type on structure type .... ??? ....

?? "zeta-type" as monoidal wrt multiplicative convolution ??? .... ???? .....euler factor ...... ?????? ..... ????? ..... ???? freeness here ???? ..... ?? logarithm of zeta-function ???? .... ?? "euler summand" .... ?????? ......

?? confusion about "euler summand" here .... ??? some sort of 1/(1-x) = 1 + x + ... flavor ??? ..... ????? ..... ?? seems ad hoc .... ??? .... confusion .... ???? ....

?? monoidalness wrt both additive and multiplicative convolution ..... ??? ....
?? quasitopos vs localed topos ?? ... ???? .....

Thursday, March 29, 2012

?? "picard variety" ... ???? .... ??? "picard stack" .... ???? ..... ?? line bundle over such stack given by "the fiber over point p" .... ???? .... ??? .... ??? .... "duality" .... ??? ....

?? smooth dynamical system with all orbits nice and closed .... ???? renormalized to be all same period ??? .... ??? .... orbit space .... bundle of orietned circels over it .... ??? ....

?? special case where the smooth dynamical system is the "geodesic system" of a riemannian manifold ....
?? "line bundle over x with lie alg structure on section space" for x given finite discrete space .... ??? .... moduli stack .... ????? ....
?? piston ... chaos ... ergodic .... conservative .... ??? ....

?? .... symplectic ... ?? global vs local ... ??? all vector fields look alike generically locally ... all hamiltonian ones look alike locally ... ?? "completely integrable" .... ????? non-generic stuff ... classification of equilibrium points ... ??? ..... conjugacy class / adjoint orbit in linear symplectic group ... ???? .....

?? "renormalization" .... ????? .....

?? "blow-up of singularity" .... ??????? ........

?? "kam" ... ???? .....

?? general idea of system of ideal gas reservoirs connected by pistons ... ??? .... how qualitative behavior varies with complexity (?? in vague intuitive sense ?? ...) of system ... ??? ...
?? linear structure type carried by finite set ... "assign to set s direct sum over [space of semi-simple comm k-alg structures on s] of ..." ... ??? ....

?? l-function .... ???? ...

?? extent to which we've been through this before ..... ???? .....

??? upward vs downward .... factor ... irrep ... vs ... ?? bigger thing ... ?? .... ?? underlying vs free .... induced ..... ???? .....

?? euler factor structure type here ... ???? .....
?? concrete explicit approach to relating two approaches to quasicoherent sheaves over p_1 ....

graded module / "proj" approach, vs borel rep approach ...

_1d object l embedded in 2d object x_ <- _1d object embedded in v_ -> _1d object mapped to v_

?? right adjoint games here ??? ... ?? co-/monad games ??? ....

?? _1d object l mapped to 2d object x_ .... ???? .... ???? again (... ?? ...), vague hint of relationship to "bruhat classification" here .... ???? .....
?? configuration variables x,y ... ?? vanilla riemannian structure here ... "conservative force field" ... ???? .....

?? hmmm, for 1 variable _any_ force field seems conservative ??? .... ????? .... hmmmm .... ????? ..... ???? "self-interaction" ???? .... ????? ....

f(x,y) ....

f_x(x,y) ... ??? ....

f_y(x,y) ...

x'' = g(x,y)

y'' = h(x,y)

g_y(x,y) = h_x(x,y) ???? ....

?? "momentum-conservativeness" .... ??? .... "equal and opposite reaction" ... ???? .... ?? connection ... ??? flatness ??? .... ???? ..... ???? ...... ?? level slip ??? ... ???? .....

?? x'' = y

?? y'' = x ??? .....


x'' = y^2

y'' = 2xy ??? .....

?? "work done by ... on ..." .... ???? ....

?? wait a minute ... ??? .... ?? "h1' = -w12' * p1" .... ??? .... idea of "velocity-dependent force" here ??? ... ?? following philosophy of "h1 already being an energy ..." ... ??? .... .... ??? .... ?? "magnetism" ??? .... ????? .....
?? originality in teaching as more important than originality in research ... ??? ... path vs endpoint ... ?? ....

?? "blank slate" ... "symmetry-breaking" ..... ?? neoteny in human evolution .... ?? mental flexibility vs mental pre-preparedness ... ??? .....
?? "conservative force field" ... ?? riemannian gradient vector field of smooth function .... ??? ..... modification / generalization to "heat bank" case ?? .... ?? "equality of mixed partials" .... ???? .... ?? "work ..." .... ??? ....

?? "tensor" in diff geom, and ... ??? "associated graded" ... ??? .... ?? "fake higher tangent space" .... ???? .....

?? "interaction" ... ?? "conservative interaction" .... ???? .... ?? constraint vs "force" ... ?? "energy term" ... ??? ....

Wednesday, March 28, 2012

?? walking composition alg sym mon algebroid .... ??? .... ?? fd hom spaces ??? ....

?? "biquadratic" ... ??? 4-linear, symmetric in first two and (separately ...) in last two ... ?? ...

?? "polynomial morphism in sym mon algebroid" .... ???? ..... ?? base change ?? .... ???? ....

?? composition algebra and / vs "alg with involution with real aspect standard" .... ??? ..... ??? killing .... ???? .....

?? walking "alg with involution with real aspect standard" sym mon algebroid .... ???? .....

?? walking "object with symmetric bimorphism" symmetric mon cat ... ??? morphism m -> n as ... ?? injection with partition into pairs on complement of image ??? .... ??? .... ?? general idea of ... ??? comm monoidal structure type giving notion of morphism like this (?? ...) ??? .... ????? ......
?? "conservative force-gorce field" .... ??? .... heat bank gorce counterbalancing configuration force ... ??? ....

?? energy function on heat bank .... ?? ....

?? ... "kinetic" .... ??? "macrokinetic vs microkinetic" .... ???? ......

Tuesday, March 27, 2012

?? what hamiltonian flow looks like near equilibrium point ... ?? ... conjugacy classes in linear symplectic group ??? .... ???? ..... ??? what contact flow looks like near euqilibrium point .... ????? .... ???? .... ???? ....
1

piston hamiltonian ... mathematica simulation ....


2

?? "relatively propositional ag theory" .... ?? variants thereof ... ??? ... lawvere and slice topos ... "point" of (?? "total stack" of ?? ...) given quasicoherent sheaf ... modules of comm monoid object in ag theory ... ?? co-module of co-monoid ... ?? .... ??? ......


3

"ordinary-toric" ag morphism ... affine case ... working out ... ??? ...


4

??? etale htpy group and so forth of toric variety .... "cohomology" ... "glueing" ... ??? ....



5

?? "the line where the cubic polynomial associated to the elliptic curve lives" .... ???? .....

6

mention of "blowing up singularity of pde ..." i saw .... ??? ...

Monday, March 26, 2012

?? 2 ways to try to calculate heating rate of gas compressed by piston .... ??? ..... ?? energy change of gas particles vs of piston wall .... ???? ... ?? agree ?? .... ???? ....

?? multiple gas reservoirs ... ??? ....
x'' = -x + h/x

h = k - (x^2 + x'^2)/2

?? .....

?? some interesting results from mathematica so far ... ?? ...

?? hamiltonian / poisson / symplectic structure here (...) ??? .....

?? bit about ... ?? extra info in symplectic (?? ...) system beyond (?? ...) "2th order dynamical system" ... ??? .... ?? energy .... ???? ..... ??? action ... ???? ...

?? "friction" ... ??? ...

?? "dispersion" ... ??? ....

?? energy function" ... "hamiltonian" ... "dynamics" ... ??? ....

?? fiberwise constant hamiltonian ??? .... ????? ......

?? some kind of symplectic /poisson reduction ?? .... ???? ....

Saturday, March 24, 2012

?? idea now of ... ??? making sure to continuously eject just enough ballast to keep ascending .... ??? ....

?? pressure as function of height ..... then .... ???? .....

?? not enough ??? ... ?? pressure as function of height and total heat ?? ... ??? .... ??? but ... ??? raises further questions ??? .... ????
(m1,v1,h1,m2,v2,h2)

m1' = m1

m2' = m2

h1' + h2' = h1 + h2 - f(s)

v1' = v1 + s

v2' = v2 - s

h1'/v1' = h2'/v2' + f'(s)

?? pun on "'" here .... ???? .....
?? talking with curtis pro about jacobi fields and riemannian submersions and so forth ...

?? space / stack of geodesics of a riemannian manifold .... ???? .... interpretation of jacobi field in thois context .... ?? case of torus ??? .....

?? special case of 3-sphere ... relationship to pgl(4) incidence geometry .... ???? .....

?? getting "positive contact flow" from riemannian metric ... ?? how close this process is to being reversible ... ??? ....

?? trying to relate minimalish example of riemannian submersion with non-integrable horizontal distribution to nice simple contact distribution .... ??? .... ??? ....

?? situation where wavefront advances to geodesic .... ???? ...... ?? hmm, only in 2d case ... and special even there ?? ... ???? ...

?? gradient of scalar curvature .... ?? ....

?? closed geodesics .... ??? ....

?? for which surfaces are all geodesics closed ??? .... ??? ....

??? vague memory about "bott periodicity" and geodesics and "jacobi ..." .... ???? .....

?? jacobi field as special case of .... ??? ... "deformation of trajectory of dunamical system" .... ???? ....
?? hmmm ... ??? thermodynamic second law as extra constraint needed to determine behavior ?? .... ???? ....

?? hot-air balloon example ... ?? use of stat-mech assumptions .... ???? ..... ?? reversibility ??? .... ??? ....

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

Friday, March 23, 2012

?? so ... besides making the freight of infinite density so that it's potential energy will be purely gravitational, we probably want to make the atmosphere's potential energy purely _non_-gravitational ... again, to try to get things as simple as possible ... ?? so we're back to a sort of piston, but with the emphasis on there being some sort of "load" on the divider .... ??? and again, idea that ... ??? instead of focusing on "cycle" idea .... ??? keep "delivering loads" until no obvious "waste heat" is left .... ???? ..... ???? .... ?? same temperature and pressure ??? .... ??? ... ?? ...

?? silly idea of mystical "digital" interpretation of sequence of loads of size (?? ...) 2^[-n] .... ??? ....

?? so ... one silly idea that we had was that just mixing samples together might somehow produce macroscopic work, whereas now it seems pretty different ... you have to do something as the mixing occurs .... "impose a load on the divider between them" ... ??? .... ???? .... ???? .....

?? hmmm ... ?? kinetic vs potential approach here (...) ??? ....

Thursday, March 22, 2012

?? theoretical efficiency of hot-air balloon ....

?? "dropping passengers / freight off ... " .... ??? ....

?? slight subtlety here about gravitational potential energy vs "gravitational-hydrostatic potential energy" ???? .... ??? is that going to screw things up ??? .... ???? ..... ?? definition of potential energy difference here coicncide with definition of .... ???? .....

??? trying to keep "hydrostatic" aspect straight from "thermodynamic" aspect ?? .... ??? despite .... ???? .... ?? "interaction" ... ??? ....

?? hmmm, this whole issue pretty much avoided by taking the freight to be infinite density ... so their potential energy is purely gravitational instead of hydrostatic ... ??? ....

?? "re-(?? ...)use any waste heat" .... ???? ....

?? "reversibility" .... ??? ... ?? "piston" .... ???? .... ??? ....

?? "one person's freight (or in extreme case one person) as another person's ballast" ... ?? ....

?? limit where starting (not ending .....) volume of balloon approaches zero ..... ??? ......

Tuesday, March 20, 2012

?? dimensionful / non-pure quantity as section of line bundle .... ???? .....

?? "locally eliminable stuff" vs "locally trivial bundle" .... ??? .... ?? confusion about _non-_[locally trivial] bundles in "holomorphic" case ... continuous "jacobian variety" ... ??? .... ???? .....
?? group of "rolling actions" ??? .... ???? .... ?? metric structure ?? .... ?? ...

?? does this really make any sense ??? .... ??? what if "axis you're instructed to roll wrt isn't in rolling position" ??? .... ???? ..... ?? trying to get flag from partial flag ... ??? ....

?? "cohort" ... ?? .... dodecahedron ... ????? .....

??? derek wise .... ??? ...

?? coordinate convention ... map .... rolling ....

?? "torsoroid" ... ??? ....

?? mirror ball ??? ....

??? funhouse mirror .... ?????? .....

?? cayley-dickson functoriality action ... aut(quat) |-> aut(oct) ... ??? ... ?? how manifested on rolling ball ??? ...

?? "moving frame" ....

?? so(3,4) .... ???? distinct functors _g2-torsor_ -> _so(3,4)-torsor_ ?? ... ??? ... ?? try to make explicit ... ???? ....

?? one fr via octonions, other via rolling ball .... ??? .... ??? "gambit" ... ??? ....

?? maybe ... ??? valued in _vsp_ ... ??? .... ??? ....
?? non-trivial ideal class group ... ?? for example affine cubic over c .... ??? .... "point at infinity" "projective completion" point here as "irrelevant" ??? .... ????? .... ????? ....

?? g2 as both affine and projective variety ... ?? .... line bundle over g2 .... ???? ..... ???? .....

Monday, March 19, 2012

mass, volume, temperature .... ??? ....

(m1,v1,k1)+(m2,v2,k2) = (m1+m2,v1+v2,k1+k2)

p = t*m/v ???

t = k/m ???

?? p = k/v ... ??? ...

(1,1,1) + (1,1,2) = (2,2,3)

1 2 3/2 ..... ?????? .....

(1,1,1) + (2,1,1) = (3,2,2) .... ????? ....

?? idea that this is saying that mixing at same pressure is "foolish" ??? .... ?? or we're doing something foolish, again .... ???? ....

?? no conversion of heat to anything else yet .... ??? ....

?? "... cycle ..." .... non-closed 1-form ... ???? .... ??? "work" .... ???? ....

?? "piston" .... ??? ....

?? theoretical efficiency of hot-air balloon .... ???? ....

(m1,v1,h1), (m2,v2,h2)

m1' = m1 ...

m2' = m2 ...

v1' + v2' = v1 + v2

h1' = h1
h2' = h2

(?? these to be modified in some way relating to "work" ... ?? ....)

h1'/v1' = h2'/v2'

(1,1,1) (1,1,2)

(1,2/3,1) (1,4/3,2)

?? .....

?? work function of volume change .... ????? ....
(i,1) (i,i)

(i,-1) (i,-i)

(i,i) (i,1)

(i,-i) (i,-1)

(i,j) (i,-k)

(i,-j) (i,k)

(i,k) (i,j)

(i,-k) (i,-j)

(j,j) (j,1)

(j,-j) (j,-1)

?? hmm, seems like no real-linear equivalence here .... ??? .... to what extent does this spoil certain plans involving so(3,4) ?? .... ?? i mean, which of them are spoiled ?? ... ???? ...
?? "birational geometry as guide to glueing ..." ... ??? toric case ?? ... but non-toric too ! .... ?? "good scheme" .... ?? as having good birationalizaiton ..... ??? .....

?? relationship to cockett's ideas ... ???? .....
?? rig of mass, volume, temperature specifications .... other rigs here .... ??? level slip ??? ... gibbs-boltzmann .... ??? .... ?? "statistical mechanical system of paths ..." .... ?? "path-integral" .... ??? ....

?? standard deviation vs entropy .... ??? ....

?? split octonion basis adapted to rolling ball .... ???? ..... ?? quaternions .... ???? .....

Sunday, March 18, 2012

?? idea of instead of tweaking to close the gap between the two distributions, might be able to tweak the split octonion product by an elt of so(3,4) to get one whose nil subalgebra distribution matches the naive rolling distribution ..... ???? .....

?? (g,e) -> (g, g/e) .... ??? "homogeneous" .... ??? live at level of 1d isotropics ... 2d isotropics .... ???? ....... ???? .....
?? so ...

(i,1) (j,-1) (k,1) (i,-1) (j,1) (k,-1)

... ??? looks like probably what i'd call "tweaked" version of grand tour ??? .... ?? which huerta is calling "untweaked", maybe ?? ... ?? but in any (?? ...) case rightly complaining about it .... ??? ....
?? exterior alg vs operator alg of 2d vsp ... ??? deformation relationship ??? ..... ?? "geometric algebra" .... ???? .....
?? 4d distribution on g2 config space ??????? ..... ????? ....... i mean, mystery (?? ...) of why there doesn't seem to be one ... ??? .... ?? so(3,4) .... ???? relationship between "2 rolls away" for g2 and certain orientation for so(3,4) .... ???? ..... ??? perp of 1d null subspace ... conceptual / geometric interpretation thereof ..... ???? .....

?? ok, so seems real likely that the 2-rolls-away subalgs in the split octonions are in fact _exterior algs_ rather than split quaternion .... ?? should have been pretty obvious, and huerta and i probably once talked about it pretty long ago .... ?? but i _did_ try checking last week or so the anticommutator on my canonical 2-rolls-away algebra and it seemed non-tirivial ... which now seems like must be a mistake; i'll try checking up on that check ....

ok, here's where i tried the check, quoting from here:

?? meanwhile, anticommutator of (i,1) and (j,1) .... ??? as (k+1,i-j) + (-k+1,j-i) = (2,0), so that's a positive constant, so good representatives a,b should be (i,1),(-j,-1) .... ???? .... ?? so then a+b is (i-j,0) ..... ????? ..... hmmmmm ...... ????

?? so .... ?? can we find our mistake here ??? .... hmm, i _still_ seem to be getting (2,0) here, using wikipedia convention .... ?? .... ?? and even using other convention ?? ...

hmm, huerta's email might give some explanation of the glitch here ... ?? involving tweak map ??? ..... ????? ..... ???? .....


?? anyway, lots of stuff threatening to make sense here ... including connections with so(3,4) and tweak map ..... ???? .....

?? split octonion product in 2-rolls-away / exterior algebra case as giving the (?? unique ...) pivot point witnessing the 2 rolls ..... ???? ....

?? so(3,4) .... 5d space of 1d isotropics ("projective light cone" ...), 7d space of 2d, 6d space of 3d .... ???? some of all those showing up here in interesting ways .... ????? ......

?? does "graph of isometry" property get screwed up in complex case ???? ...... ???? ..... ???? biquaternions as not division algebra ..... ???????? ....... not sure i said the (??? ....) right thing here ..... ?????? .... ?? some correction (?? ...) huerta pointed out to me at some point .... ????? .... ...... ?? stuff we thought about in connection with homogeneous coordinate alg of certain flag varieties around october 2011 ... trying to prepare for talk which as it turned out didn't go much in that direction ... ??? maybe around here ???? .... isometry as invertible ..... ???????? ........ ?? flag variety that (?? over the reals ??? ...) mimics group in way but is actually far from being alg gp ...... ??????? ...... ????? ...

?? exterior subalg of split octonions .... ??? .... 2-roll path vs 1-non-roll path ... so(3,4) ... ??? .....
?? functor from _g2-torsor_ to _so(3,4)-torsor_ ... ?? projective light cone of imaginary part of split octonion algebra .... ???? ..... ?? what this looks like in rolling ball picture ??? .... ??? ....the "new lines" .... ????? .... 7d space thereof ... ??? i don't really suspect that it's the special split quaternion subalgebras, i don't think .... ???? ... [later that day : hmmmmm .... ???? .....] ..... ?? maybe do suspect that it's "the naive guess" ... ??? ignoring tweak map ..... ???? .... ???? ....

?? given pair of g2 configurations ..... ??? .... ?? not connected by geodesic rolling trajectory ..... ??? getting path between them via [?? split quaternion subalg ... ???? ....] vs by [linear subspace .... ??? ..... ???? ....] ... ??? .... ??? very different ??? ..... ???? .... ??? ....

?? wait ..... ???? dot vs cross here ??? .... ?? not generic configuration pair, but pair satisfying dot but not cross constraint ..... ???? ..... ?? is this "2 rolls away" ???? ..... ????? ..... ???? .....
5 + 5 - 2 .... ??? = 8 dim space of generic split quaternion subalgs ... ??

5 + 4 - 2 = 7 dim space of special split quaternion subalgs [correction later that day : exterior subalgebras .... ??? ....] ..... ???? .... ... hmmmm ..... ???? ...... ?? so(3,4) ... ???? ....

Saturday, March 17, 2012

?? map taking

a b
c d

to

b -a
d -c

, exchanging (...) traceless with symmetric ... ?? as a complex structure ??? ..... ????? ......

???? su(1,1) here somewhere ????? .... ????? .....

?? or u(1,1) ... ?? quaternions as 2d rep of u(2) ... ?? split quaternions as 2d rep of u(1,1) ??? ... ?? .....
dot / cross .... anticommutator / commutator .... cayley-dickson ... anti-involution ... "anti-scalar product" .... ??? ....

heisenberg ... "mystical 2n+1 stuff" ... cayley-dickson .... ?? contact structure ??? ....

?? sub-[projective lightcone] in general ... split quaternion subalg in split octonions in particular .... ??? .....

?? with / without tweak, standard rolling trajectory as special case of null subspace / partial isometry graph .... ???? .... ?? split quaternion algebra as _not_ such .... ??? .... some confusion ?? ... ?? "lightcone subcone" vs nil / null such ... ??? ....

?? distribution vs its collection of "geodesic rolling trajectories" .... ??? .... ?? as basically equivalent, and latter as more convenient / better for certain purposes ... ??? ..... ?? ask huerta about this ?? ... ?? to what extent we tried this approach before .... ??? ...

?? "diaconescu's and giraud's thms together" ... ???? .... presheaves on small category .... ?? .... ??? canonical grothendieck topology ... ?? "covering of non-flat / non-representable thing by canonical presentation ..." ??? .... ???? ..... ???? .....

?? coherence confusion ??? ....
?? so(3,4) (?? and / or pso(3,4) ?? .... ???? ....) ... grassmanian of 2d isotropics .... ?? subvariety consisting of nil subalgs of split octonions, vs subvariety consisting of ... ?? rolling trajectories for fermion rolling on projective plane with 1:3 radius ratio ... ?? whether these two subvarieties are "equivalent" in certain sense .... ??? so(3,4)-equivariantly, maybe ??? .... ???? ..... ?? and what yes / no ( / ?? ...) answer would mean, conceptually ..... ???? .....

?? first subvariety here as, what, cubic ??? .... ??? ... not quite sure actually .... ??? for some reason vaguely reminds me of "generalized pluecker embedding" issues .... ??? .....
?? symmetry vs structure ... ?? learning not to try to empathize (?? too much ?? ...) with, say, a bat, vs ... learning that if you do then things are going to seem pretty weird .... ??? .... ???? ....

Friday, March 16, 2012

?? _g2-torsor_ -> _so(7)-torsor_ .... ??? .....
?? g2 .... ??? tweak map .... ??? uniqueness up to so(7) elt ??? .... ??? ... conceptual argument for why split quaternion lines can't change if nil lines don't .... ??? ....

?? whether we _did_ accidentally omit tweak in latest animation ??? ..... ??? ... ?? built in to "displayconfig2" or not ?? .... i was assuming yes, i think ... ??? ....

?? any tweak map analog in lower dimensions ??? .... ??? ....

?? "boost" ... ????? ..... geometry of isometries .... ???? ...

?? so ... looks like tweak map is definitely built into displayconfig2 ... and i tried various alleged split quaternion lines ... looks to me like they could all be non-skid ... the one that's supposed to be pure spin seems to be pure spin ... so though some aspects of it i didn't quite get ...

?? what happens if you use our intended algorithm except the anticommutator isn't normalized the right way ?? ... ?? because that's what might be happening ?? ... hmmm .... ??? maybe try more explicit normalization of the anticommutator .... ???? ....

??? at the moment for my alleged canonical "special cayley-dickson" line, i seem to be getting for some reason that exactly halfway through the loop the rolling ball is "dot-centered" ... and very centered in the whole picture .... and at a nice angle ... ???? ....

?? _is_ the tweak map telling us about some mismatch between g2 geometry and so(7) geometry, or conversely is it telling us about some _better_ way to see (...) so(7) geometry ?? ....
?? topos of filteredly cocontinuous set-valued frs .... ??? ... "flatness axioms" and "higher flatness axioms" .... ??? non-coherent or worse ??? .... ???? .... ?? "flat" vs "filtered" ..... ?? "filtered" (?? ...)expressed in terms of cover ???? ..... ??? level slip concerning property of diagram schemes vs property of preserving colimits of those shapes .... ???? but then ... ??? certain patterns that _do_ recur (?? ...) here .... ???? "object st homming from it preserves colimits of given kind" .... ??? situations where this corresponds to not being a colimit of that kind in any non-trivial (?? ...) way ... ????? ..... ?? then also .... ??? "flat" as related to "filtered" ....... ????? ...... ((higher ...) flatness/filteredness axioms .... ??? non-coherent or worse ??? ....) ... ??? when tensoring with preserves limits vs when homming from preserves colimits ???? .... ??? ... suggestive .... ??? .... ??? try to straighten out .... ??? ....

?? diaconescu's thm here ... ??? still not very "internal" approach, though .... ???? .....

?? freyd's "extension logic" (?? ...) here (?? ...) .... ????? ..... ?? relationship to "grothendieck topology" and "cover" ??? ..... ?? filteredness (?? for example ?? ...) .... ???? .... ?? categorification .... ???? higher (?? more than one way ???? .... ???? ....) topos ... ??? ..... ???? ... "arbitrary length quantifier alternation strings" .... vs "geometric" ??? ... ????? ......

Thursday, March 15, 2012

?? at the moment the g2 animation seems to be saying that the special aplit quaternion lines exhibit skidding, but ... ?? maybe just a convention mismatch ?? ... ... ?? argument that we have one solid generic "line" that's non-skid, and that that implies that all the other "lines" are non-skid as well .... ??? ...
?? confusion about .... ??? ..... universally inverting object in ag theory ... ?? try working out some concrete examples .... ??? ....

?? "blow-up" .... ??? "rees ...." .... ??? .... "normal cone" .... ?? ....

?? "normal cone" ... "renormalize" .... ????? .....

?? filtered vs graded .... ??? .....
?? g2 .. idea for getting split quaternion line animation to work ...

?? toric geometry ... ?? way categorified comultiplication (?? ...) issue seems moot in "classical" (=?= unstacky ?? ...) case but possibly annoyingly un-moot in stacky case .... ????? .... ?? "ordinary" vs "toric" approaches to stacky toric case .... ???? ..... ?? but wait, categorified comultiplication here (post-affine toric geometry ... ??? ...), right ??? ... because of inessential diagonal of accidental topos ... ?? so what are we suggesting here ???? ... ??? toric convolution as substitute for categorified comultiplication ??? .... ???? .... ?? or did we have some other idea ??? .... ???? i mean, we did have _some_ ideas, like going to derived level .... ??? .... ?? but was there something more particular here (?? ...) ??? ....

?? are we leaning here towards ... ?? taking plain old smccc doctrine as only good (?? ...) idea for post-affine development ??? ... ?? .... ??? .... ?? property vs structure .... ??? .....

?? project of [trying to understand left universal property of fe projective line in setting uncluding categorified comult] as pretty badly spoiled under this (...) approach ... ??? ...

?? that weird special case where cartesian product of topos coincides with tensor product .... ???? ..... ??? ....

?? grothendieck topology on free finite limit completion (?? .... ?? ....) of small cat .... ??? .... ?? organizing hierarchy of such ... various landmarks and sub-regions .... ???? .... ?? limit vs colimit games on both formula and model side ??? .... ???? ..... ?? "as few/many co-/limit axioms as possible" ... ???? .... ?? "flat" ... "canonical" ... "filteredly cocontinuous" .... ???? .....

?? pair of subgroup inclusions h1 >-> g <-< h2 ... ?? special case where h2 is 1 ... corresponding "restriction" ag morphisms between rep cats .... (??? exactly what sort of reps ??? ..... ?? exactly what sort of group are we / should we be dealing with here ??? .... ???? "algebraic" ??? .... ??? .... ?? ?? with enough reps ... ???? .... ??? ....) .... weak pushout of such .... ?? this as big relationship (discussed with alex to some extent ... ?? also chris rogers ?? ... ??? ... baez ... ??? ...) between consecutive (?? ...) pet obsessions of mine ... : categorified hecke operator (...), and doctrine theory .... ??? .... for example specialize not only h2 = 1 but also g = gl(2) and h1 = borel .... ???? ... projective line as h1,h2 double coset space here .... ???? .... ?? h,1 double cosets as "h-structures" .... ???? "take rep cat of borel and freely adjoin isomorphism from "the big (= 2d ...) object" to actual given concrete 2d vsp ... as acted on trivially by borel .... ???? "universally split the 2d rep" ... ???? ..... ???? effect on "k-theory" ???? .... ???? ....

?? "freely adjoining isomorphism in given hom-space in ag theory" ... ?? "syntactic" approach ??? .... ??? .... ?? ?? analogy to bit about ... ??? adjoining morphism in given hom-space in topos ??? ... ??? .... not sure how accurately i'm remembering that bit ....

Wednesday, March 14, 2012

?? topics for huerta ... ?? ...

?? fixed points of tweak map .... ??? 2-rolls away .... ??? 4d cone ... ??? this as maybe indicating tweak map isn't linear, though maybe i should just check directly .... ??? .... ?? trying to understand tweak map conceptually .... ??? so (7) ?? ....

?? associativity for r,i1,i2 ... ?? preserved by cayley-dickson ???? ... ???

?? mystical "2n+1" stuff ... schanuel's open interval ... ?? binary numbers and ruler markings ... ??? ....

?? structure constants in terms of binary numbers ... ??? ...

?? evil one .... ???? ....

action vs grading ... grading approach to anti-involution and cayley-dickson .... ???? ....

a 1 a
-1 0 1
-a -1 -a

s 1 s
1 1 1
s 1 s

those are just casual guesses so far ...

?? albert's parameter here ???

?? trying to count degrees of freedom and degrees of wasted freedom here ... ??? ....

?? cartesian square showing up here ??? .... ???? ....

incidence geometry .... new "line"s ... reconsideration of orientation classification macro-micro relationship ... ?? trying to understand and resolve disagreements here ??? ....

Tuesday, March 13, 2012

?? action version of super-algebra ??? ....

?? lives even at set level ??? .... ??? alternative symmetry on cartesian product of z/2-sets ??? .....

?? braided version with z-sets ??? .....

(x,y) |-> (y',x) .... ??? reminding me of particularly _recent_ stuff ??? ... ??? but what was it ??? ....

?? hmmm ... ?? seems to have been that "right-adjoint slow-motion" idea .... ??? ....
?? 4d tangent cone of "2-rolls away" as containing 2d zariski tangent space of "1-roll-away" ... ??? does that allow us to take the 5d ambient tangent space and mod out by the 2d subspace, and have the 4d cone survive as a 2d cone ??? ..... ??? hmm, maybe not ... ?? could still try to project it down to the 3d quotient space, but .... hmm, it's only codimension 1 so pretty much any smearing is going to be total, right ?? .... ??? .... i'd been hoping that there was some way to get something here ... and that it would fit with how it seems very natural to get to _generic_ via roll-spinning ... ??? .... ?? had visual image of 1d line going through "middle" of cone .... ?? but now it seems like whole picture might be screwed up here .... ??? .....

?? maybe vague intuition here on right track ... ??? .... ?? "spin as proportionate to overspin" ... ??? if you know what i mean .... ??? ... ?? "yaw as proportionate to roll" .... ????? .... "proportionate" in some vague sense that should be made precise .... ??? ... ?? "yaw as proportionate to "difference" between roll and "speed"" ... ??? .... ?? so far still just trying to describe tangent cone .... ??? ....

?? actually i may have gotten "pitch" and "roll" here mixed up, for somewhat understandable reasons ... ?? but anyway, this raises the question of whether maybe both pitch and roll should enter into the situation somehow ..... ????? ...... ???? .....

?? "fixed points of tweak map" ??? .... ???? ....
??? algebra with anti-involution * with the *-invariant part associative and commutative .... ??? as module with quadratic form and anti-commutative bilinear operation ??? ....

hmmm .... ??? "qfs" as special case .... ????? ...... ??? "killing form" ???? ..... ??? possible extra laws to impose here .... ???? "killing law" ??? ..... ???? leibniz identity .... ??? .....

?? "enveloping algebra" .... ????? ..... ???? ..... exterior algebra .... ???? .....

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

?? bit about axiomatic (?? ...) characterization of cayley-dickson endofunctor ... ?? incorporating some of above ideas ??? .....
?? ag morphism that's both "toric" and "flat" ?? .... ??? possibility of getting from this "ag morphism going the other way ... using the toric convolution products" ??? ..... ???? .... ??? .... adjointness games .... ????? tendency towards order 2 or order 4 here ???? .... ???? ....

?? ... confusion ...

?? .... just wanted to state some stupid guesses here .... ?? and suggest might be worth looking at situation more carefully ... ??? relationship to topos picture ?? .... ??? .....
??? "holomorphic" =?= "definable" ??? .... ???? .....
?? trying to give nice axiomatic (...) characterization of cayley-dickson process as endofunctor of _alg with involution_ riding endofunctor "_ # k^2" ... equipped with embedding form identity functor .... ??? .... ??? .... ??? ....

?? confusion here about acting (that is, the z/2 action given by the involution ...) vs grading .... ??? ....

?? ok, probably some of the confusion was from some people saying "algebra with involution" for "algebra with anti-involution" ... which might arguably be defensible, though also arguably indefensible, since true "algebras with involution" are probably quite important in their own way ...

?? anyway, does this explain why for a moment i was expecting the product of imaginaries to be real, despite being somewhat familiar with "ij = k" ... ??? ....

?? but ... in the anti-involution case is there anything vaguely like a substitute for the product of imaginaries being real ??? .....

(xy)* = x* y* vs (xy)* = y* x* .... ??? ....

x imaginary <=> x* = -x ... ?? ...

(xy)* = (-x)(-y) = xy .... ???? ....

??? commutators and anticommutators here ??? ..... ??? anticommutators of imaginaries as real, commutators of imaginaries as imaginary ????? .... ???? ..... ??? did we in fact bump into "anticommutators of imaginaries as real" the other day, without quite completely noticing it ??? .... hmmmm ..... ????? ..... ?? in connection with .... ???? jordan product of traceless operators ..... ???? ..... ???? ..... ????? ..... ????? traceless vs symmetric ... ?? again ?? ... ??? .... ??? ....

???? commutator of real and real .... ??? or of real and imaginary .... ????? ......

x,y imaginary wrt anti-involution * .....

(xy+yx)* = (-y)(-x) + (-x)(-y) .... ??? so anticommutator of imaginaries is real ???

(xy-yx)* = (-y)(-x) - (-x)(-y) .... ??? so commutator of imaginaries is imaginary ???

x,y real wrt anti-involution * ....

(xy-yx)* = yx-xy ... ???? so commutator of reals really is imaginary here ???? ..... ???? so an associative alg with anti-involution gives a graded jordan algebra and a _shiftedly_ graded lie algebra ??? .... ..... ????? ..... and non-associative case ..... ?????? .....

?? relationship to other grading shifts ????? ..... ????? .....

??? generalizations to algebra with twisted (?? ...) order n endomorphism ??? ....

?? generalizations to algebras of arbitrary vsp-enriched operad (?? ...) ?? ... ??? ...

?? anyway, back to acting vs grading .... ??? .... ?? duality between them as somewhat broken ??? .... ?? in maybe interesting way ... ??? vague feeling of relationship to recent observations about ... ?? very different galois reps associated to (?? "fourier" ??? ...) dual bistable hopf algs ... ???? ..... ?? anyway, grading as always special case of acting via duality, but not generally vice versa .... ???? ......

?? nevertheless, might be interesting to look at grading version of cayley-dickson process .... ???? ..... ??? ..... ?????? ....... ....

?? conversely (??? ...), might be interesting to look at acting version of "superalgebra", trying to squeeze out some extra generality .... ???? .....
?? integral spinor idea ... in arbitrary dimensions .... ??? "bott periodicity" ??? .... ????? ......

??? witt ... ???? .....

?? "tracelessization" .... ?? vs "symmetrization" ... ??? games with 1/2 .... ???? ......

a -b
c -d

d b
-c -a

a-d ..... ???? ....
?? "vertical" or "fiberwise" tangent bundle of map f : m -> n .... ?? as equalizer of t(f) : t(m) -> t(n) and t(m) -> m -> n -> t(n) ... ??? ....

?? special case where f is t(t(m)) -> [t(m) direct-sum t(m)] .... ?? vertical tangent bundle of this f as naturally equivalent to tangent bundle of m ... ???? ...

?? this as trying to get at heart of "lie bracket" from certain viewpoint ... ??? .... ?? maybe heart of some other things too ??? ...

?? torsors of tangent bundle here ?? ... ??? any interesting "cohomology" aspect here ??? ....

?? thinking about above in connection with ... ??? ... the "3" / "roll-spin" invariant distribution of the "2,3,5" of the g2 rolling ball geometry as not the tangent distribution of any invariant binary relation .... ??? ..... ?? and since not, then trying to systematize geometric process by which it _does_ arise (?? from "macroscopic" stuff ... ?? extent to which we've thought about / understood this before ?? ... ?? relationship to huerta's concerns / complaints ?? .... ??? .....

?? so ... ??? tangent cone of "2-rolls-away" schubert variety based at point of g2 configuration space ... ?? as some 4d cone ??? ..... ??? how related to 2,3,5 distributions ???? .... ?? ... again, seems pretty unfamiliar for some reason .... ???? ..... ?? if this _is_ what huerta's been complaining about then i don't think that they described it very well .... ??? .....

?? systematization mentioned above ... ??? "higher tangent spaces" and / or "auxiliary geometric figures" .... ?? relationship therebetween ... ???? ....

?? "there's an obvious way to identify the projective light cone of the imaginary split octonions with the configurations of a rolling ball ... ?? but the business of the "tweak map" means that we're _not_ using that obvious way, but a somewhat less obvious way ..." .... ??? .... ??? .... ?? still confused about this .... ?? and about what it might say about relationships of g2 to so(7) and so(8) and so forth .... ?????? ..... ????? does tweak map live in so(7) ?????? ..... ???? ..... split form, i mean ... ?? .....

Monday, March 12, 2012

?? 4d space of g2 points 2 rolls away from given one ... ?? vs 3d vector space of "roll/spin" tangent vectors .... ???? .... ??? did we ever have a story about this ??? ..... ???? ....
hmm, in the g2 animation mathematica program we seem to use a cayley-dickson product convention "(a,b)(c,d) = (ac+db*,a*d+cb)" which might differ from the one that i've been using recently that i copied from wikipedia ...
?? rep of clifford algebra of ["discriminant" quadratic form on binary quadratic forms] on k^2 given by isomorphism (?? ...) of clifford-jordan algebra to underlying jordan algebra of operators on k^2 ... ??? .....

?? tricky vs non-tricky aspects of left-universal properties involving jordan algebras ... ?? on input vs on output sides ... ???? .... ?? ...

?? hmm, sounds probably related to story about "integral spinors" that i used to know ...... ????? ....

?? trying to dredge up vague memory .... ??? relationship to "rationalness of the circle" ??? .... ???? ..... ??? ..........

?? ... ?? any relationship to a1 dynkin diagram coincidence ?? ... ?? .... ?? hmm, maybe yes ?? ... gl(2,2) as split quaternions ... ?? gets non-standard anti-involution here, with traceless as "imaginary" .... shift-gradedness of anticommutator wrt (?? ... fourier duality grading ...) anti-involution .... making imaginaries into anticommutator algebra .... ?? meanwhile traceless corresponding to symmetric ... so transport the anticommutator algebra across that bridge and ... ?? maybe that's the symplectic lie algebra structure on the symmetric bilinear forms ??? .... ???? .... ???? .... ?? "quadrant" picture of symplectic lie bracket here ??? ..... ???? .... ???? ...
?? various ideas about particular motivations for introducing "up to a particular level" (?? ...) new colimits of affine algebraic varieties .... ??? ....

?? "modifying the original colimits as little as possible so as to get "points over fields (?? ...)" functors to preserve (?? some ?? ...) colimits" .... ???? .... ?? some sort of "atomicness" (??? ...) of field spectrums .... ???? .....

?? achieving toposness ... ?? but then rejecting it .... ?????? ...... ????? .... morphisms vs objects .... ?? "allowing only flat (?? ...) glueing" ??? ..... ???? ....

??? borrowing colimits from ag theories ...... ?????? ......

?? recycling various ideas mentioned here in "higher" cases ... higher versions of non-/affine .... ??? ....

?? (?? "co-" ???)minimal topos completion (?? ...) of _affine scheme_ .... ????? .....

??? ideas (from (?? ...) toric case ?? ...) about .... ??? comparing [theory morphism] concept to morphism concept defined by ... ?? "recycling glueing of objects concept from "absolute" to "relative" case ..... ????? ..... ?? fixed vs variable base .... ???? ..... ??? 2-colimit ??? ..... ???? ...... ..... ?? affine base vs non-affine base ..... ????? ..... ???? ....
?? "locally but not globally trivializable bundle" vs "locally but not globally eliminable stuff" ... ?? ... ruling out (?? "skyscraper"-ish ?? ... ?? ...) local pathology of bundle, vs ruling out stackiness of stack .... ??? .... ??? ... ??? relationship ??? .....
?? winnie-the-pooh as passive-aggressive version of animal farm ... ?? ...

?? ulica kubusia puchatka ...

?? "passive-aggressive" =?= "passive resistance" ?? ...

?? us army's term for people who not only resisted their enslavement but resisted helping the army identify them as a resister ...

Sunday, March 11, 2012

?? ambiguity between "passive" and "ovs" ... ??? ....
?? so consider the subalgebra of the split octonions generated by the nilpotents (i,1) and (j,1) ...

?? seems like it's probably a split quaternion subalgebra ... ?? but perhaps essentially different in nature from the more standard (?? ...) one generated by (i,1) and (i,-1) ???....

?? to what extent does the "tweak map" screw up ideas about ... ??? certain hopefully nice conceptually simple relationships of g2 geometry to so(7) and so(8) and so forth .... ???? ..... ?? maybe not that much ?? ... ?? maybe all (?? ...) that stuff lies comfortably in the background, against which the g2 lines are carved out (?? in a funny way connected with the annoyance of the tweak map ?? ...) as the foreground .... ???? ...

?? a-b .... ??? ... ?? vs a+b ?? ......

?? confusion here about 2 ray halves of line ??? ?? hmm, maybe it's pretty obvious that as soon as both halves of the signature go up to 2 then the cone doesn't break cleanly in two pieces .... ?? ...

?? some sort of funny game involving relating discriminant of binary quadratic form to determinant of traceless operator whose matrix is related by "exchanging the main and anti-main diagonals" ?? .... ????? .....

a b
b c

b -a
c -b

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

ac-bb ??? .... ???? ....

?? "checkerboard" aspect here ?? ... ?? allowing b to be half-integer ??? ... ??? .... ?? gauss vs someone else ??? ...... ?? jordan product here ?????? ..... ????? .... ??? jordan vs lie here ??????? ....... ??? clifford-jordan algebra and traceful operators .... ???? ....

?? clifford-jordan algebra of binary quadratic forms wrt discriminant, vs underlying jordan algebra of associative algebra of 2-by-2 matrixes .... ???? ..... ?? "cone" .... ???? ... "positive" / "future" .... "light" ... ??? ... "sky" .... ????? ....

b -a
c -b

e -d
f -e

be-cd -ae+bd
bf-ce -af+be

eb-fa -db+ea
ec-fb -dc+eb

be-cd+eb-fa 0
0 -af+be-dc+eb

?? that does seem to confirm it by gruesome calculation ... ?? that the clifford-jordan product of binary quadratic forms is essentially the same as the anticommutator of traceless operators on k^2 .... ???? .....

?? and determinant of traceless operators on k^2 is the "quadratic (?? de- ??)polarization" of the bilinear operation "anticommutator" ??? .... ???? .... not sure i said that right ... or in what ways it can/should be generalized .... ??? ...

??? hmm, i was going to ask about analog for unsplit integral quaternions of what we're doing here with split integral quaternions .... ?? but then perhaps this opens a can of pleasant worms ... ?? rather than just split and unsplit quaternions in the integral case, maybe there's a whole interesting family of them ... corresponding to different "discriminants" ... ???????? ...... ?????? .....

?? underlying jordan algebra of d-integral quaternions ..... ???? .... ?? "d-discriminant of binary quadratic form" ???? ...... ?????? ....

?? checkerboard aspect here ... ??? interesting lattices in the quaternions .... ???? ... conway .... ??? ....

?? ideas about "discriminant of binary quadratic form as universal such form, almost sort of ..." .... ???? .... ?? connections here ??? ..... ???? how did that (?? ...) go .... ??? "genus zero analog of taniyama conjecture" .... ????? ..... ???? ..... ???? ....

?? "spin factor" stuff ... ?? generalizing some stuff above ?? ... ??? did i see huerta do stuff like that once ??? ..... jordan algebra .... ???? ..... ?? hmm, arbitrary "spin factor" vs one associated with ..... ????? division algebra ???? .....

??? "berezinian" ..... ???? ..... ????? .....

?? "d-integral quaternions" for discriminant value d vs "q-integral quaternions" for quadratic form q .... ????? ....

??? "witt ..." .... ???? ....

?? trying to get back to the octonions here .... ?? ...

?? "positive indefinite" binary quadratic form .... ??

p 0
0 0

0 0
0 q

?? corresponding traceless operators .... ??? ....

0 -p
0 0

0 0
q 0

?? ..... anticommutator of traceless operators on k^2 ...?? ... ??as "polarization bilinear form of determinant quadratic form" .... ???? ....

?? hmmm, seem to be getting anticommutator here as _negative_ ..... ???? .... (is this due to some annoying convention mismatch somewhere ?? ... ??? ...)

?? ab+ba and bc+cb > 0 => ac+ca > 0 ??????? .... ?? maybe not quite right way to say it ... ??? .... ?? mixed vs indefinite confusion here ??? ...

?? anticommutator of (i,1) and (i,-1) vs of (i,1) and (j,1) ... ??? ....

?? once that's straightened out ... ?? that is, scaling so the anticommutators are negative .... ???? .... ?? then try ... ?? conjugating by exponential of (a+b)*k ... ??? ....

?? "alternative" = "2-at-a-time associative" ?? ... ??? so don't have to worry about associativity here ??? ....

anticommutator of (i,1) and (i,-1) as .... ??? (-2,-2i) + (-2,2i) = (-2,0) ... so that's a negative constant, so this seems to be a "good" pair a,b of representatives .... so then a+b = (2i,0) .... ??? ... ??? then conjugating by exponentials of this ..... ??? ... (r+si,0) with r^2 + s^2 = 1 ... (but this norm condition as pretty irrelevant due to freedom to rescale ... ??? ...) ?? we think that that's going to give ... ?? ....

?? meanwhile, anticommutator of (i,1) and (j,1) .... ??? as (k+1,i-j) + (-k+1,j-i) = (2,0), so that's a positive constant, so good representatives a,b should be (i,1),(-j,-1) .... ???? .... ?? so then a+b is (i-j,0) ..... ????? ..... hmmmmm ...... ????

?? exponentials of (i-j,0) ... (r+s(i-j),0) .... ?? dual (r-s(i-j),0) .... ??? ....

(r+s(i-j),0)*(i,1)*(r-s(i-j),0) = ??

= (ri - s + sk,r + si - sj)*(r - si + sj,0) = (rri - rs + rsk + rs + ssi - ssj + rsk - ssj - ssi,rr + rsi - rsj + rsi - ss + ssk - rsj - ssk - ss) = (rri - 2ssj + 2rsk,rr - 2ss + 2rsi - 2rsj)

???? r,s = 1,1 .... ??? .... (i-2j+2k,-1+2i-2j) .... ??? ... ?? doesn't seem easy to visualize offhand ??? .....

?? "maximal associative subalgebra" .... ???? .... ?? includes unsplit quaternions though ??? ....

????? ... "modifying jordan product by declaring new neutral element" ... ??? "jordan triple system" ... ??? ... bounded symmetric domain .... ??? .....

??? "homming of qf spaces" .... ??? siegel .... ?? ... bit about "discriminant of binary quadratic form as almost sort of universal such" .... ??? ..... ?? hecke operator .... ??? .... ??? ....

Saturday, March 10, 2012

?? vague idea of using symplectic case of dynkin incidence geometry to study "ordinary" case with figures of different dimension all living together .... ??? .... ?? discrete sum of (ordinary ...) grassmanians, with modular lattice structure .... ?? vs something non-discrete ?? ... ???? .... ??? cohomology rings here .... ???? .... ???? ...

?? maybe some confusion here about ... compact situation vs .... ???? ..... ?? or ... ??? linear vs projective ??? ... ??? .... ?? arnold ... ???? .... ?? vs affine ??? .... ??? ....

?? "wave breaking" .... ??? ....

?? natural contact structure on coadjoint partial flag manifold .... ???? ....

?? ... started (?? ...) thinking about this (...) stuff in connection with ... ?? stuff like ... bruhat classification of g2 point-pairs, and "how the split octonion subalgebras generated by nilpotents x,y vary as x,y vary ....." .... ???? ....
?? g2 point-point orientations .... ??? .... in terms of split octonion subalgebra .... ??? .... ?? some missing from my classification so far .... ??? ...

?? hmm, or maybe there aren't any (or not too many) missing and i just miscounted .... ??? ... partial-partial .... ??? ....

Friday, March 9, 2012

?? bit about something like galois rep arising from stable hopf alg ... ?? stable hopf alg here as morally "affine algebraic abelian group" ??? .... ???? ....

"algebraic k-module" .... ?? possibility of k being external constant, or itself an algebraic comm ring ...... ????? such as "_the_ comm ring" ..... ????

?? example of algebraic module over algebraic comm ring "the comm ring" ... ???? ....

?? "the comm ring" ... ??? affine algebraic comm ring ... spec(k[x]) .... ?? affine algebraic module over it ???? .....

?? spec(k[x,y]) as another affine algebraic comm ring ... ?? in fact affine algebraic comm alg over the previous one .... ??? .... ?? with the comultiplication (x |-> x1*x2, y |-> y1*x2 + y2*x1) ...... ?????? ....

?? vague memory of ... ??? slightly exotic hopf alg (?? ...) example we thought about ... ?? and coincidentally read someone else write something about .... ?? galois flavor .... ??? ....

?? "finite" sub-[algebraic group] of gl(1) or of some particular abelian variety, for example .... ??? ....

gl(1) as affine algebraic ... spec(k[x,y]/xy-1) .... spec(k[x]/x^n-1) ... ?? as obtained by modding out by (?? "hopf" ??? ... ???? ....) ideal ... ?? ....

?? affine algebraic subgp of non-affine algebraic gp .... ???? .....

?? hopf alg arising as gp alg ... of external constant gp ..... ??? .... ?? fourier duality confusion here ???? ..... ???? ....

???? algebraic fourier dual of external constant gp as not external constant ??? ..... ???? .....

?? k[x]/x^2-1 vs k^2 ???? .... ?? well, maybe in that particular case there's no difference .... ??? but try .... ????? k[x]/x^3-1 vs k^3 .... ???? ...... whether these are "algebraic fourier dual" ??? .... ?? that is, measures on z/3 vs functions on z/3 .... ??? z-valued ???? .... ?? evaluation (??? .... or maybe "integration" more apt terminology here ??? .... ??? ......) pairing .... ???

?? "split" ... ?? ....

??? "algebraic (?? "fourier" ??? ....) dual" of finite affine sub-gp of abelian variety ???? ....... ????? ....... ?????? ......

??? "applying any functor to k-bar as giving action of abs galois gp of k" .... ????? .....

?? functions : external constant :: measures : twisted ...... ????? ..... ??? in "external constant" case ??? ..... ?????? ......

Thursday, March 8, 2012

?? split quaternions inside split octonions ... ?? "least skid trajectory" ??? .... nilpotents in split quaternions .... ??? ....

?? "least skid" as not quite right idea since you can get anywhere without skidding ... ??? .... ??? maybe "rolling while spinning" ... ??? .... ??? invariant distribution .... ????? ... ?? whether still not really getting explicit (?? ...) geometric interpretation of most generic octonion product yet, because of that (... ??? ... ??? ....) ???? ..... bruhat classes ... their tangent spaces .... ???? .....
?? various things you could try to do with an elliptic curve ... interrelationships among them .... ??? .....

?? construe its spectrum over a finite field as ... ?? special case of spectrum of hyperelliptic curve over finite field .... ??? to which you could try to apply "quadratic reciprocity with global number field q replaced by global function field f_q(x)" .... ???? .....

?? how close did that come to parsing ??? ....

?? get galois rep over z/p by taking p-torsion points on it .... ??? .....

.... ??? ....

?? l-function ... ???

?? zeta function .... ??? ...
?? module vs stable alg vs stable hopf alg of k ... ?? as alleged source of "galois rep" ... of abs galois gp of k ..... ???? .....

?? "galois stack" ....

?? trying to make ag theory out of modules or of stable hopf algs here .... ??? .... ?? relationship ??? ..... ????? ..... ????? ......

Wednesday, March 7, 2012

?? 1d rep of abs galois gp of gaussian field ... ?? index 2 in abs abs galois gp ?? .... ?? 2d induced rep .... ???? ....
?? equivalence of categories between .... ??? invertible (?? as more than just property ?? ... ??? ..... hmm, i seem to be "going beyond gpd regime" here ... ??? ....) modules of z/n, and "n-cyclotomic fields" ??? .... ?? given in reverse direction by taking the nth roots of unity, with their multiplication structure ... as the addition in the invertible module ... ??? ....

?? generalize .... ??? .... understand .... ??? ....
?? (peculiar ... ?? ...) relationship between ellerman's "partition logic" and "unary predicate logic" ?? ...

?? binariness of equality .... ???? ....

?? theory where model is point of given stone space ... ???

?? theory where model is set over (?? or surjectively ?? ... ??? or injectively ?? ...) given stone space ... ?? ...

Tuesday, March 6, 2012

?? so ... certain strategy may be close to working here ... elliptic curve with complex multiplication .... trying to use as somewhat degenerate (but hopefully stepping-stone to less degenerate ...) example of stuff diamond and shurman seem to be talking about .... ???? ....
?? non-toric analog of classification of geometric morphisms between accidental toposes ... ?? analog of "relatively affine" (??? ... ?? cohmology ??? ... ??? ...) and / or of "flat" .... ??? .... ???? .....
?? "toric ag with exponentiation" ??? ....

?? "product-exponentials algebra" .... ??? as "comm monoid with hom to its own endomorphism monoid" ??? .... ??? is that all it is / should be ??? .... ?? categorifiying, and/or throwing addition into the mix ??? .... ??? quandle echos here ??? .... ?? lawvere / schanuel / joyce / freyd / yetter ...

?? "action of underlying comm monoid of pe alg x, compatible in certain way with x's "exponentiation" action of u(x) on itself" .... ??? in certain way imitating way exhibited by u(x)'s cayley action ???? .....

?? x's exponentiation action of u(x) on itself as nicely / canonically extending (...) to action of u(x) on (comm monoid, action) pair (u(x), cayley(u(x)) ?? ....

?? by ... ??? thinking of "cayley" as lawvere theory morphism ?? ...

?? with left adjoint "cayleyfication" ?? ... ??? ... (?? idea for actual systematic terminology here ??? .... ???? ....) ....

?? concretely .... ???? .....

?? hmm, so are we suggesting here that there should be _another_ ("exponentiation" ??? ....) action of m on a required, besides the usual ("multiplication" ...) one ??? ....

(a1 * m1) ^ m2 = (a1 ^ m2) * (m1 ^ m2) ???? ....

?? "semi-direct product using inner action as actually direct" trick here (??? ....) ??? ... ???? ... ???? ....

?? .... schanuel .... ?? tarski .... ??? ....

?? semi-direct product as weak and / or maybe lax colimit ?? .... ?? presheaf topos over weak / lax colimit site category as ... ??? ....
?? "slow-motion" (?? by given factor ...) left-adjoint endofr on dynamical systems .... ?? left-exact ??? ... ?? "toric flat ..." ... ??? ....

?? "invertible" case here ???

?? cartesian square of slow-motion of walking fixed point, vs slow-motion of cartesian square of walking fixed point ?? ....

?? possible ways in which we might have been already supposed to sort of know this ?? .... ??? .... ?? ....

?? right adjoint slow-motion of x as x^2 with (a,b)' = (b,a') ?? ... ???? .... ?? something quandle-/braid-ish here ??? ..... ???? ..... ???? ... ?? quandle where conjugation of a by b doesn't depend on b ?? ... ??? .... ??? ....

Monday, March 5, 2012

?? l-function of galois rep ..... ?????? .... for example p-torsion of lemniscate .... ???? .....

?? business about ... trying to understand abelian langlands via diaomnd and shurman .... ??? certain key ideas ... ?? apparent toy example of "automorphic rep" .... ???? .....
?? "rigid" and "robust" as opposite kinds of "strong" for structures ... guaranteeing others's survival (like mine canary ...) vs surviving oneself ... under action by permutations .... ???? .... strong in breaking symmetry, vs in ... ??? getting along with it ??? .... robust structure as adaptable, flexible, springy ... opposite of rigid ... rigid = brittle ...

?? brittle ... ?? "fragile" ... "symmetry-breaking" ambiguity ... who gets "blame" / "credit", the structure or the permutation, when the latter doesn't preserve the former .... ??? .... ??? contravariance here ???? ..... ????? ....

?? strong in "breaking" other, vs in remaining "unbroken" oneself ... with perspective shift on who breaks who .... ??? .... ??? co-operation vs competition here ??? .... ?? failure as loss for both, vs [[as win for one over other] vs [as win for other over one]] ???? ..... ??????? .....


??? idea for categorification of certain flavor of "galois connection" ??? ... ?? from boolean-valued question "is f(x) = x ?" to set(?? ...)-valued question "what are the morphisms f(x) -> x ?" ..... ???? ..... ?? "galois connection" ... "idempotent monad" .... "adjunction" .... "monad" .... ???? .... ???? confusion about specialness (???? .... ?? of ... ?? particular flavor of galois connection .... ???? ....) here ??? .... ???? ....
?? taking topos-flavored (?? or other ...) tannakian reconstruction seriously in artin reciprocity context .... ??? ...

?? "interesting" extension of gaussian integers as further splitting same primes already split; otherwise just re-using previously available splitting opportunities .... ???? ....

??? dyadic roots of unity .... ????? .... galois group .... ??? ....

?? adele ... idele .... ???? ....

?? mult gp of profinitization of comm ring (for example z ... ??? ....), vs profinitization of mult gp .... ??? .....

?? "inclusion of generators" ..... ????? ..... ???? ....

Sunday, March 4, 2012

?? "absolute galois group of f_q(x)" .... ??? .... ??? extent to which we've thought about this before ??? .... ??? why _wouldn't_ we have ??? .... ??? .... ?? certainly thought about trying f_q itself as analog of z ... and ran into problem of some stuff seeming too degenerate, so alleged interesting non-abelian stuff doesn't seem to materialize .... whereas we've certainly heard about global function fields as somewhat nicer (?? more "regular" ??) analogs of global number fields .... ??? .... ??? function global vs number global, vs local vs global .... ???? ..... ???? .... also "function local" ????? ..... ??? ... ?? level slips .... ??? ....

?? conceptual role of change of special subring in function global case ??? .... ???? .....

?? tannakian (?? ....) approach to abs galois gp of f_q(x) .... ???? ..... ?? classification of "(?? semi-simple ... ??? ...) finite comm f_q(x)-algs" .... ???? pointed finite comm f_q-algs .... ??? ..... ?? tate motive ?? .... as formal inverse to .... ??? .....

?? decategorification here .... ??? concepts of "zeta function" and "l-function" ..... ????? ......

?? started thinking about this stuff as result of previous post about ... ?? relationship between certain ways of thinking about points of lemniscate over finite fields .... ??? ..... ??? .....
?? solutions over finite fields to y^2 = x^3 - x (for example ...) ... ?? various possible reasons for being interested in such ... ?? relationship among such reasons ... ?? ....

?? so .... ??? let's imagine for a moment that we have some idea of what the finite abelian group of points on the lemniscate is over some or maybe all finite fields ... ??? .... ?? then what might we try to do with this info ??? ....

?? so ... ?? might try to "turn it around", in sense of ... ??? for given p (say ... ?? ...), consider all those finite fields for which that finite abelian group includes a factor of z/p .... ???? .... "case where the point ab gp has p-torsion" .... ???? ...

?? meanwhile .... ???? try applying "quadratic reciprocity" over function field over finite field .... ???? .... ..... ?????? ......
?? over on the "p-torsion point" side of things, we have the alleged method of looking at the elliptic curve over finite fields .... ??? ... ?? anything analogous over on the "x-ideal" side of things ??? ..... .... ????? .... ??? ....
?? counting nth roots of unity in finite fields ..... ?????? ......

n=2 .... ??????
??? idea of ...... ????? understanding galois representation aspect of p-torsion points of given elliptic curve (?? as p varies ??? ....) via .... ????? studying that elliptic curve over finite fields ...... ??????? ...... ?????? ......

??? relationship to something todd said .... ???? ....... ??? ....
?? convolution idempotent .... ??? .... ??? ...

?? haar measure of subgroup .... ??? ..... ??? ... vs gaussian flavor ??? ....

??? non-archimedean vs archimedean ??? .... ???? ....

Saturday, March 3, 2012

?? quasicoherent sheaf over the lemniscate, for example .... ???..... ?? getting explicit .... ??? ....
?? green structure type .... "filtration with semi-simple (?? ...) k-alg structure on associated grades" for k given number ring ... ???? ... ?? analyzing filtration / extension in terms of torsors ??? .... ??? bit about zeta function of number ring and evaluation at integers (?? ...) and torsors (?? of ... ??? additive group ?? ....) and "tate motive" (?? ...) .... ???? ..... ???? .....

?? "original" jugendtraum .... ("absolutely absolute" case ... ?? ...) ... ?? nth roots of unity as forming sort of "galois representation" .... (?? for example n = p ... ??? ....) ???? ..... ?? thinking of extension field as covering space ... ?? f_p-vsp structure on fibers of covering space .... ?? cyclotomic field vs group alg of group of nth roots of unity .... ???? ..... ?? this group / hopf algebra as corresponding to "galois representation" .... ???? philosophy of "any ag object defined over comm ring k as giving galois rep of abs galois gp of k" ?? .....

?? explicitly working out this example in bunch of ways ..... ??? .... ??? also analogous example over imaginary quadratic base .... say gaussian integers .... 2-torsion of the lemniscate .... ???? .... ???? .... explicit hopf alg ..... ??? .... ??? ....

??? kan (?? ...) extension of "galois representation" ... ?? from complete fields to incomplete ones ... ??? ....

?? galois rep as structured "galois set" vs vice versa ..... ???? .... ?? working over finite field to get the vice versa to work better, vs working over c (?? ...) to take advantage of certain nice properties ..... ??? "getting fourier transform to work better" ??? .... ???? ..... ??? then decategorification here .... l-function as special (??) zeta function vs vice versa ..... ????? "niceness" of (certain ??) l-functions vs typical zeta functions ????? ..... ???? "character of 1d irrep" .... ???? ..... ??? periodicity .... ???? ..... ???? simple reciprocity patterns ..... ???? hmmm, bit gunnarsen tried to explain to me .... ??? check for notes on that ???? ... ?? "... coefficients of modular form ..." as stretching-it case of "reciprocity pattern" .... ????

?? galois rep .... l-function ..... ??? "how dimension varies as finite field varies" .... ????? ...... ???? .... action ... invariant subspace .... ???? ... equalizer .... kernel ....

???? "galois stack" ?????? ....... ???? vague (?? ...) idea of "thing that has l-fn as coherent sheaf of some sort ..." ???? ..... ??? relationship between "galois stack" and "ag object defined over comm ring k" (?? as manifestly abs.galois(k)-equivariant ... ???? ....) idea ...

??? 4th roots and gaussian integers, 3rd and eisenstein .... ?? jugendtraum approach here ... imaginary quadratic base .... ?? certain 1/3 or 1/4 size subgp of mult gp of .... ???? ..... ???? "alternation between randomness and order" ..... ???? .... ?? "correlation ..." ?? ....

???? "primitive" ..... ???? level slip ???? ..... ???? ...... ?? "imprimitive" ??? .... ????? ...



?? problematicness of modding out real quadratic field as additive subgp of r or c because of its non-discreteness ??? ..... ????? .....

?? understanding spectrum of given number ring .... ?? just "in itself" (?? ...), or as "galois set" ...... ???? ...... ????? .....
?? ... so ... ?? one silly trap that we seem to fall into sometimes is .... ??? .... ?? trying to identify galois correspondent subfield of subgroup with ... ?? subfield generatewd by one element corresponding to each subgp elt ... ??? level slip .... ?? duality flip .... "fourier duality" .... ???? ....

Friday, March 2, 2012

x imaginary quadratic number ring ... ?? (x # c) / x .... "additive" complex lie gp .... ??? algebraic ?? ... ?? in somewhat unambiguous way ?? ... iso to (c,+) + "elliptic curve with complex multiplication" ??? .... ??? ....

?? confusion here ?? .... ??? non-dependence on multiplication .... ???? .... ?? ....

x -> c^2 .... a+bj |-> (a+bj, a-bj) .... ???? .... ?? (2a,2bj) ??? .... maybe just silly mistake before ??? .... ????? ...

?? subcyclotomic but not cyclotomic field .... ?? fourier duality ... ?? .... ???? .....

??? tensor product of abelian varieties .... hom of them .... ??? shimura & taniyama book .... ??? ....
?? "x-ideal" ... "x-real" (?? "x-actual" ??? ...) ... ?? x "ramification
degree" ... ?? elliptic vs modular ... ??? ..... jugendtraum .... ???? ..... ?? "sub-x-ideal" .... ???? ..... ???? .....

?? see paper discussion from this morning ... ?? contravariance of galois correspondence as hint about "fourier duality" ??? .... ?? algebraic geometry ... ?? .... toric .... ??? ..... ???? .....

?? bit about mere subgroup / subfield correspondence vs more detailed thing .... ??? .....

?? "theory of relation from 1 to 1 which locally is either graph of map one way or the other" ... ???? .... ?? relation vs co-relation ?? .... ?? .... ?? ag vs g doctrine ... ??? ....

?? toric categories .... in some sense ... their universal properties ... in various senses .... ??? try to straighten soem of this out .... ???? spectrum of exotic toric stack over arbitrary comm ring ... ?? "richer doctrine" .... ????? ..... ag morphism with right adjoint strong monoidal wrt toric convolution .... ??? various adjunctions here ... comonoids .... ???? .....
?? definite integral secretly amounting to calculation of coordinate of torsion point on algebraic group .... ??? ..... ?? from 0 to fraction of pi ?? .... ?? ...

?? quartic _part_ of torsion-points field of c mod gaussian integers .... ??? ....

?? jargon for what i think i mean by "torsion-points field" here ??? .... ?? "moduli field" ??? ... ??? .... ?? mystical significance .... ??? .... ??? .....

?? "trivial part of ..." ... ??? ... "redundant ..." .... "repeat of earlier ..." ... ??? how such redundancies / trivialities fit in / together ... ??? ....
?? tree rings ... ?? as perhaps helpfully in twilight zone between discrete and continuous ... (also resonating with person growth example .... ??? do humans have any at all detectable seasonal "human rings" in their bones ?? ... ?? hmm, also oak tree elevator idea ??? ..... ??? actual problem of tree growth with (?? for example ...) rings of equal thickness ?? ... ?? issue of perhaps surprisingly robust solution method by dimensional consideration ... ???? ....) ?? maybe similar examples ??? .... ?? ice layers under certain circumstances .... ??? .... mystical relationship between "differential" and "boundary" .... ??? ....

?? "horizontal vs vertical blocks" issue in ftc intuition ... ?? relationship to "cross-dimensional cavalieri principle" ??? .....

?? can tree rings be measured non-invasively nowadays ???
?? blackness of black hole ... ??? ....

Thursday, March 1, 2012

?? "locally but not globally eliminable stuff" ... =?= "locally but not globally trivializable bundle" .... ????? .....

?? "purely topological / homotopy-theoretic / global" aspect, vs ... ??? .....

?? "flat" .... ???? .....

?? "doctrine" (?? ....) associated with derived level ....

?? "what derived category of quasi(?? ...)coherent sheaves sees" ... ??? vague memory about sudden realization about this (?? ... ??? maybe ... ?? "alg k-theory" ... ?? .... ?? ...) .... ??? which way ?? ... ???? .... bit about ??? underlying module of ideal as retaining knowledge of ideal in anomalous(?? ...)ly coarse way in co-dimension 1 case ..... ???? ..... ???? ....

?? attempted idea of .... theory of derived level doctrine as coarse, "homotopy-like" invariant ..... ?????? ....
?? duality .... reciprocity .... ??? .... ?? "symbols" of certain sorts as matrix coefficients for some sort of fourier transform ?? .... ???? ...

?? nth roots of unity ... multiplicative group of z/n ..... ???? ......

?? fourier duality between multiplicative group of quotient of number ring, and corresponding (?? ...) torsion subgroup of abelian variety .... ????? ..... ???? .... ???? galois ambiguity .... naturality ...... ?????? ...... ??? confusion here ???? ..... ???? attempted bit about jugendtraum as saying something like "galois correspondent of x as practically same thing as x" ... ??? except with duality twist ???? .... ??? well, but why not just "untwist" ???? ... =?= use different basis (???) ??? ........ confusion ... ???? ...... ?? hmmm, try to straigthen out .... "quadratic part of given cyclotomic extension" .... ???? ..... ????? .......

?? "artin symbol" ..... ?????? "kronecker symbol" ?????? ...... "jacobi symbol" .... ???? .....

?? subgroup / quotient group relationship ... with self vs with fourier dual .... ???

?? allegation that .... ???? ... ?? explicit fourier duality approach developed by tate reduces exceptionalness of archimedean places ?? .... ???? ....

??? toric geometry as relative of fourier duality .... ???? ....
?? so .... idea that difference in flavor between "(absolutely...) absolute" and "imaginary quadratic" cases of jugendtraum is because of always ramified at -1 vs never so .... ?? and that this in some way lets you get away with thinking purely in terms of multiplicative groups of quotient rings (?? ...) in imaginary quadratic case .... ??? .... ??? though also so in the absolutely absolute case, apparently, but for some presumably slightly different reason ??? ..... ???? ....

?? abelian extension vs radical extension .... visual intuition about "subdivision" here .... ??? ... threatening to make things _ridiculously_ simple ... ?? "the subfiled practically _is_ the subgroup" .... ??? ..... ?? but .... ??? ......

?? gaussian integers .... quartic reciprocity .... ??? .... also eisentstein and cubic .... ???? .... ?? "reciprocity" ..... ??????? .......

?? "theta constants" ...... ???? "hodge structure" .... ??? .... ???? ....

?? field generated by coordinates (?? ...) of all rational subdivision points of all algebraic groups .... ??? ..... ???? ..... ?? non-abelianness here ??? ....

?? "existentially closed groups" and "galois theory" ??? ..... ???? .... "algebriac closure" ... ??? ... ??? ....?? "duality" .... ???? .... ???? .....
?? arbitrary sym mon cocontinuous fr between accidental toposes certainly gives _some_ sort of ag morphism between toric varieties, specializing to alg comm monoid hom in affine case, so seems reasonable to define this as _some_ sort of "morphism of toric varieties" .... ??? but could still try to prove that this is equivalent to something like "locally a c_u-fan" ..... ???? .....
?? forgetful ag morphism from p^n to "[n+1]-dim object ew 1-dim good subobject" ...
thinking of this as something like ... t := [geometric codomain] theory here .... geometric domain theory as something like "t-oidal ag theory" .... ???? then for some reason wondered about similar idea with forgetful ag morphism from "cartan torsor" to "borel torsor" or perhaps vice versa ... ??? .... hmmm .... ??? .... "forgetting one of a generic pair of flags" .... ????? .... not much in reverse direction ??? .... hmmmm ..... ??? cartan reps as borel-reps-oidal ??? .... ?? in way amounting purely to imposition of extra equations ??? .... ??? ....
?? abelian extensions of z .... ??? ...

?? multiplicative group of z/20 ... ?? acting (?? "tautologically" ?? ... ?? ...) on underlying additive group of z/20 .... ????? ..... as related to absolute galois group of z acting on 20th roots of unity ..... ???? .....
?? todd ... ?? possible topic list ... ??? ...

?? disjunction ...

"locally but not globally eliminable stuff" ....

"theory of relation from 1 to 1 which is either a map from 1 to 1 or vice versa" ... ???

"skyscraper" ... anomalously big over closed subvariety ... ??? ... vs opposite ... ??? ... ??? .... ?? "localization" ?? ... ??? ... ??? ... relationship ... ?? which way ?? ... ??? .... ??? ....




?? jugendtraum .... ??? ....

?? "tautological" rep of ... ?? "relative ideal class group" .... ???? .....