visualization of rank 2 real algebraic buildings

?? disclaimer about ... not "serious" ??? ?? "this is a talk about visualization so it's intended to be a bit more on the fun side than the serious side ... besides i already did it so i might as well inflict some of the fun on you ..." ... ??? ....

1 concrete visual stuff ... a2, b2, g2 .... s^0, s^1 (gimbal ... penrose ..), s^2 X s^0 (rolling ball) .... concrete relationship to bi-multi-homogeneous coordinate algebra ... (??how to handle g2 "tweak map" issue ?? ...?? ...) .... warnings/jokes about tradeoff between real and complex alg geom ... for example naive real view thinks projective quaternions form a group .... ??? ....

?? idea even in real case quaternion zero hints at problem ... ??? .....

g2 crown grassmanian vs quincunx grassmanian ... (?? analogous names for b2 and even a2 ?? ... ?? triangle and line ?? too many other conceptual associations ?? ...) ... quincunx grassmanian as projective light cone and maybe worth explaining how to analyze into revolution and rotation state ... ?? crown grassmanian as traditionally somewhat hard to get at ?? .... ?? maybe not so bad, depending on ... ??? above "tradeoff(...)"-flavored issues again ... ???? ... trying to minimize dimension of projective embedding .... ???? .....

?? funny switch with b2 tradeoff issues ?? ... ??? it's some of the b2 quincunx (??? ...) variables that you're tempted to eliminate, whereas ... ??? morally it's the b2 square irrep that's "causing the problem" ... namely latent non-invertible complexifed quaternions (?? "biquaternions" ... ??) .... ?? anything like this in g2 case ??? ....


?? gimbal-axis as "skewer" ... independent rotations around it of globe and gimbal ... ??? globe as sun=hot .... roast gimbal on spit ... ???.... don't use this (...), but ...

gyre and gimbal ...gyroscope from nested gimbals ?? ....


?? from octonions to rolling ball via quaternions ??? ... ?? octonionic approach to bi-multi-homogeneous coordinate algebra here ... ?? cheating to use so(7) ?? ...

?? no "folding" stuff on this talk ??? .....

?? building as easier to visualize than it's symmetry group ??? ....??? except for maximal compact subgroup ??? .... ???? is that exactly correct in all 3 cases ???? ..... ?? trickiness of visualizing "boost" in b2 case ??? .... "mixing time and space = gimbal and 2-sphere" ... ??? .... ??? flat (2,3) vs conformal (1,2) here .... ???? ....


?? a2 animation ... point (with doppelgaenger ...) moving geodesically on 2-sphere ... ?? comment about geodesic structure (without parameterization) as much weaker than metric structure ?? ?? or maybe that's sufficiently obvious (when in right mood ...) not to mention ... ???? .... ??well, etymology (?? ...) of "geodesic" might justify comment .... ??



2 toric stuff ??? .... kaleidoscope as fan .... ??? ..... ?? schubert singularities ... ??? .... light cone, penrose ... 2,3,5 ... roll, spin, skid ... (root system picture here ...)


3 pretend to justify appropriateness for this conference ,,, ?? very short ...

theory .... propositional : predicate :: scheme/variety : stack .... ????? .....

(emphasize screwed-up-ness of alg geom (??) terminology here ?? ....)

?? slightly (?? ...) different theories here (...) depending on how may gradings you use ...

?? proj ... ????.... "tannakian" version ??

doctrines ...

theory of projective planes, for example .... live interestingly in more than one doctrine ....

tannakian program ...

!! ask alex about talk length ... ?? also mathematica at university ...


???? trying to limk together .... rolling ball and octonions .... rolling ball and quaternions .... quaternions and b2 flag variety bi-multi-homogeneous coordinate algebra ... b2 and b3 ... b3 and g2 ... g2 and octonions and rolling ball .... g2 fvbmhca and rolling ball ... ???? ... in some combination .... ??? ...


?? "steering wheel" ... ?? in trying to explain intutitie sense of flag concept (rank 2 case ...) and "generic flag pair" .... "6 degrees ..." ....