gl(2,f_3) ... ?? aka e7 mckay group = "binary cubic group" ... ??? ....
?? maximal toruses .... ???? .....
?? split ... diagonal ...
?? unsplit .... ??? ....
gl(1,f_9) -> gl(2,f_3) .... ??? ....
?? bit about ... ??? "nice" .... ??? way for field structure to get along with basis ?? ... ??? ... ??? ..... ?? or "nice generator" ... ??? whose powers form ... ?? a basis, or some particularly nice sort of basis ??? ....
?? "necklace" .... ?? free lie algebra ... ?? lyndon .... ???? ...... ??? ....
?? more combinatorics .... ???? .....
?? apparently i once wrote about "field structures on the vector space [f_2]^[2^[n-1]] for which the frobenius automorphism is the obvious rotation operator" ... ??? .... ??? "aperiodic necklace" ??? .... ???? .... ???? .....
?? "and its multiplicative identity is the vector each of whose components is the multiplicative identity of f" ... ??? ....
???? hmmm, ramification here ??? ... ???? .....
?? hmmm, apparently there _was_ some special thing about the case of [f_2]^[2^n], but more generally i was interested in [f_2]^n .... ??? "i'll call a field structure on the vector space f^n over a finite field f "good" iff its frobenius automorphism is the obvious rotation operator and its multiplicative identity is the vector each of whose components is the multiplicative identity of f" ... ???
?? "when f = f_p with p prime we can straightforwardly identify the good field structures on f^n with the irreducible factors of x^p^[n-1] + x^p^[n-2] + ... + x^p^1 + x^p^0 - 1 over f_p that are of maximal degree and have linearly independent roots" ... ??? .....
f_3[x]/x^2=2 .... ???? .....
?? p=3, n=2
?? x^3 + x - 1
?? wild ramification and associated graded .... ????? ...... "tangent cone" ... ??? ....
?? well , let's at least get _some_ explicit unsplit maximal toruses here ... ??? ....
x^2 = 2 ... ???
(a+bx)(c+dx) = (ac+2bd)+(ad+bc)x .... ????? .....
c 2d
d c
?? generator here ???? ..... ????? ......
?? hmmm, for generator maybe _neither_ c nor d should be zero ??? ..... ???? .... ??? mystically suggestive ??? .... ???? .....
12
11
01
20
11
21
20
02
21
22
02
10
22
12
?? "cycle" in 4! ... ??? ... ?? vs split maximal torus here, which .... ??? fixes 2 points and cycles just the other 2 ... ??? ..... ??? so can unsplit maximal torus be thought of as fixing two "imaginary" points, and is this a useful idea ??? .....
?? consider real case (?? again ??) here ... ??? ... ?? gl(1,c) -> gl(2,r) as ... ??? "fixing two complex points" ??? ... ??? ??? which two ??? .... ?? obvious naive guess as ... ?? i and -i as points of riemann sphere (as cp^1) .... ???? ...... ?? seems to work ??? ..... ???? ..... ?? moebius transformation preserving equator and preserving north pole = moebius transformation preserving equator and metric structure (?? and orientation ??? ....) on it ..... ???? .... ?? one pole vs two here ... ??? borel vs cartan .... ???? ... ???? ..... ??? kaleidoscope ....... ????? ..... ????? .....
?? functoriality of complexification here ??? .... ??? ....
?? some confusion here ?? ... ?? q^2-1 vs (q-1)^2 .... ??? vs ... q^2-1 vs q^2-q ???? ..... ????? ..... ????? ....... ?? check recent numerology posts where i might have gotten into such confusion ?? ...
?? 4 "real" points ... ?? 10 "complex" .... ??? ..... 10 = 1 + 1 + 4 + ?4? ... ???? .... ?? = 1 + 1 + 8 .... ???? ....
?? so given character of unsplit maximal torus, try splitting the torus, inducing the character, then applying parabolic induction, then trying to "de-split" ...... ?????? ......
Monday, April 30, 2012
?? second term as swing vote between "not induced" first term and "induced" third term ?? ... ??? ... accusing bump of using "induced" in peculiar way ?? ..... ??? trying to systematize / make more precise such peculiar and/or non-peculiar ways .... ?? "collection of all ways of seeing rep r as induced / quasi-induced" .... ?? and / or universal such .... ???? ...
?? maximal toruses in gl(3) ... trying to match up with kinds of irreps we seem to see in classification .... ??? young-diagrammatic flavor here ?? ....
?? q^3-1 ...
?? (q^2-1)(q-1) ...
?? (q-1)^3 ...
342*336*294 - (1^2 + 56^2 + 343^2)*6 - ((57*6)^2)*21*6 - (57^2 + (57*7)^2)*30 - ((57*8)^2)*20 =
33784128 - 724716 - 14737464 - 4873500 - 4158720 = 9289728 = 2^14 * 3^4 * 7
(q^3-1)*((q^2-1)(q-1))*((q-1)^3) = ??? .....
??? hmmm .... ??? _something_ being parameterized by "one character for "each" maximal torus" ????? ...... ????? .....
?? double life of these particular q-polynomials ... ?? well, or maybe triple, or single .... ???? .... ?? i was going to say "flag combinatorics vs ... this new (?? ...) stuff that we're doing now" ... ?? but ... maybe more unified, as to do with "factorization" of group gl(3) .... ???? ... ?? is there something about .... ???? subgroup pair of complementary index, with corresponding "decomposition" ??? ...
?? hmm, just remembering now that we didn't quite straighten out the q-polynomial for the cuspidal sector .... ???? ......
?? maximal toruses in gl(3) ... trying to match up with kinds of irreps we seem to see in classification .... ??? young-diagrammatic flavor here ?? ....
?? q^3-1 ...
?? (q^2-1)(q-1) ...
?? (q-1)^3 ...
342*336*294 - (1^2 + 56^2 + 343^2)*6 - ((57*6)^2)*21*6 - (57^2 + (57*7)^2)*30 - ((57*8)^2)*20 =
33784128 - 724716 - 14737464 - 4873500 - 4158720 = 9289728 = 2^14 * 3^4 * 7
(q^3-1)*((q^2-1)(q-1))*((q-1)^3) = ??? .....
??? hmmm .... ??? _something_ being parameterized by "one character for "each" maximal torus" ????? ...... ????? .....
?? double life of these particular q-polynomials ... ?? well, or maybe triple, or single .... ???? .... ?? i was going to say "flag combinatorics vs ... this new (?? ...) stuff that we're doing now" ... ?? but ... maybe more unified, as to do with "factorization" of group gl(3) .... ???? ... ?? is there something about .... ???? subgroup pair of complementary index, with corresponding "decomposition" ??? ...
?? hmm, just remembering now that we didn't quite straighten out the q-polynomial for the cuspidal sector .... ???? ......
Sunday, April 29, 2012
?? associated graded object of ideal power filtration of ideal
in z ..... ??? .... ???? ....
p = 5, say ....
?? then consider, say, extension z[i] .... ??? does this come along for the ride in some interesting way ??? .....
?? polynomials over z/5 .... ???? ....
-1 .... ???? .....
?? ideal power filtration of certain (??which??) ideal in gaussian integers .... ????? ......
?? adjoining square root of -1 to f_5[x] ??? .... ??? is that what happens here ??? .... ???? .... ?? how "involved" is x here ??? .... ???? ... ?? maybe not much ?? .... ?? and generally doesn't get any more involved in other cases either ?? .... ???? ... ?? hmmmm ..... ???? ramification ??? .... ?? ideal power filtration of ideal associated with 2 in gaussian integers .... ????? ..... ?? ideal power filtration associated with 5 in golden integers .... ?????? ..... ????? .....
"symbol" ... ???? ... "principal symbol" ??? .... ??? "secondary symbol" ???? ..... ?? "obstruction" ??? ... ??? ... ???? ..... ???? "hessian" ???? .... ???? ..... ??? .....
?? "y^2 = x^3" ..... ???? ......
?? well, one source of confusion about possibility of fitting gl(2,f_3) into mckay correspondence is ... ?? "binary tetrahedral group" (=?= "e6" ???) only covers _rotations_ of tetrahedron .... ??? .....
?? on the other hand, what about e7 ?? ... ??? still doesn't seem to have right number of irreps .... ?? what about number of even irreps ??? .... ?? still seems wrong ?? .... ?? ....
?? hmm, maybe extra dot is key here ??? ... ?? neighbors of extra dot as "spinor" rep ??? ....
?? e7 with extra dot .... ??? ....
?? ok, this seems like it might make sense .... ?? also mesh with child's drawing stuff ??? ..... ???? .....
?? on the other hand, what about e7 ?? ... ??? still doesn't seem to have right number of irreps .... ?? what about number of even irreps ??? .... ?? still seems wrong ?? .... ?? ....
?? hmm, maybe extra dot is key here ??? ... ?? neighbors of extra dot as "spinor" rep ??? ....
?? e7 with extra dot .... ??? ....
?? ok, this seems like it might make sense .... ?? also mesh with child's drawing stuff ??? ..... ???? .....
?? gl(2,f_2) ... ??? ....
?? split maximal torus ??? .....
?? hmm, unsplit maximal torus here as maybe anomalous ??? .... ?? because quadratic extension of f_2 as anomalous ..... ???? .....
?? but is there something funny about split maximal torus here as well ?? ... ??? ....
?? cuspidal vs non-cuspidal here .... ????? .....
?? trivial rep and 2d irrep as non-cuspidal here ... other 1d rep as cuspidal ...
?? making it seem like split maximal torus here should be z/2 or something .... ??? ....
?? bit todd pointed out about .... ?? bn-pair "n" here (?? ... ??? 2 vs 3 ??? ....) as not precisely normalizer of ..... ???? ..... ????? .....
?? actual maximal abelian subgps of 3! ... ?? z/2 and z/3 ..... ????? ..... ?? z/3 as unsplit maximal torus, maybe ??? .... ?? relationship to other 1d rep ..... ?????? ........
?? z/2 as "point" stabilizer here ???? ... ?? abelian parabolic .... ???? .....
?? maybe lots of anaomalies here ... unclear how related to each other ... ?? try looking at gl(2,f_3) instead ... ??? ....
?? some anomalies which might become less anomalous on "passing to algebraic closure" / "treating as algebraic group" .... ??? .....
?? split maximal torus ??? .....
?? hmm, unsplit maximal torus here as maybe anomalous ??? .... ?? because quadratic extension of f_2 as anomalous ..... ???? .....
?? but is there something funny about split maximal torus here as well ?? ... ??? ....
?? cuspidal vs non-cuspidal here .... ????? .....
?? trivial rep and 2d irrep as non-cuspidal here ... other 1d rep as cuspidal ...
?? making it seem like split maximal torus here should be z/2 or something .... ??? ....
?? bit todd pointed out about .... ?? bn-pair "n" here (?? ... ??? 2 vs 3 ??? ....) as not precisely normalizer of ..... ???? ..... ????? .....
?? actual maximal abelian subgps of 3! ... ?? z/2 and z/3 ..... ????? ..... ?? z/3 as unsplit maximal torus, maybe ??? .... ?? relationship to other 1d rep ..... ?????? ........
?? z/2 as "point" stabilizer here ???? ... ?? abelian parabolic .... ???? .....
?? maybe lots of anaomalies here ... unclear how related to each other ... ?? try looking at gl(2,f_3) instead ... ??? ....
?? some anomalies which might become less anomalous on "passing to algebraic closure" / "treating as algebraic group" .... ??? .....
?? "every rep is induced, just not all the way" ... ???? ....
?? "central character" vs "parameterization by characters" .... ???? ....
?? irrep parameterized by character of split maximal torus as induced ... ?? but is the inducement from that torus, or from something related but somewhat larger ?? .... ????? .......
?? unsplit maximal torus of real lie group ....... ????? .... ?? "cayley" (?? ...) representation of (c,*) on underlying real vsp of c ..... ????? .....
?? cayley representation of (f_q^n,*) on underlying f_q-vsp of f_[q^n] .... ????? .....
?? maximal torus as not central .... center of simple group as trivial ... ?? must be forgetting some key idea here ..... ????? .... ?? ok, one key idea is gp center vs ring center .... ???? ... but bump _is_ talking only about gp centers here, right ?? .... ???? .....
?? hmm, they make it sound like they're talking about gp centers, but it seems like they're secretly really talking about ring centers .... ???? .....
?? center of eneveloping algebra / gp algebra .... ??? vs .... ??? of quotient thereof ... ?? associated with rep ??? .... ???? .... ?? or of just plain assoc alg .... ??? .... ???? ..... ??? ....
?? "central character" vs "parameterization by characters" .... ???? ....
?? irrep parameterized by character of split maximal torus as induced ... ?? but is the inducement from that torus, or from something related but somewhat larger ?? .... ????? .......
?? unsplit maximal torus of real lie group ....... ????? .... ?? "cayley" (?? ...) representation of (c,*) on underlying real vsp of c ..... ????? .....
?? cayley representation of (f_q^n,*) on underlying f_q-vsp of f_[q^n] .... ????? .....
?? maximal torus as not central .... center of simple group as trivial ... ?? must be forgetting some key idea here ..... ????? .... ?? ok, one key idea is gp center vs ring center .... ???? ... but bump _is_ talking only about gp centers here, right ?? .... ???? .....
?? hmm, they make it sound like they're talking about gp centers, but it seems like they're secretly really talking about ring centers .... ???? .....
?? center of eneveloping algebra / gp algebra .... ??? vs .... ??? of quotient thereof ... ?? associated with rep ??? .... ???? .... ?? or of just plain assoc alg .... ??? .... ???? ..... ??? ....
Saturday, April 28, 2012
[for david yetter]
hi...
thanks for trying to help me get accepted at ksu, but it looks like it isn't going to work out ...
i hope that it's clear why it makes no sense for me to accept the compromise proposal that i've been offered ... for me to teach under those conditions is a guarantee of failure, while at the same i know that if given a fair chance under more reasonable conditions i can demonstrate that i can be a highly competent and successful teacher ....
hi...
thanks for trying to help me get accepted at ksu, but it looks like it isn't going to work out ...
i hope that it's clear why it makes no sense for me to accept the compromise proposal that i've been offered ... for me to teach under those conditions is a guarantee of failure, while at the same i know that if given a fair chance under more reasonable conditions i can demonstrate that i can be a highly competent and successful teacher ....
?? "multiplicative group of k[x]/x^2=5" (for example ...) as algebraic group .... ?? to what extent and in what context we've thought about stuff like that recently ... ?? .... ?? part of "alg comm ring" bit ?? .... ?? ...... ??? ..... ?? in part hoped to relate zeta function to l-function ... ??? ..... ??? .....
?? asking this now because of reading bump .... ??? .... "unsplit torus" ... algebraic group over f_q that "splits" to give torus over f_[q^n] ... ?? more abstractly, alg gp over z ..... ????? ...... ???? .....
?? confusion here about ... ??? naive size count ..... ??? .... (q-1)^n vs q^n-1 .... ???? ......
?? gaussian integers .... ???? ..... reciprocity law .... splitting pattern .... ???? ..... ???? .... ???? .... counting solutions ....
?? counting homomorphisms from gaussian integers, vs counting dimension of space of homomorphisms from .... ???? "formal gaussian integers" .... ???? ..... ????? ...... ?????? ....... ????? ...... zeta vs l .... ????? .... ???? ....... ???? .....
?? studying reps of gl(2,f_q) .... "unsplit torus" intrudes .... ??? .... gl(1,f_[q^2]) -> gl(2,f_q) .... ???? ....
??? studying reps of gl(2,z) ... ??? gl(1,z[i]) -> gl(2,z) ...... ???? ..... ???? .....
?? gl(1,z[i]) -> gl(2,z) as algebraic stabilizer subgroup of what structure on what stuff ??? ..... ???? ... hmmm .... ??? stuff here as "2d object" ???? ..... ???? .... stabilized structure as .... ????? ......
vs gl(1,z^2) -> gl(2,z) .... ????? ...... ???? .... ?? "splitting" ... ??? bit about "complex splitting" / "complex idempotent" .... ??? .... ?? .... ??? level slip ??? .... ??? ..... ???? ...... "splitting" here as "generic flag-pair" ??? .... ???? ..... ????? ..... ?? "twisted splitting" ??? ..... ???? .....
???? "lattice" ???? ...... ???? .... "extra dot" .... ???? brown vs pressley and segal .... ???? ......
?? asking this now because of reading bump .... ??? .... "unsplit torus" ... algebraic group over f_q that "splits" to give torus over f_[q^n] ... ?? more abstractly, alg gp over z ..... ????? ...... ???? .....
?? confusion here about ... ??? naive size count ..... ??? .... (q-1)^n vs q^n-1 .... ???? ......
?? gaussian integers .... ???? ..... reciprocity law .... splitting pattern .... ???? ..... ???? .... ???? .... counting solutions ....
?? counting homomorphisms from gaussian integers, vs counting dimension of space of homomorphisms from .... ???? "formal gaussian integers" .... ???? ..... ????? ...... ?????? ....... ????? ...... zeta vs l .... ????? .... ???? ....... ???? .....
?? studying reps of gl(2,f_q) .... "unsplit torus" intrudes .... ??? .... gl(1,f_[q^2]) -> gl(2,f_q) .... ???? ....
??? studying reps of gl(2,z) ... ??? gl(1,z[i]) -> gl(2,z) ...... ???? ..... ???? .....
?? gl(1,z[i]) -> gl(2,z) as algebraic stabilizer subgroup of what structure on what stuff ??? ..... ???? ... hmmm .... ??? stuff here as "2d object" ???? ..... ???? .... stabilized structure as .... ????? ......
vs gl(1,z^2) -> gl(2,z) .... ????? ...... ???? .... ?? "splitting" ... ??? bit about "complex splitting" / "complex idempotent" .... ??? .... ?? .... ??? level slip ??? .... ??? ..... ???? ...... "splitting" here as "generic flag-pair" ??? .... ???? ..... ????? ..... ?? "twisted splitting" ??? ..... ???? .....
???? "lattice" ???? ...... ???? .... "extra dot" .... ???? brown vs pressley and segal .... ???? ......
?? "maximal torus" .... ?? "toric variety" .... ??? ..... galois aspect of toric variety .... ???? ..... gl(1) as torus .... gl(1) and "abelian langlands" .... ???? .... ?? torus as variety vs as alg gp ... ??? ...
?? kaleidoscope as fan .... ???? ..... maximal torus ..... ????? .......
??bike path to(wards) san bernardino .... ????? ......
?? kaleidoscope as fan .... ???? ..... maximal torus ..... ????? .......
??bike path to(wards) san bernardino .... ????? ......
a?? so having gotten some of the numerology a bit straighter let's try thinking a bit more about gl(2) .... ??? .....
q=2
6 = (1^2)*1 + (1^2 + 2^2)
q=3
48 = (2^2)*3 + (1^2 + 3^2)*2 + (4^2)*1
q=4
180 = (3^2)*6 + (1^2 + 4^2)*3 + (5^2)*3
?? hmmm ... ?? seemingly obvious naive numerological guess about which part of middle term lives over pgl(2) .... ?? seems to not work ??? .....
?? pgl(2,f_3) .... ???? .... ??? acting faithfully and transitively on 4 points of projective line .... ???? just 4! ?? ....
4 ?? 1
31 ?? 3
22 ?? 2 ?????
211 ?? 3
1111 ?? 1
gl(2,f_2) 3 irreps ...
gl(2,f_3) 8 irreps ...
gl(2,f_4) 15 irreps ....
... ???? .....
?? in principle it's just the cuspidals here (?? gl(2,f_q) for generic q ...) that we don't "understand" so far ?? ....
??? tensoring with 1d reps here (...) ... ??? ... ?? whether this puts some order into the variety of cuspidals here ... ?? ....
?? hmmm, trying to understand these 1d reps better ... ?? hmm, by 2 approaches ....
1 abelianization ...
2 ?? "vector bundle over projective line assigning to 1d subspace x f(x)#g(v/x) where f and g are functors from _1d f_q-vsp_ to _1d c-vsp_" .... ??? .... ?? then understanding "q-braiding" here and modding out by it .... ??? .... "hecke operator / algebra ..." .... ????? .....
?? abelianization and determinants ???? ....
?? abelianization of gl(n,field) vs of pgl(n,field) ?? .... ??? .....
?? abelianization of gl(n,field) as gl(1,field) ??? ..... ????? ....
?? "algebraic k-theory" ... ???? ...... ?????? ..... ???? ....
?? is "projectivization" always finer than "abelianization" ??? .... ?? meaning ... ?? ... when you abelianize gl(n,field), does it always happen that the constants get killed off ?? ..... ???? .... ?? meaning, maybe (?? ...) that the determinant of a constant is ...... ???? ......
gl(2,f_3) .... ????
?? here it really is true that the constants are determinant 1 ??? ..... ???? ....
?? but that seems like an extremely special case .... ?? ....
?? did i ask the question anywhere close to correctly ?? ... ???? ...
?? maybe we had an original intuition here (?? ...) that was right-track ... ?? that _of course_ determinant wouldn't survive too well to projective level .... ??? .... ??? .....
?? for some reason i feel tempted to say "determinants of constants form an obstruction to determinant surviving to projective level" .... ???? does that make _any_ sense ??? .... ???? ....
?? take a look at 1d irreps of gl(3,f_q) as somewhat classified in recent post here ... ??? ... ??? see if fits with "determinant" idea ??? .... ???? ....
?? yes, does seem to fit fairly straightforwardly ?? ... only 1d irreps that show up seem to be "applying trivial 3-box young diagram to gl(1)-cuspidals" .... ???? .... ??? hmm .... ??? idea that .... ?? abelianization of gl(n) is gl(1) via determinant (?? which idea maybe i'm beginning to remember some subtleties to ... ?? maybe only in beyond-field case ?? ... anyway never mind that for a moment ... ?? ...) as embodied here as "process of applying trivial n-box young diagram to gl(1) irrep" ... ??? .... ???? .... ????? ... ?? suggestive somehow ?? ... ???? .....
?? maybe enough progress here to satisfy original goal of being ready to take a look at what bump says about this stuff ... ?? though not yet running into any obvious way in which kinds of things bump seemed to be about to say ("induced rep from maximal torus rep" ?? ... ??? ...) show up ..... ???? .....
?? size numerology of maximal toruses here ??? ...
?? field = f_q .... ??? but ... ??? trying to see where field = c (or r ....) fit in here .... ??? .....
?? "maximal torus" in gl vs pgl vs sl case .... ???? .....
?? diagonal torus of gl(n) ..... ???? .....
q=2
diagonal torus 1 ... gl(2) 6
?? so 6d induced rep ?? ... ?? but maybe breaks into smaller irreps ??? ...
?? ok, so when bump talks about "induced reps", maybe they're really talking about "corresponding" irreps .... ?? how much does that help to straighten stuff out here ??? ..... ????? .....
?? so one stupid guess that doesn't quite make sense yet is something like ... ?? ... ?? bump is setting aside the cuspidal reps (my first term above ... ?? ....) for the moment (?? ...), and associating my second term with "unsplit torus", and my third term with "split torus" .... ...... ?? really not seeing it yet .... ????? .....
q=3
diagonal torus 4 .... gl(2) 48
??? 12 ..... ???? .... 4*3 ???
q=4
180 / 9 = 20 = 5*4
15*12 = 5*3*4*3 ..... ????? .....
?? "generic flag pair" .... ????? ......
?? vector bundle over [flag variety]^2 ??? .....
??? alleged other sort of "maximal torus" ..... ????? .....
?? splitting of unsplit torus ..... ????? ......
?? trying to associate "vector bundle over [flag variety]^2" with my third term as ... ?? seeming rather iffy and relying on special coincidences ... gl(2) vs gl(n) for more general n .... ???? .... ???? ....
?? mckay correspondence for binary tetrahedral group ..... ???? .... ?? is that e6 ?? .... bipartite graph .... ???? ...
?? having some trouble trying to get mckay correspondence numerology to work here .... ????? ... (actually having trouble at the moment getting it to work anywhere, so don't hold it against here too much .... ?? ...)
?? elt of gl(2,f_q) that .... ?? arises from .... ??? putting structure on [f_q]^2 making it into an f_[q^2] ... ??? and then .... ??? .... ?? picking generator for cyclic group gl(1,f_[q^2]) ... ?? and considering group it generates ... ??? hmmm, i guess that that's pretty much like saying ... ?? consider gl(1,f_[q^2]) -> gl(2,f_q) arising in hopefully obvious way .... ???? ....
?? so one stupid guess is that that's what an "unsplit (?? maximal ... ?? ....) torus" is ... ??
?? "ramified torus" ???? .... ???? gl(1,f_q[t]/t^2) -> gl(2,f_q) .... ?????
?? gl(1,z/p^2) -> ????? ..... ?? analog for q in place of p ??? .... ????? .... ??? ...
?? split torus and (q-1)^2, vs unsplit torus and q^2-1 ??? ..... ?? anything like that show up in our classification yet ??? .... ???? ......
?? hmm, almost getting a numerological glimmer of ... ?? cuspidal : unsplit :: non-cuspidal : split .... ???? ... try testing it out a bit further ... ?? or maybe generically .... ??? .....
(q^2-1)*(q^2-q) = (q-1)^2*(q*(q-1)/2) + (1^2 + q^2)*(q-1) + (q+1)^2*((q-1)*(q-2)/2)
?? the glimmer argument is something like ... ?? ... dividing left-hand side by unsplit torus size gives q^2-q, and factors of that get squared in first term ... ?? more specifically, the factor q-1 does .... ??? whereas dividing left-hand side by split torus size gives q^2+q, and factors of that get squared in last two terms .... ??? .... ?? so _is_ this a good hint of what bump's talking about ??? .... ???? .....
?? well, bump does say this :
The representations parametrized by maximal split tori are induced representations, those parametrized by nonsplit tori must be constructed by some other method.
... so yeah, it seems like we're on the right track here .... though in retrospect maybe it should have been somewhat obvious ... ?? ...
?? counting characters here .... ??? also trying to develop correspondences with real and complex cases .... ???? ....
?? not quite getting the non-squared numbers to correspond nicely to numbers of characters yet ..... ??? ..... hmmm, but ... ?? might be possibilities .... ??? "kaleidoscope folding" ?? .... ??? ....... ??? ..... ?? or sub rather than quotient ?? .... ?? "... chamber ..." ... ???? .... ??? hmmm, try adding together the raw counts for the second and third terms .... ??? .... ?? does seem to have some relationship to "(q-1)^2", while raw count for first term seems to relate to "q^2-1" .... ???? ..... ???? .....
?? wait a minute ... ?? might be right track, but some confusion here ... ?? between "parameterization by" as given by inducement (? which sort of seemed to work numerologically ... ?? ...), vs bit about those parameterized by nonsplit toruses as "_not_ induced" .... ???? .....
q=2
6 = (1^2)*1 + (1^2 + 2^2)
q=3
48 = (2^2)*3 + (1^2 + 3^2)*2 + (4^2)*1
q=4
180 = (3^2)*6 + (1^2 + 4^2)*3 + (5^2)*3
?? hmmm ... ?? seemingly obvious naive numerological guess about which part of middle term lives over pgl(2) .... ?? seems to not work ??? .....
?? pgl(2,f_3) .... ???? .... ??? acting faithfully and transitively on 4 points of projective line .... ???? just 4! ?? ....
4 ?? 1
31 ?? 3
22 ?? 2 ?????
211 ?? 3
1111 ?? 1
gl(2,f_2) 3 irreps ...
gl(2,f_3) 8 irreps ...
gl(2,f_4) 15 irreps ....
... ???? .....
?? in principle it's just the cuspidals here (?? gl(2,f_q) for generic q ...) that we don't "understand" so far ?? ....
??? tensoring with 1d reps here (...) ... ??? ... ?? whether this puts some order into the variety of cuspidals here ... ?? ....
?? hmmm, trying to understand these 1d reps better ... ?? hmm, by 2 approaches ....
1 abelianization ...
2 ?? "vector bundle over projective line assigning to 1d subspace x f(x)#g(v/x) where f and g are functors from _1d f_q-vsp_ to _1d c-vsp_" .... ??? .... ?? then understanding "q-braiding" here and modding out by it .... ??? .... "hecke operator / algebra ..." .... ????? .....
?? abelianization and determinants ???? ....
?? abelianization of gl(n,field) vs of pgl(n,field) ?? .... ??? .....
?? abelianization of gl(n,field) as gl(1,field) ??? ..... ????? ....
?? "algebraic k-theory" ... ???? ...... ?????? ..... ???? ....
?? is "projectivization" always finer than "abelianization" ??? .... ?? meaning ... ?? ... when you abelianize gl(n,field), does it always happen that the constants get killed off ?? ..... ???? .... ?? meaning, maybe (?? ...) that the determinant of a constant is ...... ???? ......
gl(2,f_3) .... ????
?? here it really is true that the constants are determinant 1 ??? ..... ???? ....
?? but that seems like an extremely special case .... ?? ....
?? did i ask the question anywhere close to correctly ?? ... ???? ...
?? maybe we had an original intuition here (?? ...) that was right-track ... ?? that _of course_ determinant wouldn't survive too well to projective level .... ??? .... ??? .....
?? for some reason i feel tempted to say "determinants of constants form an obstruction to determinant surviving to projective level" .... ???? does that make _any_ sense ??? .... ???? ....
?? take a look at 1d irreps of gl(3,f_q) as somewhat classified in recent post here ... ??? ... ??? see if fits with "determinant" idea ??? .... ???? ....
?? yes, does seem to fit fairly straightforwardly ?? ... only 1d irreps that show up seem to be "applying trivial 3-box young diagram to gl(1)-cuspidals" .... ???? .... ??? hmm .... ??? idea that .... ?? abelianization of gl(n) is gl(1) via determinant (?? which idea maybe i'm beginning to remember some subtleties to ... ?? maybe only in beyond-field case ?? ... anyway never mind that for a moment ... ?? ...) as embodied here as "process of applying trivial n-box young diagram to gl(1) irrep" ... ??? .... ???? .... ????? ... ?? suggestive somehow ?? ... ???? .....
?? maybe enough progress here to satisfy original goal of being ready to take a look at what bump says about this stuff ... ?? though not yet running into any obvious way in which kinds of things bump seemed to be about to say ("induced rep from maximal torus rep" ?? ... ??? ...) show up ..... ???? .....
?? size numerology of maximal toruses here ??? ...
?? field = f_q .... ??? but ... ??? trying to see where field = c (or r ....) fit in here .... ??? .....
?? "maximal torus" in gl vs pgl vs sl case .... ???? .....
?? diagonal torus of gl(n) ..... ???? .....
q=2
diagonal torus 1 ... gl(2) 6
?? so 6d induced rep ?? ... ?? but maybe breaks into smaller irreps ??? ...
?? ok, so when bump talks about "induced reps", maybe they're really talking about "corresponding" irreps .... ?? how much does that help to straighten stuff out here ??? ..... ????? .....
?? so one stupid guess that doesn't quite make sense yet is something like ... ?? ... ?? bump is setting aside the cuspidal reps (my first term above ... ?? ....) for the moment (?? ...), and associating my second term with "unsplit torus", and my third term with "split torus" .... ...... ?? really not seeing it yet .... ????? .....
q=3
diagonal torus 4 .... gl(2) 48
??? 12 ..... ???? .... 4*3 ???
q=4
180 / 9 = 20 = 5*4
15*12 = 5*3*4*3 ..... ????? .....
?? "generic flag pair" .... ????? ......
?? vector bundle over [flag variety]^2 ??? .....
??? alleged other sort of "maximal torus" ..... ????? .....
?? splitting of unsplit torus ..... ????? ......
?? trying to associate "vector bundle over [flag variety]^2" with my third term as ... ?? seeming rather iffy and relying on special coincidences ... gl(2) vs gl(n) for more general n .... ???? .... ???? ....
?? mckay correspondence for binary tetrahedral group ..... ???? .... ?? is that e6 ?? .... bipartite graph .... ???? ...
?? having some trouble trying to get mckay correspondence numerology to work here .... ????? ... (actually having trouble at the moment getting it to work anywhere, so don't hold it against here too much .... ?? ...)
?? elt of gl(2,f_q) that .... ?? arises from .... ??? putting structure on [f_q]^2 making it into an f_[q^2] ... ??? and then .... ??? .... ?? picking generator for cyclic group gl(1,f_[q^2]) ... ?? and considering group it generates ... ??? hmmm, i guess that that's pretty much like saying ... ?? consider gl(1,f_[q^2]) -> gl(2,f_q) arising in hopefully obvious way .... ???? ....
?? so one stupid guess is that that's what an "unsplit (?? maximal ... ?? ....) torus" is ... ??
?? "ramified torus" ???? .... ???? gl(1,f_q[t]/t^2) -> gl(2,f_q) .... ?????
?? gl(1,z/p^2) -> ????? ..... ?? analog for q in place of p ??? .... ????? .... ??? ...
?? split torus and (q-1)^2, vs unsplit torus and q^2-1 ??? ..... ?? anything like that show up in our classification yet ??? .... ???? ......
?? hmm, almost getting a numerological glimmer of ... ?? cuspidal : unsplit :: non-cuspidal : split .... ???? ... try testing it out a bit further ... ?? or maybe generically .... ??? .....
(q^2-1)*(q^2-q) = (q-1)^2*(q*(q-1)/2) + (1^2 + q^2)*(q-1) + (q+1)^2*((q-1)*(q-2)/2)
?? the glimmer argument is something like ... ?? ... dividing left-hand side by unsplit torus size gives q^2-q, and factors of that get squared in first term ... ?? more specifically, the factor q-1 does .... ??? whereas dividing left-hand side by split torus size gives q^2+q, and factors of that get squared in last two terms .... ??? .... ?? so _is_ this a good hint of what bump's talking about ??? .... ???? .....
?? well, bump does say this :
The representations parametrized by maximal split tori are induced representations, those parametrized by nonsplit tori must be constructed by some other method.
... so yeah, it seems like we're on the right track here .... though in retrospect maybe it should have been somewhat obvious ... ?? ...
?? counting characters here .... ??? also trying to develop correspondences with real and complex cases .... ???? ....
?? not quite getting the non-squared numbers to correspond nicely to numbers of characters yet ..... ??? ..... hmmm, but ... ?? might be possibilities .... ??? "kaleidoscope folding" ?? .... ??? ....... ??? ..... ?? or sub rather than quotient ?? .... ?? "... chamber ..." ... ???? .... ??? hmmm, try adding together the raw counts for the second and third terms .... ??? .... ?? does seem to have some relationship to "(q-1)^2", while raw count for first term seems to relate to "q^2-1" .... ???? ..... ???? .....
?? wait a minute ... ?? might be right track, but some confusion here ... ?? between "parameterization by" as given by inducement (? which sort of seemed to work numerologically ... ?? ...), vs bit about those parameterized by nonsplit toruses as "_not_ induced" .... ???? .....
?? trying to figure out new plan now that ksu plan seems to have fallen through ...
?? stony brook ?? ...
?? baez / ucr ?? ... seems like real bad idea ....
?? huerta ??? ....
?? robin cockett ?? ....
?? categories mailing list .... ??? other vaguely similar ideas about possible places to "advertise" ... ??? .....
?? stony brook ?? ...
?? baez / ucr ?? ... seems like real bad idea ....
?? huerta ??? ....
?? robin cockett ?? ....
?? categories mailing list .... ??? other vaguely similar ideas about possible places to "advertise" ... ??? .....
Friday, April 27, 2012
6 = 1 + 4 + 1
48 = 1 + 9 + 1 + 9 + ...
28 ... ???? ....
15*12 = (1 + 16)*3 + 129
?? 129 = 3 * 43 .... ????? ....
?? 2d f_3 vsp .... ???
?? 4 1d subspaces ...
3d irrep ...
?? 2 cuspidals of gl(1,f_2) ?? ....
?? green convolution of 2 1d cuspidals ..... ????? .....
?? "take 2d f_3 vsp v, take 1d subspace x, take a-structure on x and b-structure on v/x" .... ????? ....
0
[]
1
[1]
2
[2] [11]
3
[3] [21] [111]
4
[4] [31] [22] [211] [1111]
5
[5] [41] [32] [311] [221] [2111] [11111]
??? gl(2) ....
?? cuspidal ??
?? green convolution of 2 different gl(1) cuspidals ...
?? apply 2-box young diagram (?? either [2] or [11] ?? ...) to gl(1) cuspidal ... ???? ....
c2 ....
c1 X c1 - c1 ... ????
c1 X 2 ... ???? ....
??? hmmmm .... q=2 .... ???? cj = 1 so c1 X c1 - c1 = 0 ??? .... ??? ...
6 = 1^2 + 2^2 + 1^2 .....
48 = 1^2 + 3^2 + 1^2 + 3^2 + 4^2 + 12 .... ???? 12 = (2^2)*3 ??? ....
15*12 = (1^2 + 4^2)*3 + (5^2)*3 + 54 ..... ???? 54 = (3^2)*6 ??? ....
24*20 = (1^2 + 5^2)*4 + (6^2)*6 + 160 .... ???? 160 = (4^2)*10 ??? ....
48*42 = (1^2 + 7^2)*6 + (8^2)*15 + (6^2)*21 ?????? .....
2016 = 300 + 960 + 756
?? so naive guesses seem to be checking out here .... ??? ....
7*6*4 = 168
26*24*18 = 11232
63*60*48 = 181440
?? "monocuspidal" :
gl(3) cuspidal .... ??? ....
gl(1) cuspidal with 3-box young diagram applied ... ??? ...
?? "bicuspidal" :
gl(2) cuspidal green-convolve gl(1) cuspidal
(gl(1) cuspidal with 2-box young diagram applied) green-convolve gl(1) cuspidal
?? "tricuspidal" :
gl(1) cuspidal green-convolve gl(1) cuspidal green-convolve gl(1) cuspidal
q=2 ... ??? ...
168 = 1^2 + 1^2 + ??? ....
?? nothing, point, flag .... ???? .....
?? induced vs irrep ... categorified gram-schmidt .... ???? ....
?? nothing, point - nothing, flag - point*2 + nothing
1, q^2+q+1, (q^2+q+1)*(q+1)
1, q^2+q, (q^2+q+1)*(q-1)+1 = q^3 ??? ....
q=2 ...
1, 6, 8 ???? ....
168 = (1^2) + (1^2 + 6^2 + 8^2) + (7^2)
1 + (1 + 36 + 64) + 49
?? could it be that there are 2 3d gl(3) cuspidals here ???? ..... ????
168 = (3^2 + 3^2) + (1^2 + 6^2 + 8^2) + (7^2)
11232 = (?^2)*?? + (1^2 + 12^2 + 27^2)*2 + ((13*2)^2)*3*2 + (13^2 + 39^2)*1
11232 - (1748 + 4056 + 3380) = 3738 = 2*3*7*89 ?? .... must be arithmetic mistake somewhere .... ???? ....
?? hmmm, so maybe the mistake was ... to neglect that for the case where you use a distinct pair of gl(1) cuspidals, the pair are distinguishable ...... ??? (maybe figure out what i mean by difference between "distinct" and "distinguishable" here sometime ??? ...) ... so that 1690 should be doubled to 3380 ... ?? and now the arithmetic seems a bit more encouraging ??? .... 2048 left over ... = 2^11 .... ??? .....
??? so that could fit with 2 32d gl(3,f_3) cuspidals ?? ... ?? or 8 16d ones ??? ... ?? ... or 32 8d ..... ???? or 128 4d, or 512 2d, or 2048 1d .... ???? .... ?? which seems more plausible offhand ??? ....
2 1d gl(1,f_3) cuspidals ....
3 2d gl(2,f_3) cuspidals ...
8 16d gl(3,f_3) cuspidals ... ???? .....
...
1 1d gl(1,f_2) cuspidal ....
1 1d gl(2,f_2) cuspidal ....
2 3d gl(3,f_2) cuspidals .... ???? ....
...
q-1 1d gl(1,f_q) cuspidals ....
q(q-1)/2 [q-1]d gl(2,f_q) cuspidals .... ????? ...
.... ???? .....
181440 = (?^2)*?? + (1^2 + 20^2 + 64^2)*3 + ((21*3)^2)*6*3 + (21^2 + (21*4)^2)*6 + ((21*5)^2)*1
181440 - 13491 - 71442 - 44982 - 11025 = 40500 = 2^2 * 3^4 * 5^3
?? so .... ???? gl(3,f_4) cuspidals ... ??? maybe 5 90d ??? or 20 45d ?? .... ?? or 45 30d ?? .... or 125 18d ?? .... or 180 15d ?? .... or 405 10d ... or 500 9d ... or 1125 6d ... or 1620 5d .... or 4500 3d ... or 10125 2d ... or 40500 1d ... ???? ......
124*120*100 = (?^2)*?? + (1^2 + 30^2 + 125^2)*4 + ((31*4)^2)*10*4 + (31^2 + (31*5)^2)*12 + ((31*6)^2)*4
1488000 - 66104 - 615040 - 299832 - 138384 = 368640 = 2^13 * 3^2 * 5
342*336*294 - (1^2 + 56^2 + 343^2)*6 - ((57*6)^2)*21*6 - (57^2 + (57*7)^2)*30 - ((57*8)^2)*20 =
33784128 - 724716 - 14737464 - 4873500 - 4158720 = 9289728 = 2^14 * 3^4 * 7
?? some sort of categorified gram-schmidt here ?? .... ???? .... ?? try to get q-polynomials for various sectors here ... ??? ... ?? ...
48 = 1 + 9 + 1 + 9 + ...
28 ... ???? ....
15*12 = (1 + 16)*3 + 129
?? 129 = 3 * 43 .... ????? ....
?? 2d f_3 vsp .... ???
?? 4 1d subspaces ...
3d irrep ...
?? 2 cuspidals of gl(1,f_2) ?? ....
?? green convolution of 2 1d cuspidals ..... ????? .....
?? "take 2d f_3 vsp v, take 1d subspace x, take a-structure on x and b-structure on v/x" .... ????? ....
0
[]
1
[1]
2
[2] [11]
3
[3] [21] [111]
4
[4] [31] [22] [211] [1111]
5
[5] [41] [32] [311] [221] [2111] [11111]
??? gl(2) ....
?? cuspidal ??
?? green convolution of 2 different gl(1) cuspidals ...
?? apply 2-box young diagram (?? either [2] or [11] ?? ...) to gl(1) cuspidal ... ???? ....
c2 ....
c1 X c1 - c1 ... ????
c1 X 2 ... ???? ....
??? hmmmm .... q=2 .... ???? cj = 1 so c1 X c1 - c1 = 0 ??? .... ??? ...
6 = 1^2 + 2^2 + 1^2 .....
48 = 1^2 + 3^2 + 1^2 + 3^2 + 4^2 + 12 .... ???? 12 = (2^2)*3 ??? ....
15*12 = (1^2 + 4^2)*3 + (5^2)*3 + 54 ..... ???? 54 = (3^2)*6 ??? ....
24*20 = (1^2 + 5^2)*4 + (6^2)*6 + 160 .... ???? 160 = (4^2)*10 ??? ....
48*42 = (1^2 + 7^2)*6 + (8^2)*15 + (6^2)*21 ?????? .....
2016 = 300 + 960 + 756
?? so naive guesses seem to be checking out here .... ??? ....
7*6*4 = 168
26*24*18 = 11232
63*60*48 = 181440
?? "monocuspidal" :
gl(3) cuspidal .... ??? ....
gl(1) cuspidal with 3-box young diagram applied ... ??? ...
?? "bicuspidal" :
gl(2) cuspidal green-convolve gl(1) cuspidal
(gl(1) cuspidal with 2-box young diagram applied) green-convolve gl(1) cuspidal
?? "tricuspidal" :
gl(1) cuspidal green-convolve gl(1) cuspidal green-convolve gl(1) cuspidal
q=2 ... ??? ...
168 = 1^2 + 1^2 + ??? ....
?? nothing, point, flag .... ???? .....
?? induced vs irrep ... categorified gram-schmidt .... ???? ....
?? nothing, point - nothing, flag - point*2 + nothing
1, q^2+q+1, (q^2+q+1)*(q+1)
1, q^2+q, (q^2+q+1)*(q-1)+1 = q^3 ??? ....
q=2 ...
1, 6, 8 ???? ....
168 = (1^2) + (1^2 + 6^2 + 8^2) + (7^2)
1 + (1 + 36 + 64) + 49
?? could it be that there are 2 3d gl(3) cuspidals here ???? ..... ????
168 = (3^2 + 3^2) + (1^2 + 6^2 + 8^2) + (7^2)
11232 = (?^2)*?? + (1^2 + 12^2 + 27^2)*2 + ((13*2)^2)*3*2 + (13^2 + 39^2)*1
11232 - (1748 + 4056 + 3380) = 3738 = 2*3*7*89 ?? .... must be arithmetic mistake somewhere .... ???? ....
?? hmmm, so maybe the mistake was ... to neglect that for the case where you use a distinct pair of gl(1) cuspidals, the pair are distinguishable ...... ??? (maybe figure out what i mean by difference between "distinct" and "distinguishable" here sometime ??? ...) ... so that 1690 should be doubled to 3380 ... ?? and now the arithmetic seems a bit more encouraging ??? .... 2048 left over ... = 2^11 .... ??? .....
??? so that could fit with 2 32d gl(3,f_3) cuspidals ?? ... ?? or 8 16d ones ??? ... ?? ... or 32 8d ..... ???? or 128 4d, or 512 2d, or 2048 1d .... ???? .... ?? which seems more plausible offhand ??? ....
2 1d gl(1,f_3) cuspidals ....
3 2d gl(2,f_3) cuspidals ...
8 16d gl(3,f_3) cuspidals ... ???? .....
...
1 1d gl(1,f_2) cuspidal ....
1 1d gl(2,f_2) cuspidal ....
2 3d gl(3,f_2) cuspidals .... ???? ....
...
q-1 1d gl(1,f_q) cuspidals ....
q(q-1)/2 [q-1]d gl(2,f_q) cuspidals .... ????? ...
.... ???? .....
181440 = (?^2)*?? + (1^2 + 20^2 + 64^2)*3 + ((21*3)^2)*6*3 + (21^2 + (21*4)^2)*6 + ((21*5)^2)*1
181440 - 13491 - 71442 - 44982 - 11025 = 40500 = 2^2 * 3^4 * 5^3
?? so .... ???? gl(3,f_4) cuspidals ... ??? maybe 5 90d ??? or 20 45d ?? .... ?? or 45 30d ?? .... or 125 18d ?? .... or 180 15d ?? .... or 405 10d ... or 500 9d ... or 1125 6d ... or 1620 5d .... or 4500 3d ... or 10125 2d ... or 40500 1d ... ???? ......
124*120*100 = (?^2)*?? + (1^2 + 30^2 + 125^2)*4 + ((31*4)^2)*10*4 + (31^2 + (31*5)^2)*12 + ((31*6)^2)*4
1488000 - 66104 - 615040 - 299832 - 138384 = 368640 = 2^13 * 3^2 * 5
342*336*294 - (1^2 + 56^2 + 343^2)*6 - ((57*6)^2)*21*6 - (57^2 + (57*7)^2)*30 - ((57*8)^2)*20 =
33784128 - 724716 - 14737464 - 4873500 - 4158720 = 9289728 = 2^14 * 3^4 * 7
?? some sort of categorified gram-schmidt here ?? .... ???? .... ?? try to get q-polynomials for various sectors here ... ??? ... ?? ...
?? you can substitute equal functions for functions and equal values for values, but the trouble starts when you try to substitute one function for another just because they have equal vlaues at one point ... ??? .... ?? "evaluation at x" as operator vs as conceptual blurring / mistake .... ??? .....
?? "graph" over time of "president of usa" vs of "barack obama", for example ... ?? photographs .... ???? ...
zeno's (non-)moving arrow paradox ...
referential opacity paradox ...
f'(x) = 0 mistake .... ??? .....
?? chain rule ... ???? ....
?? "graph" over time of "president of usa" vs of "barack obama", for example ... ?? photographs .... ???? ...
zeno's (non-)moving arrow paradox ...
referential opacity paradox ...
f'(x) = 0 mistake .... ??? .....
?? chain rule ... ???? ....
Tuesday, April 24, 2012
Sunday, April 22, 2012
Saturday, April 21, 2012
?? "mr a-or-b" vs "mr if-such-and-such-then-a-else-b" .... ??? .... ??? vs "mr if-such-and-such-then-a-or-b-else-c-or-d" ..... ????? .....
?? exponentiation of structure types .... ???? ..... ???? ......
?? plain disjunction vs case-breakdown .... ???? .... possibility of latter dealing with higher tuples better ??? ... ??? ..... correlation between case-breakdowns ... ?? arbib-manes ... monad m ... "m-fuzzy elt of s" .... ?? case m(s) = probability measures(s) (?? or maybe possibility measures is more relevant here ... ?? ...) vs case m(s) = s^k ..... ????? .... ????? ...... ??? "sections of trivial bundle over k" ... ??? ..... monad from adjunction ...
?? exponentiation of structure types .... ???? ..... ???? ......
?? plain disjunction vs case-breakdown .... ???? .... possibility of latter dealing with higher tuples better ??? ... ??? ..... correlation between case-breakdowns ... ?? arbib-manes ... monad m ... "m-fuzzy elt of s" .... ?? case m(s) = probability measures(s) (?? or maybe possibility measures is more relevant here ... ?? ...) vs case m(s) = s^k ..... ????? .... ????? ...... ??? "sections of trivial bundle over k" ... ??? ..... monad from adjunction ...
?? field f .... ??? subfield k .... a1,...,an in f, b in f .... ?? "can b be defined / expressed in terms of stuff from k tw a1,...,an ?" .... ????? .....
?? lagrange extrapolation .... ????? ....
?? some confusion here about ... ?? "eyeglasses" phenomenon ??? .... ambiguity between aj and background elt in k .... ???? ....
?? try to straighten out .... ???? .....
?? c in f(x1,...,xn) .... f(a1,...,an)= b ..... ???
???? ......
?? lagrange extrapolation .... ????? ....
?? some confusion here about ... ?? "eyeglasses" phenomenon ??? .... ambiguity between aj and background elt in k .... ???? ....
?? try to straighten out .... ???? .....
?? c in f(x1,...,xn) .... f(a1,...,an)= b ..... ???
???? ......
Friday, April 20, 2012
?? hmmm, so i think that i'm getting a clearer idea of how particular version of galois's principle game should go ... single-player web app ... wallpaper groups ... superimposed pair of decorations ... clicking (?? ...) where you predict second decoration (?? or maybe part thereof ?? ....) will show up ..... ??? ......
?? to define x in terms of y when x is more symmetric (= more robust ...) than y, just "make sure that the names that you give everything are y-correct, and then describe x naively, exhaustively in terms of those names" ..... ???? ....
?? but this seems very gambit-ish .... ?? so how does it relate to deterministic / constructive / "disjunction" / "lagrange extrapolation" methods ??? ....
?? well, so maybe it's like this : the way that the gambit-ish method becomes deterministic is by conscientiously holding back and not assigning names to things more precisely than you can actually discern the identities of the things ...
?? hmmm, idea here of bohr's commandment as "anti-mule" measure ...... ???? ...... ???? ......
?? so for example, if a/b distinction is indiscernible to you, then instead of randomly assigning name a to one of them and b to the other, you conscientiously refer to either of them as "mr a-or-b" ..... ???? ....
(for some reason this reminds me of peter sellers and keenan wynn ...)
(with complication of higher tuple classes not being just tuples of singleton classes .... ?? "non-exactness" of some sort ??? ..... ????? .....)
(?? vague feeling here about .... "squeezing out flab" ... "decategorification" .... ???? maybe not that vague ..... ???? ..... hmmmm ....... ????? ....... julian barbour ...... ????? ..... "passion" .... ???? .... "desiccated ... shriveled up ..." .... ?? .... mule ..... ???? ....)
?? "disjunction" level slip here ??? .... within equivalence class vs between equivalence class ..... ????? ..... both somewhat relevant .... ???? .... ??? hmmm, different "cases", each defined disjunctively ... ??? so maybe less of a slip than i was thinking a moment ago ..... ???? .... ?? some "feynman path integral / slide rule" phenomenon here ?? .... ??? sum of products vs sum of exponential of sums .... ????? ......
?? deterministicization here as .... ??? "integration wrt haar measure" flavor ??? .... ?? integrating "acceptable labeling" constraint characteristic function over all possible labelings .... ????? ....... ????? .....
?? christian / muslim ....
?? usa / ussr ..... von neumann .... mccarthy ... john that is ... their idea of mathematicians's fondness for symmetry as leading to unhealthy level of support for mad doctrine ... ?? mad doctrine as birthday-cake solution on steroids .... ??? tom-and-jerry ... eating own-tail-sandwich ...
?? actually .... ???? funny _non_-symmetric aspect of birthday-cake solution, yet somehow it really does embody symmetry idea ..... ????? ...... "one being symmetric for the two of them" ..... ????? ..... ?? variant where instead of other chooses, flip a coin ..... ???? "other chooses" variant as sharper in some ways, but .... ???? ..... ???? .....
?? brown / arena / gorn / other .... ??? hunger games ... ?? ...
?? not yet to point of really getting lagrange extrapolation bit to mesh perfectly with bohr's commandment bit ??? ..... ?? bit about lagrange extrapolation not just in "function" case .... ?? exmple that i did for gunnarsen's students ... "number" case .... ???? .... ?? elt f of k[x] as structure on k, related to [certain other sorts of structure on k, namely input data for lagrange extrapolation, yielding f as output] .... ???? .... ??? _is_ that really the way it works ??? .... ??? really try to check more carefully .... ????? .....
?? "find any y-correct frame ..." vs "find all y-correct frames ..." .... ??? .....
?? but this seems very gambit-ish .... ?? so how does it relate to deterministic / constructive / "disjunction" / "lagrange extrapolation" methods ??? ....
?? well, so maybe it's like this : the way that the gambit-ish method becomes deterministic is by conscientiously holding back and not assigning names to things more precisely than you can actually discern the identities of the things ...
?? hmmm, idea here of bohr's commandment as "anti-mule" measure ...... ???? ...... ???? ......
?? so for example, if a/b distinction is indiscernible to you, then instead of randomly assigning name a to one of them and b to the other, you conscientiously refer to either of them as "mr a-or-b" ..... ???? ....
(for some reason this reminds me of peter sellers and keenan wynn ...)
(with complication of higher tuple classes not being just tuples of singleton classes .... ?? "non-exactness" of some sort ??? ..... ????? .....)
(?? vague feeling here about .... "squeezing out flab" ... "decategorification" .... ???? maybe not that vague ..... ???? ..... hmmmm ....... ????? ....... julian barbour ...... ????? ..... "passion" .... ???? .... "desiccated ... shriveled up ..." .... ?? .... mule ..... ???? ....)
?? "disjunction" level slip here ??? .... within equivalence class vs between equivalence class ..... ????? ..... both somewhat relevant .... ???? .... ??? hmmm, different "cases", each defined disjunctively ... ??? so maybe less of a slip than i was thinking a moment ago ..... ???? .... ?? some "feynman path integral / slide rule" phenomenon here ?? .... ??? sum of products vs sum of exponential of sums .... ????? ......
?? deterministicization here as .... ??? "integration wrt haar measure" flavor ??? .... ?? integrating "acceptable labeling" constraint characteristic function over all possible labelings .... ????? ....... ????? .....
?? christian / muslim ....
?? usa / ussr ..... von neumann .... mccarthy ... john that is ... their idea of mathematicians's fondness for symmetry as leading to unhealthy level of support for mad doctrine ... ?? mad doctrine as birthday-cake solution on steroids .... ??? tom-and-jerry ... eating own-tail-sandwich ...
?? actually .... ???? funny _non_-symmetric aspect of birthday-cake solution, yet somehow it really does embody symmetry idea ..... ????? ...... "one being symmetric for the two of them" ..... ????? ..... ?? variant where instead of other chooses, flip a coin ..... ???? "other chooses" variant as sharper in some ways, but .... ???? ..... ???? .....
?? brown / arena / gorn / other .... ??? hunger games ... ?? ...
?? not yet to point of really getting lagrange extrapolation bit to mesh perfectly with bohr's commandment bit ??? ..... ?? bit about lagrange extrapolation not just in "function" case .... ?? exmple that i did for gunnarsen's students ... "number" case .... ???? .... ?? elt f of k[x] as structure on k, related to [certain other sorts of structure on k, namely input data for lagrange extrapolation, yielding f as output] .... ???? .... ??? _is_ that really the way it works ??? .... ??? really try to check more carefully .... ????? .....
?? "find any y-correct frame ..." vs "find all y-correct frames ..." .... ??? .....
Thursday, April 19, 2012
?? level slip for category theorist learning alg geom ... comm ring as (giving) theory of doctrine, rather than alg of theory ... ?? though also funny duality flip here ... ?? interpretation as theory where models are modules ... ?? and formulas are .... ??? also modules (?? whole bit about ... doctrine as cateogrified lex theory ... lex theory as having "model-formula confusion" of sort .... ??? ...), or perhaps free modules ??? ..... ??? or finitary such ..... ???? ...... ????? .....
?? noncomm rings here .... ?? comm rings as monoidal such ..... ???? ...
?? model vs module vs model vs moduli .... ???? ......
?? noncomm rings here .... ?? comm rings as monoidal such ..... ???? ...
?? model vs module vs model vs moduli .... ???? ......
?? "composition algebra" ... ??? multiplication by unit length element as isometry .... ???? something like closed sym mon cat of qfs's, with unit ball or its boundary as forgetful fr .... ??? ....
?? ... metric soace as enriched cat ... ???
??? .....
?? bracketing with element as infinitesimal isometry .... ????? .... antisymmetry and "associativity between bracketing and dot product" .... ???? .....
... talking with huerta about some of this stuff .... ?? ....
?? ... "banach ..." .... ?? ...
?? ... "...-star ..." ??? ...
?? ... metric soace as enriched cat ... ???
??? .....
?? bracketing with element as infinitesimal isometry .... ????? .... antisymmetry and "associativity between bracketing and dot product" .... ???? .....
... talking with huerta about some of this stuff .... ?? ....
?? ... "banach ..." .... ?? ...
?? ... "...-star ..." ??? ...
Wednesday, April 18, 2012
?? game involving .... ??cooperation between two players .... ?? one sees two decorations, the other sees only one of them, and the one who sees both tries to tell the other how to add in the second decoration .... ??? ....
?? how tricky is it to work out good rules here ??? .... ??? ...
?? project ozma .... ??? ....
?? i still want the "social" flavor too though ....
?? how tricky is it to work out good rules here ??? .... ??? ...
?? project ozma .... ??? ....
?? i still want the "social" flavor too though ....
?? "quotient rule" ... ??
?? logarithmic derivative of f/g as f'/f - g'/g .... ??? then multiply by f/g ??? ....
f'/g - fg'/g^2 ... ??? ....
?? product rule .... ?? ...
logarithmic derivative of fg as f'/f + g'/g ; then multiply by fg ...
?? row of ratios ...... ??????? .....
?? f'/f - 1 .... ???? .....
(f'-f)/f ????? .....
?? confusion here .... ?? .... f'/f + 1 .... (f+f')/f ...... ?????? ..... ??? ....
?? derivative of logarithm vs exponential of derivstive ....... ????? ....... ????? ...... ?? exponential of derivative as seeming somewhat silly ??? .....
??? exponential of logarithmic derivative ??? ......
??? still feels like there's an idea here somewhere that i'm groping for ... ???? .....
???? "row of ratios" .... ?? [row of differences] of logarithm .... ??? .....
?? logarithmic derivative of f/g as f'/f - g'/g .... ??? then multiply by f/g ??? ....
f'/g - fg'/g^2 ... ??? ....
?? product rule .... ?? ...
logarithmic derivative of fg as f'/f + g'/g ; then multiply by fg ...
?? row of ratios ...... ??????? .....
?? f'/f - 1 .... ???? .....
(f'-f)/f ????? .....
?? confusion here .... ?? .... f'/f + 1 .... (f+f')/f ...... ?????? ..... ??? ....
?? derivative of logarithm vs exponential of derivstive ....... ????? ....... ????? ...... ?? exponential of derivative as seeming somewhat silly ??? .....
??? exponential of logarithmic derivative ??? ......
??? still feels like there's an idea here somewhere that i'm groping for ... ???? .....
???? "row of ratios" .... ?? [row of differences] of logarithm .... ??? .....
Tuesday, April 17, 2012
?? try naive "explicit" approach to "green substitution" ?? ...
?? q-analog of "take set, split into pieces, put structure of type t1 on each piece, of type t2 on set of pieces" ..... ????? ....
?? "take f_q-vsp, put partial flag on it, put type t1 structure on each grade, and put type t2 structure on ?????? of grades" ??????????? ...... ??????? .......
?? "parabolic induction" .... ???? ....
?? q-analog of "take set, split into pieces, put structure of type t1 on each piece, of type t2 on set of pieces" ..... ????? ....
?? "take f_q-vsp, put partial flag on it, put type t1 structure on each grade, and put type t2 structure on ?????? of grades" ??????????? ...... ??????? .......
?? "parabolic induction" .... ???? ....
?? "baez" level slip .... more rigid structure vs larger subspace of point space .... ??? ..... ??? relationship to vsp vs comm ring / field ??? .... ???? .....
?? ag theory of line object l with embedding into k^n and (say ...) comm ring structure on cokernel ... ??? is there a nice explicit gca here ?? ... ??? .....
?? doctrine / theory interpretation of exponentiation of "schemes" / "stacks" .... ???? .....
?? idea that .... ??? if "everything's homotopy-/derived-flat" (?? ...) and flatness promotes exponentiableness, then ... ?? everything (maybe especially with some "finiteness" ??? .... ??? ...) tends to be homotopy-/derived-exponentiable ??? .... ???? .... ?? ....
?? ag theory of line object l with embedding into k^n and (say ...) comm ring structure on cokernel ... ??? is there a nice explicit gca here ?? ... ??? .....
?? doctrine / theory interpretation of exponentiation of "schemes" / "stacks" .... ???? .....
?? idea that .... ??? if "everything's homotopy-/derived-flat" (?? ...) and flatness promotes exponentiableness, then ... ?? everything (maybe especially with some "finiteness" ??? .... ??? ...) tends to be homotopy-/derived-exponentiable ??? .... ???? .... ?? ....
?? substitution as application of universal property of free on one (object ...) generator ... ?? green substitution as application of universal property of free on one (?? q-braided object ??? .... ??? .....) generator .... ????? ..... ????? .....
?? getting feeling here of .... ??? hard to believe _really_ universal here .... ???? .... ?? need to check ... ???? ...
?? prasad ... ?? subring as contravariant algebraic analog of quotient space ... kaleidoscope-invariant polynomials ..... ??? ... various conceptual roles thereof ????? ...... ???? (?? somewhat ???) canonical free generating set here ?? ... ??? hazy memories .... ?? "casimir ..." ???? .... ????? ......
?? characteristic and normal subgroups of absolute galois groups ??? .... ???? ..... ?? hmm, some silly duality flipping here ... forgetting basic contravariance of galois correspondence for a bit .... ???? .....
?? rep of gl(n,k1) over k2 ... ??? "field-jumping functor" .... ???? .... ??case where ... ?? some part of representation theory is somewhat k1-independent ... ??? .... ?? overlap with case of "tautological rep" ??? ... ???? ... ???? .....
?? getting feeling here of .... ??? hard to believe _really_ universal here .... ???? .... ?? need to check ... ???? ...
?? prasad ... ?? subring as contravariant algebraic analog of quotient space ... kaleidoscope-invariant polynomials ..... ??? ... various conceptual roles thereof ????? ...... ???? (?? somewhat ???) canonical free generating set here ?? ... ??? hazy memories .... ?? "casimir ..." ???? .... ????? ......
?? characteristic and normal subgroups of absolute galois groups ??? .... ???? ..... ?? hmm, some silly duality flipping here ... forgetting basic contravariance of galois correspondence for a bit .... ???? .....
?? rep of gl(n,k1) over k2 ... ??? "field-jumping functor" .... ???? .... ??case where ... ?? some part of representation theory is somewhat k1-independent ... ??? .... ?? overlap with case of "tautological rep" ??? ... ???? ... ???? .....
Monday, April 16, 2012
?? "stuff (?? ...) on ec definable purely in terms of knwoing it's an ec" ... ??? quasicoherent aheaves here ... over moduli stack of ec's .... ?? ...
?? also ... ??? definable in terms of some bit more .... ??? ... ?? such possibilities as maybe themselves showing up in original case .... ??? .... level slip ... ??? ... ??? ....
?? doctrine ... logic ... gp theory .... ???? ....
?? also ... ??? definable in terms of some bit more .... ??? ... ?? such possibilities as maybe themselves showing up in original case .... ??? .... level slip ... ??? ... ??? ....
?? doctrine ... logic ... gp theory .... ???? ....
?? try to refresh my understanding of classification of irreps of for example pgl(3,f_2) and / or gl(3,f_2) .... ??? partly in preparation for reading bump's stuff about gl(2,f_q) .... ???? ....
?? so ... ?? also thinking a bit here about "pgl(3,f_1)" aka 3! ... ?? whose irreps are classified by :
...
..
.
.
.
.
?? which also classify partial flag types here .... ???? ....
?? hmmm ..... ??? ....
point, line
point
line
-
?? 2 middle ones as equivalent in some sense ??? .... ??? ....
?? "categorified gram-schmidt" ... ???? .... ???? but .... ???? ..... ?? confused about how we're supposed to deal with the two "equivalent" ones here .... ???? ....
?? "combed vs uncombed young diagram" .... ???? .....
?? q=1 case .... ????? .... ?? as encouraging not worrying too much about this distinction ??? .... ??? stuff that's q-independent .... ???? .....
?? "class class" .... ???? .....
?? flag variety / conjugacy class .... ????? .....
?? "green type" .... ???? ....
?? "schur-weyl duality" .... ???? ..... ?? q-deformed .... ??? ....
------3---21--12--111
3 1 1 1 1
21 1 2 2 3
12 1 2 2 3
111 1 3 3 6
?? ....
?? parabolic induction and schur-weyl duality .... ????
?? young diagram as functorial operation .... "schur functor" ..... ???? ......
?? induced vs irrep here ..... ???? ....
?? applying schur functor to ..... ???? .....
?????? .......
?? "green substitution" .... ?? ...
?? "cuspidal rep" ... ???? what in world relationship if any to "cusp form" .... ???? .... ???? .... ???? ....
?? "applying a-box young diagram to b-box one to get [a*b]-box one ...." ??? .... ??? confusion about q-deformed case ... where q can / does enter .... ??? ...
?? so ... ?? also thinking a bit here about "pgl(3,f_1)" aka 3! ... ?? whose irreps are classified by :
...
..
.
.
.
.
?? which also classify partial flag types here .... ???? ....
?? hmmm ..... ??? ....
point, line
point
line
-
?? 2 middle ones as equivalent in some sense ??? .... ??? ....
?? "categorified gram-schmidt" ... ???? .... ???? but .... ???? ..... ?? confused about how we're supposed to deal with the two "equivalent" ones here .... ???? ....
?? "combed vs uncombed young diagram" .... ???? .....
?? q=1 case .... ????? .... ?? as encouraging not worrying too much about this distinction ??? .... ??? stuff that's q-independent .... ???? .....
?? "class class" .... ???? .....
?? flag variety / conjugacy class .... ????? .....
?? "green type" .... ???? ....
?? "schur-weyl duality" .... ???? ..... ?? q-deformed .... ??? ....
------3---21--12--111
3 1 1 1 1
21 1 2 2 3
12 1 2 2 3
111 1 3 3 6
?? ....
?? parabolic induction and schur-weyl duality .... ????
?? young diagram as functorial operation .... "schur functor" ..... ???? ......
?? induced vs irrep here ..... ???? ....
?? applying schur functor to ..... ???? .....
?????? .......
?? "green substitution" .... ?? ...
?? "cuspidal rep" ... ???? what in world relationship if any to "cusp form" .... ???? .... ???? .... ???? ....
?? "applying a-box young diagram to b-box one to get [a*b]-box one ...." ??? .... ??? confusion about q-deformed case ... where q can / does enter .... ??? ...
Sunday, April 15, 2012
?? so ... gaussian integers .... hom its spec into .... ??? hom of spec(f_3) into spec(z[x]) ???? .... ???? hmmm, is that really what we mean ??? ..... ??? extreme non-flatness of f_3 here ??? .... ???? .... ??? nevertheless suggestive ??? .... ???? ...... ?? to what extent did we notice (?? ...) this before ?? .... ??? ... ??well, ... ?? we did think about spec(f_p) -> spec( .... ???? .... ?? actually, some confusion / level slip here ... ?? maybe sort of good though ... ??? ....
?? we thought about spec(f_p) -> spec(k^3)^spec(k[i]) .... ???? ..... ?? k=c ?? ... ??? .... ?? but maybe we didn't think (?? or at least, not in this context ?? ... ??? ...) about trying to interpret spec(k^3) here ("3" actually some sort of variable .... ?? ....) as itself being something like spec(k[x])^spec(f_3) ..... ????? ..... ???? any particularly bad level slip here ??? ... ??? ... ?? "if f_3 were flat, then ...." .... ??? ....
?? was going to return to idea of trying to "direct-sum" decompose stable hopf alg here ..... ??? .... ?? ideas about components that should (?? ...) turn up .... ??? ...
?? we thought about spec(f_p) -> spec(k^3)^spec(k[i]) .... ???? ..... ?? k=c ?? ... ??? .... ?? but maybe we didn't think (?? or at least, not in this context ?? ... ??? ...) about trying to interpret spec(k^3) here ("3" actually some sort of variable .... ?? ....) as itself being something like spec(k[x])^spec(f_3) ..... ????? ..... ???? any particularly bad level slip here ??? ... ??? ... ?? "if f_3 were flat, then ...." .... ??? ....
?? was going to return to idea of trying to "direct-sum" decompose stable hopf alg here ..... ??? .... ?? ideas about components that should (?? ...) turn up .... ??? ...
??? so ... vague memory of paper giving "humble adjunction ... homming of just plain categories ..." as extreme special case of "structure / semantics adjunction" ... (?? maybe gray or someone ??? ....) ... ?? i guess now that that was all about "single-universe semantics" ... formula category vs model category ....
_cat_^op -> _cat_
x |-> _set_^x
?? "flip level slips" here ??? .... ??? ....
?? anyway, "homming into k (?? mainly k=_set_ ??) as contravariantly self-adjoint" ??? ....
?? then .... require formula category be c-complete for some class c of diagram schemes .... ???? .... ?? what should happen to model category then ???? ..... ????? ....
?? not really sure i'm that close yet to what they wrote in that paper .... ??? ....
?? maybe try working out "products" case to get more of an idea .... ??? ....
?? _should_ colimits show up here too ?? ... ??? ... i meant in formula cat, along with limits, but then also, model cat .... ???? ...
?? multi-universe semantics .... ??? "naturality" .... ???? ....
??? "underlying set of model" .... ???? .... "concrete operation" .... ??? "naturality" .... ???? ..... "structure theory of object in functor category" ..... ?????? ..... ???? .... monad .... ???? ..... ?? where monad wants to "be interpreted" / "algebras" ..... ???? .....
?? "threshold where model cat and formula cat look same" .... ??? .... ???? .... ??? "contravariance" here ?? ... ??? .....
_cat_^op -> _cat_
x |-> _set_^x
?? "flip level slips" here ??? .... ??? ....
?? anyway, "homming into k (?? mainly k=_set_ ??) as contravariantly self-adjoint" ??? ....
?? then .... require formula category be c-complete for some class c of diagram schemes .... ???? .... ?? what should happen to model category then ???? ..... ????? ....
?? not really sure i'm that close yet to what they wrote in that paper .... ??? ....
?? maybe try working out "products" case to get more of an idea .... ??? ....
?? _should_ colimits show up here too ?? ... ??? ... i meant in formula cat, along with limits, but then also, model cat .... ???? ...
?? multi-universe semantics .... ??? "naturality" .... ???? ....
??? "underlying set of model" .... ???? .... "concrete operation" .... ??? "naturality" .... ???? ..... "structure theory of object in functor category" ..... ?????? ..... ???? .... monad .... ???? ..... ?? where monad wants to "be interpreted" / "algebras" ..... ???? .....
?? "threshold where model cat and formula cat look same" .... ??? .... ???? .... ??? "contravariance" here ?? ... ??? .....
?? bit about ... exposure to striking examples of left universal properties of algebraically structured categories, then realizing that much simpler (?? but as a whole also ultimately striking in their own way ... ?? ...) but somewhat analogous examples form secret subtext of alg geom ... ?? is "simpler" really right idea here ?? ... ??? maybe "tradeoff" ??? .... those original (...) striking examples that i was exposed to tended to be "rich at doctrine level but poor at theory level" ... ?? some other examples vice versa .... "poor" not really pejorative here; very "poor" doctrine as allowing its theories to be interpreted in great generality .... ??? .... ?? and somehow encouraging (??? ...) development of great variety of theories, some complicated .... products doctrine, limits doctrine, limits and colimits doctrine .... ??? .... (?? not to overdo it though ... ??? ... greater expressiveness of limits plus colimits over just limits, for example, as ... ?? also somehow encouraging development .... ??? .... ???? ....)
?? so maybe idea should be to mention both some "rich doctrine poor theory" and some "poor doctrine rich theory" examples (?? explicitly noting ocntrast to some extent ?? ....) as prelude to ag situation ..... ???? .....
?? so maybe idea should be to mention both some "rich doctrine poor theory" and some "poor doctrine rich theory" examples (?? explicitly noting ocntrast to some extent ?? ....) as prelude to ag situation ..... ???? .....
?? diamond and shurman make it sound like that "system of eigenvalues ... (??of hecke operators ??? ... ???? ...)" stuff is supposed to be some (ill-explained ...) general prescription for getting l-series (?? ...) from "automorphic representation", somehow specializing in certain (?? "modular" ??? ....) case to ... ?? fourier series of modular form ??? .... ???? ...
??? "fourier series of automorphic form" ???? ..... ???? .... ??? .... ??? higher dimension .... ??? ....
??? "fourier series of automorphic form" ???? ..... ???? .... ??? .... ??? higher dimension .... ??? ....
?? confusion about ... ??? putting k-vsp structure on g-set as one component of perm rep over k .... ???? maybe extremely k-dependent .... ??? ..... ??? free / underlying adjunction here ...... ???? ......
??? vs "constant part" ... ????? hmmm, in classical quadratic reciprocity case these two seem very opposite ..... ????? .... no wait .... 2 vs 3 here ... z/2 as mult gp of z/3 .... ??????? ...... ?????? ..... confusion ....... ???????? ..... ?????? .....
?? multiple k-vsp structures on same g-set ... ?? zero must always be a fixed point .... ??? ...
?? "l-adic" ....
?? showing up in connection with jugendtraum .... ???? ....p-torsion ...
?? "local" langlands .... ??? .... ?? galois rep over local field, vs "euler factor" ... ?? galois rep for galois gp _of_ local field .... ????? ..... ???? ... ??? .....
?? "geometric langlands" .... ??? ......
??? vs "constant part" ... ????? hmmm, in classical quadratic reciprocity case these two seem very opposite ..... ????? .... no wait .... 2 vs 3 here ... z/2 as mult gp of z/3 .... ??????? ...... ?????? ..... confusion ....... ???????? ..... ?????? .....
?? multiple k-vsp structures on same g-set ... ?? zero must always be a fixed point .... ??? ...
?? "l-adic" ....
?? showing up in connection with jugendtraum .... ???? ....p-torsion ...
?? "local" langlands .... ??? .... ?? galois rep over local field, vs "euler factor" ... ?? galois rep for galois gp _of_ local field .... ????? ..... ???? ... ??? .....
?? "geometric langlands" .... ??? ......
?? "homogenizing variable" .... ???? .....
??? presentation ... ?? weighting of the generators .... ???? giving line bundle ???? ...... ???? any divisor lurking here ??? ..... ??? .... ?? divisor of the homogeneizing variable ?? ....
?? change of weighting here ??? .... ?? effect on line bundle ?? ....
?? guaranteed line bundles here ... ??? ....
?? ever get trivial bundle this way ??? .... ??? ...
?? completion .... ???? .....
??? total presentation ???? .... ????? ..... ???
??? presentation ... ?? weighting of the generators .... ???? giving line bundle ???? ...... ???? any divisor lurking here ??? ..... ??? .... ?? divisor of the homogeneizing variable ?? ....
?? change of weighting here ??? .... ?? effect on line bundle ?? ....
?? guaranteed line bundles here ... ??? ....
?? ever get trivial bundle this way ??? .... ??? ...
?? completion .... ???? .....
??? total presentation ???? .... ????? ..... ???
Saturday, April 14, 2012
?? whether stuff that we've been playing around with recently is more like "local" langlands conjecture than "global" ?? ... ??? "abelian" case ?? ... ???? .....
?? extent to which when thinking about "quasicoherent sheaves" as adjoint to "spectrum" we tried to think of it as a special case of syntax / structure as adjoint to semantics .... ??? .... ??? ....
?? decategorified analog ??? .... "spectrum" ... ???? .....
?? invertiblization of cusp forms .... ??????? ....
?? object (?? .... ??? ....) in ag theory of cqa's given by "p-torsion elements" .... ????? ..... ???? .... ????? ...... ?? p=2 test case ... ??? ....
?? extent to which when thinking about "quasicoherent sheaves" as adjoint to "spectrum" we tried to think of it as a special case of syntax / structure as adjoint to semantics .... ??? .... ??? ....
?? decategorified analog ??? .... "spectrum" ... ???? .....
?? invertiblization of cusp forms .... ??????? ....
?? object (?? .... ??? ....) in ag theory of cqa's given by "p-torsion elements" .... ????? ..... ???? .... ????? ...... ?? p=2 test case ... ??? ....
Friday, April 13, 2012
?? "polynomial with coefficients in sections of anti-tautological line bundle over p^1 ...... ????? .....
?? by degree ....
??? 0 ....
constant ....
degenerate ... no roots ... "inconsistent" ... 0=1 .... all fibers empty .... ??? ....
1
?? c + [ax+by]z = 0 ... ... ?? don't seem to have "homogeneous" aspect straight here yet .... ??? .....
?? variable / unknown / new generator z in grade 1 ... with x and y ..... ????? .....
?? equation to live in grade n ... ?? say n=3 for now .... ??? ....
?? coefficient of z^0 should live in grade n ??
?? coefficient of z should live in grade n-1 ??
...... ???? ......
az + bx+cy = 0
az^2 +[bx+cy]z + [dx^2+exy+fy^2] = 0 .... ???? .....
??level slip here ... ?? ... concerning whether we're thinking of p^1 as analogous to moduli stack of ec's (?? and contemplating adding extra structure to the ec's ... ?? probably more or less what we thought we had in mind when we started here ?? ...), or as analogous to particular ec .... (????? case where that particular one is the generic one ??? .... ????? ......) .... (and contemplating it being double cover of (??? the / a ????? .... ???? ....) p^1 ... (?? identity analogy of p^1 hsowing up here ??? ....) ... ??? or also, now that i think about it, contemplating "hecke-flavored" covering maps to / from other ec's .... ???? ..... ?? connected of course with extra structure mentioned in other parenthetical remark above ... ?? lots of level-slipping here ... ???? ....) .... ??? .....
?? ag theory of line object l equipped with g2 : l^2 -> 1 and g3 : l^3 -> 1 ....
?? l as "generic cqa" ... more or less ...
?? other objects and so forth in here ... connected with fleshing out ec corresponding to cqa .....
?? ag theory of line object l equipped with g2 : l^2 -> 1, g3 : l^3 -> 1, x,y : 1 -> 1, y^2 = x^3 + g2*x + g3
?? ....
?? this theory as "ag propositional" relative to the first one ??? ... ?? and thus ??? ... ?? corresponding to commutative monoid in the first one ?? ... ?? hopefully in evident way .... ???? ....
??? intermediate ag theory here ... ??? .... ag propositional relative to first, second ag propositional relative to it in turn ... ??? ....
line object l, g2 : l^2 -> 1, g3 : l^3 -> 1, x : 1 -> 1
?? corresponding to hopefully evident comm monoid object in first theory .... ??? hmm, evidently really external comm monoid, manifestly .... ???? ..... (?? conflict / confusion here ??? ... ??? .... ?? maybe not too bad ?? ... ??? how much did we used to know about for example taking x as covering projection instead of y ?? .... ???? usual special "2" confusion about "taking y ..." as equivalent to "forgetting y ..." and thus to "focusing on x ..." .... ????? ....)
?? ag theory of "commutative algebra of polynomials in x" .... ??? ..... ?? trying to express property that "spectrum is discrete 2 except at 4 special points ..." .... ???? ..... ???? .... ?? i was going to say that that "externality" conflict and confusion that i mentioned was happening here, but ... ??? maybe it's really ok ??? ... ?? via ... ?? y ??? .... ??? ... well, nevertheless i'm confused ... but ... ?? somewhat hopeful of getting unconfused .... ??? ...
?? plain old (pointed ...) ec as .... lying naturally over ... not _the_ p^1, but _a_ p^1, right ??? ..... here's some confusion .... ???????????????? ....... ?? hmm, so maybe we forgot about the "intermediateness" here ... ?? we have _x_ in the theory, right ??? ...... ?? but still confusion about "x as variable vs as constsnt" .... ???? ...... ?? seems like sort of both ??? ..... ?? maybe bit about "some variables as more variable than others (aka parameters ...)" ... ?? ... ?? relationship to "intermediateness" here ??? .... ??? ..... ???? .... still confusion .... ??? ..... ????? ......
?? if straighten this out should try to work "h" and "theta function" into this ...
?? idea of program to relate structure on ec to structure on cqa as maybe becoming "tautological" (... ?? or at least ... make some progress ... ??? ...) if things get straightened out here .... ???
??? idea of actually thinking of complex p^1 as real s^2, and trying to organize theory (?? ...) this way .... ???? always use same 4 branch points and same covering torus, but vary conformal structure on 2-sphere ... and hence on torus, by pulling back ... ??? hmmm, "how much" do you have to vary the conformal structure on the 2-sphere here to get the variety (?? ....) you need ???? .... ???? ..... ?? "ellipsoidal" structures seem not to give enough variety because ... hmm, actually not sure about that ... ?? confusing / confusion here ??? ....
?? "logic programming" ... ???? ....
?? "ag theory programming" ... ??? ....
?? endo-span of p^1 given by ec .... ????? ..... ?? possibility of intuitive geometric description ... ??? "for most points, particular pair of other points ... except in 4 special cases the pair degenerates ..." ... ??? .... ?? not much symmetry to exploit though .... ??? .....
?? "branched" (??? ....) double cover of real p^1 with branch points at 0, 1, -1, infinity .... ???? .... ?? hmmm .... ??? "branching" manifesting as "zig-zagging" here ??? ..... ?? or maybe not ??? .... equator ?? 2 completely separate "perfectly formed" equators .... ??? ... perhaps right the first time ... trying to remember the picture i sketched the other day ... ok, branch point seems to manifest as "4-valent vertex" .... ???? .... hmmmm ..... ???? .... "rectangle boundary with 2 side edges for each one original" ... ?? eisenstein case ??? .... ?? ??? real coefficients vs real roots here ????? ...... ???? .....
?? hmmm, confusion between .... ?? real elliptic curve, and part of complex elliptic curve lying over real projective line under projection to complex projective line ... ?? "meridianal equators vs longitudinal equators" ... ??? real vs imaginary square roots of real quantities .... ?? definitely still some confusion here, but seems like not too far from straightening it out .... ??? .... ?? x^3-x as positive from 1 to infinity, and from -1 to 0 ..... ??????? extent to which this resolves some old confusions we had about ... ?? knowing that real ec has to be ab gp but thinking that it looked like it had singularities ... ??? ... ??? ....
?? gauss square ... ??? eisenstein rhombus ??? .... ???? ??? other rhombuses though .... other rectangles besides square ........ ?????.... ?? sides of modular triangle ... ??? ..... ?? what's the third side ??? ... ??? eisenstein and "tate" ... ??? ... ?? particular visual commonality between the parallelograms (??? ...) doesn't jump out at me .... ????? ..... ?? not sure how much i might have things screwed up here .... ???? .....
??? modularity and buckyballs .... ????? .... ??? simplex-based .... ???? ....
?? by degree ....
??? 0 ....
constant ....
degenerate ... no roots ... "inconsistent" ... 0=1 .... all fibers empty .... ??? ....
1
?? c + [ax+by]z = 0 ... ... ?? don't seem to have "homogeneous" aspect straight here yet .... ??? .....
?? variable / unknown / new generator z in grade 1 ... with x and y ..... ????? .....
?? equation to live in grade n ... ?? say n=3 for now .... ??? ....
?? coefficient of z^0 should live in grade n ??
?? coefficient of z should live in grade n-1 ??
...... ???? ......
az + bx+cy = 0
az^2 +[bx+cy]z + [dx^2+exy+fy^2] = 0 .... ???? .....
??level slip here ... ?? ... concerning whether we're thinking of p^1 as analogous to moduli stack of ec's (?? and contemplating adding extra structure to the ec's ... ?? probably more or less what we thought we had in mind when we started here ?? ...), or as analogous to particular ec .... (????? case where that particular one is the generic one ??? .... ????? ......) .... (and contemplating it being double cover of (??? the / a ????? .... ???? ....) p^1 ... (?? identity analogy of p^1 hsowing up here ??? ....) ... ??? or also, now that i think about it, contemplating "hecke-flavored" covering maps to / from other ec's .... ???? ..... ?? connected of course with extra structure mentioned in other parenthetical remark above ... ?? lots of level-slipping here ... ???? ....) .... ??? .....
?? ag theory of line object l equipped with g2 : l^2 -> 1 and g3 : l^3 -> 1 ....
?? l as "generic cqa" ... more or less ...
?? other objects and so forth in here ... connected with fleshing out ec corresponding to cqa .....
?? ag theory of line object l equipped with g2 : l^2 -> 1, g3 : l^3 -> 1, x,y : 1 -> 1, y^2 = x^3 + g2*x + g3
?? ....
?? this theory as "ag propositional" relative to the first one ??? ... ?? and thus ??? ... ?? corresponding to commutative monoid in the first one ?? ... ?? hopefully in evident way .... ???? ....
??? intermediate ag theory here ... ??? .... ag propositional relative to first, second ag propositional relative to it in turn ... ??? ....
line object l, g2 : l^2 -> 1, g3 : l^3 -> 1, x : 1 -> 1
?? corresponding to hopefully evident comm monoid object in first theory .... ??? hmm, evidently really external comm monoid, manifestly .... ???? ..... (?? conflict / confusion here ??? ... ??? .... ?? maybe not too bad ?? ... ??? how much did we used to know about for example taking x as covering projection instead of y ?? .... ???? usual special "2" confusion about "taking y ..." as equivalent to "forgetting y ..." and thus to "focusing on x ..." .... ????? ....)
?? ag theory of "commutative algebra of polynomials in x" .... ??? ..... ?? trying to express property that "spectrum is discrete 2 except at 4 special points ..." .... ???? ..... ???? .... ?? i was going to say that that "externality" conflict and confusion that i mentioned was happening here, but ... ??? maybe it's really ok ??? ... ?? via ... ?? y ??? .... ??? ... well, nevertheless i'm confused ... but ... ?? somewhat hopeful of getting unconfused .... ??? ...
?? plain old (pointed ...) ec as .... lying naturally over ... not _the_ p^1, but _a_ p^1, right ??? ..... here's some confusion .... ???????????????? ....... ?? hmm, so maybe we forgot about the "intermediateness" here ... ?? we have _x_ in the theory, right ??? ...... ?? but still confusion about "x as variable vs as constsnt" .... ???? ...... ?? seems like sort of both ??? ..... ?? maybe bit about "some variables as more variable than others (aka parameters ...)" ... ?? ... ?? relationship to "intermediateness" here ??? .... ??? ..... ???? .... still confusion .... ??? ..... ????? ......
?? if straighten this out should try to work "h" and "theta function" into this ...
?? idea of program to relate structure on ec to structure on cqa as maybe becoming "tautological" (... ?? or at least ... make some progress ... ??? ...) if things get straightened out here .... ???
??? idea of actually thinking of complex p^1 as real s^2, and trying to organize theory (?? ...) this way .... ???? always use same 4 branch points and same covering torus, but vary conformal structure on 2-sphere ... and hence on torus, by pulling back ... ??? hmmm, "how much" do you have to vary the conformal structure on the 2-sphere here to get the variety (?? ....) you need ???? .... ???? ..... ?? "ellipsoidal" structures seem not to give enough variety because ... hmm, actually not sure about that ... ?? confusing / confusion here ??? ....
?? "logic programming" ... ???? ....
?? "ag theory programming" ... ??? ....
?? endo-span of p^1 given by ec .... ????? ..... ?? possibility of intuitive geometric description ... ??? "for most points, particular pair of other points ... except in 4 special cases the pair degenerates ..." ... ??? .... ?? not much symmetry to exploit though .... ??? .....
?? "branched" (??? ....) double cover of real p^1 with branch points at 0, 1, -1, infinity .... ???? .... ?? hmmm .... ??? "branching" manifesting as "zig-zagging" here ??? ..... ?? or maybe not ??? .... equator ?? 2 completely separate "perfectly formed" equators .... ??? ... perhaps right the first time ... trying to remember the picture i sketched the other day ... ok, branch point seems to manifest as "4-valent vertex" .... ???? .... hmmmm ..... ???? .... "rectangle boundary with 2 side edges for each one original" ... ?? eisenstein case ??? .... ?? ??? real coefficients vs real roots here ????? ...... ???? .....
?? hmmm, confusion between .... ?? real elliptic curve, and part of complex elliptic curve lying over real projective line under projection to complex projective line ... ?? "meridianal equators vs longitudinal equators" ... ??? real vs imaginary square roots of real quantities .... ?? definitely still some confusion here, but seems like not too far from straightening it out .... ??? .... ?? x^3-x as positive from 1 to infinity, and from -1 to 0 ..... ??????? extent to which this resolves some old confusions we had about ... ?? knowing that real ec has to be ab gp but thinking that it looked like it had singularities ... ??? ... ??? ....
?? gauss square ... ??? eisenstein rhombus ??? .... ???? ??? other rhombuses though .... other rectangles besides square ........ ?????.... ?? sides of modular triangle ... ??? ..... ?? what's the third side ??? ... ??? eisenstein and "tate" ... ??? ... ?? particular visual commonality between the parallelograms (??? ...) doesn't jump out at me .... ????? ..... ?? not sure how much i might have things screwed up here .... ???? .....
??? modularity and buckyballs .... ????? .... ??? simplex-based .... ???? ....
?? "line ew quadratic embedding to k^n" ... ?? meaning of "embedding" in this context ??? ..... ???? ....
?? confusion about possibly contrasting kinds of "non-degeneracy" here ??? ....
?? anyway, interesting to omit embdding requirement too .... ??? ....
?? various ideas about "generic ..." / "quadratic ..." .... ???? ....
?? bit about clifford alg of free qfs on vsp .... ??? variants of this idea ?? ...
?? stuff here about .... ?? tierney / johnstone and "spectrum" (?? ...) and "higher genericity" .... ???? ......
(?? vague feeling about ... "generalized quantum double" here ... ??? ..... ???? ....)
?? space (?? ...) of quadratic forms on given vsp, vs moduli stack of qfs's .... ??? ....
?? stuff that we thought about recently in connection with baez's "walking composition algebra" stuff .... ???? .... cup / cap ... ?? partition into doublets on image complement ... ?? non-degeneracy .... ??? ....
?? quadratic map from k to k^n .... ?? vs vice versa .... ???? ...... homogeneous vs not ... ??? .....
?? l : line object ...
a : 1 -> 1
b : l -> 1
c : l#l -> 1
.... ??? .....
?? confusion about possibly contrasting kinds of "non-degeneracy" here ??? ....
?? anyway, interesting to omit embdding requirement too .... ??? ....
?? various ideas about "generic ..." / "quadratic ..." .... ???? ....
?? bit about clifford alg of free qfs on vsp .... ??? variants of this idea ?? ...
?? stuff here about .... ?? tierney / johnstone and "spectrum" (?? ...) and "higher genericity" .... ???? ......
(?? vague feeling about ... "generalized quantum double" here ... ??? ..... ???? ....)
?? space (?? ...) of quadratic forms on given vsp, vs moduli stack of qfs's .... ??? ....
?? stuff that we thought about recently in connection with baez's "walking composition algebra" stuff .... ???? .... cup / cap ... ?? partition into doublets on image complement ... ?? non-degeneracy .... ??? ....
?? quadratic map from k to k^n .... ?? vs vice versa .... ???? ...... homogeneous vs not ... ??? .....
?? l : line object ...
a : 1 -> 1
b : l -> 1
c : l#l -> 1
.... ??? .....
lah / ksu ...
taxes ....
baez stuff ... snyder, dmv ... ?? ....asf ....
??? "line ew quadratic embedding to k^n" ... ??? asf os ....
??? ?? finite extrra structure / branched covering bit in p^1 setting .... ??? .... fe ... asf os ....
?? comm monoid object in graded modules of modular forms ... ??? .... asf os .... ??? .... ?? as ... ?? manifestation of ... ??? cqa / ec equivalence .... ??? ....
?? correspondence in modularity context between "latticey" stuff on ec and on mc ... ??? quadratic forms .... ??? .... ??? also ... ?? "voronoi-ish" (??? ... ??? .....) stuff .... ??? ....
taxes ....
baez stuff ... snyder, dmv ... ?? ....asf ....
??? "line ew quadratic embedding to k^n" ... ??? asf os ....
??? ?? finite extrra structure / branched covering bit in p^1 setting .... ??? .... fe ... asf os ....
?? comm monoid object in graded modules of modular forms ... ??? .... asf os .... ??? .... ?? as ... ?? manifestation of ... ??? cqa / ec equivalence .... ??? ....
?? correspondence in modularity context between "latticey" stuff on ec and on mc ... ??? quadratic forms .... ??? .... ??? also ... ?? "voronoi-ish" (??? ... ??? .....) stuff .... ??? ....
Thursday, April 12, 2012
?? given dimensional theory, universally adjoining quantity in dimension d, satisfying "polynomial" equation of degree n .... ???? .....
?? "geometric" (??? ....) interpretation ?? .... ???? ....
?? geometric interpretation of modding out comm ring by subring ... ?? give "relative function" on each orbit ?? ... ??? ....
?? "geometric" (??? ....) interpretation ?? .... ???? ....
?? geometric interpretation of modding out comm ring by subring ... ?? give "relative function" on each orbit ?? ... ??? ....
Wednesday, April 11, 2012
?? talking to john huerta about bn-pairs and buildings and so forth ... ?? non-/anomalousness of "q=1" case .... ??? "coxeter-valued metric on flags" approach seems to work fine (?? better ?? ...) in q=1 case ... ?? "apartment-chamber geometry" approach, though ?? ... ??? to what extent did we (?? ...) actually work out such an approach ?? ... ?? simple classical-valued incidence relation ??? ... ?? possibility of coxeter diagram being somehow automatically encoded in such relation ?? .... ?? not sure such approach works at all, but ... ?? even if it does, q=1 case seems somewhat (?? ...) problematic .... ??? ...
Tuesday, April 10, 2012
?? ideas to try to pull together here ....
?? idea of "langlands conjecture" as giving some sort of classification of stable hopf algs ... ??? .... (?? idea which seems a bit in trouble at the moment .... ??? ....)
?? correspondence between nice structures on ec's and nice structures on cqa's ... ?? ....
?? stuff diamond and shurman say .... ??? .... ???? ..... ??? eigen-values/-vectors of hecke operators .... ???? ....
?? stuff ____ says .... ??? ..... ?????? ....
?? attempt to understand jugendtraum .... ??? relationship to artin reciprocity .... ??? ......
?? "congruence subgp" ....
?? "machine for turning ideal numbers into actual ones" ... ??? ....
.......... ??????? .......
?? pressley and segal ... brown .... extra dot .... ???? ....
??? tate's thesis ??? .... ??? ....
?? green convolution ... ??? ....
?? homming between qfs's .... ?? ....
?? idea of "langlands conjecture" as giving some sort of classification of stable hopf algs ... ??? .... (?? idea which seems a bit in trouble at the moment .... ??? ....)
?? correspondence between nice structures on ec's and nice structures on cqa's ... ?? ....
?? stuff diamond and shurman say .... ??? .... ???? ..... ??? eigen-values/-vectors of hecke operators .... ???? ....
?? stuff ____ says .... ??? ..... ?????? ....
?? attempt to understand jugendtraum .... ??? relationship to artin reciprocity .... ??? ......
?? "congruence subgp" ....
?? "machine for turning ideal numbers into actual ones" ... ??? ....
.......... ??????? .......
?? pressley and segal ... brown .... extra dot .... ???? ....
??? tate's thesis ??? .... ??? ....
?? green convolution ... ??? ....
?? homming between qfs's .... ?? ....
?? trying to understand correspondence between extra structure on cuboquadratic alg and on elliptic curve .... ??? ....
??? get cuboquadratic algebra from elliptic curve by ... ?? well, first, seems to make sense to take elliptic curve to have basepoint ... to match small aut gp of cqa ... ?? ... ?? then reverse the branched cover process that gave rise to the ec ... ?? maybe by modding out by inverse involution ??? .... ?? any nice explicit "algebraic" way of doing this ?? .... i was thinking in terms of ... ?? taking subring of functions invariant under inversion ... but perhaps somewhat trickier ... gradedness .... ?? how "theta" line bundle over ec gets along with inversion .... ????? .....
(?? vaguely reminds me of stuff about ... ?? various (??) kinds of "toric" structure on line bundle ... and ... ??? "quantum double" .... ??? .... ??? ...)
?? ... anyway ... thinking perhaps at somewhat naive level ... seems like you can mod out by inverse to get projective line with basepoint ("at infinity" ... ?? ...), aka affine line ..... ??????? ...... ??? translations of which give 1d vsp ??? ..... ???? so that sort of seems to do it .... this 1d vsp is more or less the cuboquadratic alg, we think .... ???? ....
??? might also be able to think of this in some "infinitesimal" way ... ?? take tangent space at some point (..... ???? .....) ..... ??? .....
??? certain sort of pair of cuboquadratic structures, maybe ??? .... ???? .....
?? or ... ?? one elliptic curve being unbranched cover of another .... ???? ..... ?? composite with usual branched cover of sphere by ec .... ??? .....
??? "hecke modular curve" ??? .... ??? "hecke modular function" ???? .... ???? .... relationship to "hecke operator" .... ???? ..... ??? "correspondence" .... ???? .....
???? .......
??? get cuboquadratic algebra from elliptic curve by ... ?? well, first, seems to make sense to take elliptic curve to have basepoint ... to match small aut gp of cqa ... ?? ... ?? then reverse the branched cover process that gave rise to the ec ... ?? maybe by modding out by inverse involution ??? .... ?? any nice explicit "algebraic" way of doing this ?? .... i was thinking in terms of ... ?? taking subring of functions invariant under inversion ... but perhaps somewhat trickier ... gradedness .... ?? how "theta" line bundle over ec gets along with inversion .... ????? .....
(?? vaguely reminds me of stuff about ... ?? various (??) kinds of "toric" structure on line bundle ... and ... ??? "quantum double" .... ??? .... ??? ...)
?? ... anyway ... thinking perhaps at somewhat naive level ... seems like you can mod out by inverse to get projective line with basepoint ("at infinity" ... ?? ...), aka affine line ..... ??????? ...... ??? translations of which give 1d vsp ??? ..... ???? so that sort of seems to do it .... this 1d vsp is more or less the cuboquadratic alg, we think .... ???? ....
??? might also be able to think of this in some "infinitesimal" way ... ?? take tangent space at some point (..... ???? .....) ..... ??? .....
??? certain sort of pair of cuboquadratic structures, maybe ??? .... ???? .....
?? or ... ?? one elliptic curve being unbranched cover of another .... ???? ..... ?? composite with usual branched cover of sphere by ec .... ??? .....
??? "hecke modular curve" ??? .... ??? "hecke modular function" ???? .... ???? .... relationship to "hecke operator" .... ???? ..... ??? "correspondence" .... ???? .....
???? .......
Monday, April 9, 2012
?? 24 semi-simple comm ring structures on {a,b,c,d} ...
f2 X f2 vs f4 ....
?? depict f4 structure as for example "ab", meaning a=0 and b=1 ...
?? depict f2Xf2 structure as ... ??? ... for example, "ab", meaning a = (0,0) and b = (1,1) .... ?? ...
(?? so some of that 2+2=2X2 stuff is dimly visible here but ok to soft-pedal, i guess .... ??? ....)
?? but need way to distinguish the 2 types .... ?? 4ab and 2ab ... ??? ....
?? 2^24 as already somewhat annoyingly big ... ?? ....
?? try working with euler summand ...... ???? .... ???? .....
f2 X f2 vs f4 ....
?? depict f4 structure as for example "ab", meaning a=0 and b=1 ...
?? depict f2Xf2 structure as ... ??? ... for example, "ab", meaning a = (0,0) and b = (1,1) .... ?? ...
(?? so some of that 2+2=2X2 stuff is dimly visible here but ok to soft-pedal, i guess .... ??? ....)
?? but need way to distinguish the 2 types .... ?? 4ab and 2ab ... ??? ....
?? 2^24 as already somewhat annoyingly big ... ?? ....
?? try working with euler summand ...... ???? .... ???? .....
?? 24 semi-simple comm ring structures on {a,b,c,d} ...
?? halving ab/cd ...
12
3ac
4db
ab
c32
d14
?? "halving into ab/cd corresponds to factoring into 12/34" ... ??? ... ?? and vice versa ?? .... ??? this really is that weird thing that i noticed in high school ??? .... ?? still don't quite get all of it though ... the vice versa part .... ???? .... ?? outer automorphism of automorphism group of wreath product here ??? .... ??? _is_ this connected nicely to solution of quartic somehow ??? ....
?? 4! as resulting from making outer inner ??? .... ?? having trouble getting arithmetic to work here .... ???? ....
wreath product here as 8-elt gp ??? .... outer automorphism of order 2, evidently .... ???? .....
?? well, so maybe the semi-direct product here really is some weird 16-elt gp .... ??? ....
?? "element" vs "cross section" of split 2-elt set ....
?? "unordered dual pair of split 4-elt sets" .... ??? ....
?? number of 2+2 decompositions same as number of 2X2 decomposition because of ... outer automorphism here .... ??? .... ????? ....
?? isomorphism between subgroup of 4! preserving a 2+2 decomposition and subgroup preserving a 2X2 decomposition .... ??? .... ???? extending to ..... ???? .... ????? ..... ?? actually, extending to identity automorphism .... ??? ... ???? confusion ... ???? .....
?? is there duality between 6 halves of one 4-elt set and 6 halves of .... ??? .... ??? doesn't seem to quite make sense ??? ...
?? halving ab/cd ...
12
3ac
4db
ab
c32
d14
?? "halving into ab/cd corresponds to factoring into 12/34" ... ??? ... ?? and vice versa ?? .... ??? this really is that weird thing that i noticed in high school ??? .... ?? still don't quite get all of it though ... the vice versa part .... ???? .... ?? outer automorphism of automorphism group of wreath product here ??? .... ??? _is_ this connected nicely to solution of quartic somehow ??? ....
?? 4! as resulting from making outer inner ??? .... ?? having trouble getting arithmetic to work here .... ???? ....
wreath product here as 8-elt gp ??? .... outer automorphism of order 2, evidently .... ???? .....
?? well, so maybe the semi-direct product here really is some weird 16-elt gp .... ??? ....
?? "element" vs "cross section" of split 2-elt set ....
?? "unordered dual pair of split 4-elt sets" .... ??? ....
?? number of 2+2 decompositions same as number of 2X2 decomposition because of ... outer automorphism here .... ??? .... ????? ....
?? isomorphism between subgroup of 4! preserving a 2+2 decomposition and subgroup preserving a 2X2 decomposition .... ??? .... ???? extending to ..... ???? .... ????? ..... ?? actually, extending to identity automorphism .... ??? ... ???? confusion ... ???? .....
?? is there duality between 6 halves of one 4-elt set and 6 halves of .... ??? .... ??? doesn't seem to quite make sense ??? ...
?? "can you define multiplication of natural numbers in terms of "relatively prime" ?" as maybe good example ... ???
?? also defining anything in terms of successor ... ??? ...
?? all possible ways of decorating triangle .... ??? "special" such ... ???? hmm, confusion ... ??? .... ?? taking seriously idea of "frame-x orientations as x's" .... ?? hmmm, yet another horrible confusion here ?? ... "active vs passive" ... hard to explain without good pictures here .... ??? ....
??? each of the three frame-point orientations has a very different look when portrayed in the "collection of all frames exhibiting that orientation to given point p" style .... ??? .... ?? to what extent does this generalize ?? .... ???? .....
(?? vs ... "collection of all points exhibiting that orientation to given frame f" style ..... ????? ..... ?? hmm, that style as unhelpful (?? ...) here because whole point is that concept of frame is here (???? .....) coming equipped with canonical (??? ....) way of drawing it ..... ??? ... os ???? .... ?? hmm, hint tha answer to "to what extent does this generalize ?" above is something like ... ??? to the general context where the base carriers are plain old (?? finite ?? ...) sets, so that concept of frame has somewhat good canonical pictorial representation ... ??? or beyond that, to general context where _you've already settled on a good way to draw a frame_, and you want to use it to get good ways of drawing other structures .... ???? ...... ??? then ... ?? general idea of getting picture in which the symmetry group acts "fixed-point-free" on some orbit, so that marking single point on that orbit gives good decoration corresponding to "frame" ... ??? ...)
3 looks : "hypotenuse-to-hypotenuse", "short-side-to-short-side", "touching at only one point" .... ??? .... any nice way to single out one of these ??? ..... ???? ....
?? trying to get straight "classification of the equivalence classes of special decorations that all others can be reduced to" .... ???? nice way of getting exactly one representative of each symmetry class .... ???? ....... if that can be done ... nicely ... .... ???? .....
?? idea of having the special representative being giving each coset its own color ???? ..... color choice issue though .... ??? ....
(??no ??) paradox / confusion concerning .... ??? "distinctive" nature of "orientation classes of frames towards given x" vs "indistinctive" nature of "orientation classes of x's towards given frame" ??? .... ?????
?? also defining anything in terms of successor ... ??? ...
?? all possible ways of decorating triangle .... ??? "special" such ... ???? hmm, confusion ... ??? .... ?? taking seriously idea of "frame-x orientations as x's" .... ?? hmmm, yet another horrible confusion here ?? ... "active vs passive" ... hard to explain without good pictures here .... ??? ....
??? each of the three frame-point orientations has a very different look when portrayed in the "collection of all frames exhibiting that orientation to given point p" style .... ??? .... ?? to what extent does this generalize ?? .... ???? .....
(?? vs ... "collection of all points exhibiting that orientation to given frame f" style ..... ????? ..... ?? hmm, that style as unhelpful (?? ...) here because whole point is that concept of frame is here (???? .....) coming equipped with canonical (??? ....) way of drawing it ..... ??? ... os ???? .... ?? hmm, hint tha answer to "to what extent does this generalize ?" above is something like ... ??? to the general context where the base carriers are plain old (?? finite ?? ...) sets, so that concept of frame has somewhat good canonical pictorial representation ... ??? or beyond that, to general context where _you've already settled on a good way to draw a frame_, and you want to use it to get good ways of drawing other structures .... ???? ...... ??? then ... ?? general idea of getting picture in which the symmetry group acts "fixed-point-free" on some orbit, so that marking single point on that orbit gives good decoration corresponding to "frame" ... ??? ...)
3 looks : "hypotenuse-to-hypotenuse", "short-side-to-short-side", "touching at only one point" .... ??? .... any nice way to single out one of these ??? ..... ???? ....
?? trying to get straight "classification of the equivalence classes of special decorations that all others can be reduced to" .... ???? nice way of getting exactly one representative of each symmetry class .... ???? ....... if that can be done ... nicely ... .... ???? .....
?? idea of having the special representative being giving each coset its own color ???? ..... color choice issue though .... ??? ....
(??no ??) paradox / confusion concerning .... ??? "distinctive" nature of "orientation classes of frames towards given x" vs "indistinctive" nature of "orientation classes of x's towards given frame" ??? .... ?????
?? categorified multiplication and categorified comultiplication for structure types ... ???
"from structure type on pairs of sets, get structure type on sets" ... ??? multiplication .... ??? ... ?? special case where structure type on pairs of sets involves "no correlation" between the two sets .... ???? ....
"from structure type on sets, get structure type on pairs of sets" ... ??? comultiplication .... ????? "new structure on (a,b) as old structure on a+b" .... ??? ....
??? structure type t tw .... ??? nice morphism from comult(t) to ... ???? ..... "no correlation" structure type ........ ???? .....
?? clear up .... ???? .....
?? morphism from comult(t) to t#t ???? ..... ?????? ..... ?????? ......
"from structure type on pairs of sets, get structure type on sets" ... ??? multiplication .... ??? ... ?? special case where structure type on pairs of sets involves "no correlation" between the two sets .... ???? ....
"from structure type on sets, get structure type on pairs of sets" ... ??? comultiplication .... ????? "new structure on (a,b) as old structure on a+b" .... ??? ....
??? structure type t tw .... ??? nice morphism from comult(t) to ... ???? ..... "no correlation" structure type ........ ???? .....
?? clear up .... ???? .....
?? morphism from comult(t) to t#t ???? ..... ?????? ..... ?????? ......
?? categorified riemann zeta function ....
?? assign to each finite set set of semi-simple comm ring structures on it ... ?? n! such structures on n .... ???? ....
??? then would like .... ???? corresponding vsp-valued structure type ...... ????? ..... ?? so would give same decategorification .... ??? ....
?? assigining n!-dim vsp to n .... ??? so cardinality k^[n!] .... ???? ....
?? "structure type where structure on n is assignment to each semi-simple comm ring structure on n of map from its components to 3" ..... ??????
?? for example n = 4 .... ???? 12 one-component structures and 12 two-component ones .... ????? .... 3^12 * (3^2)^12 .... ?? whereas i was naively expecting 3^24 ???? ...... ???? just how screwed up are things here ???? .....
??? dirichlet-exponentiate structure type given by ??? field structure augmented by ...... ????? ..... ????? ............. ?????
spec(number ring) X spec(finite field) -> z^k
?? given alg ab gp, hom into it from spectrums of finite fields ... then dirichet-exponetiate that thing .... ??? .... ??? .....
???????
?? ....
?? assign to each finite set set of semi-simple comm ring structures on it ... ?? n! such structures on n .... ???? ....
??? then would like .... ???? corresponding vsp-valued structure type ...... ????? ..... ?? so would give same decategorification .... ??? ....
?? assigining n!-dim vsp to n .... ??? so cardinality k^[n!] .... ???? ....
?? "structure type where structure on n is assignment to each semi-simple comm ring structure on n of map from its components to 3" ..... ??????
?? for example n = 4 .... ???? 12 one-component structures and 12 two-component ones .... ????? .... 3^12 * (3^2)^12 .... ?? whereas i was naively expecting 3^24 ???? ...... ???? just how screwed up are things here ???? .....
??? dirichlet-exponentiate structure type given by ??? field structure augmented by ...... ????? ..... ????? ............. ?????
spec(number ring) X spec(finite field) -> z^k
?? given alg ab gp, hom into it from spectrums of finite fields ... then dirichet-exponetiate that thing .... ??? .... ??? .....
???????
?? ....
Sunday, April 8, 2012
?? alg ab gp given by ....
spec(z[0,1,2], but with "pointwise" multiplication of polynomials instead of usual ocnvolution) .... ????
?? have we ever thought about this multiplication of polynomials before ???? ...
?? comultiplication given by .... ????? ....
?? no, didn't get basis correct yet, i think .... ????? ....
-
10 01 with - = 11
10
00
01
00
00
10
00
01
with 10 =
10
10
and
01 =
01
01
....
?? so - stays multiplicative identity .... ???? ....
1,11,110,111,1100,1110,1101,1111 .... ???? ....
11
11
11
11
0,1,10,11,100,101,110,111
11
11
11
11
10
10
10
10
11
00
11
00
?? linear functionals on functions on subsets of n .... ????? .... ?? that only depend on the restriction of those functions to ... ?? finite subsets ???? ....
?? linear functionals on functions on finite subsets of n .... ????? .... ?? for which there's a finite size such that the functional only depends on the restriction of the functions to ... ?? subsets below that size ???? ....
spec(z[0,1,2], but with "pointwise" multiplication of polynomials instead of usual ocnvolution) .... ????
?? have we ever thought about this multiplication of polynomials before ???? ...
?? comultiplication given by .... ????? ....
?? no, didn't get basis correct yet, i think .... ????? ....
-
10 01 with - = 11
10
00
01
00
00
10
00
01
with 10 =
10
10
and
01 =
01
01
....
?? so - stays multiplicative identity .... ???? ....
1,11,110,111,1100,1110,1101,1111 .... ???? ....
11
11
11
11
0,1,10,11,100,101,110,111
11
11
11
11
10
10
10
10
11
00
11
00
?? linear functionals on functions on subsets of n .... ????? .... ?? that only depend on the restriction of those functions to ... ?? finite subsets ???? ....
?? linear functionals on functions on finite subsets of n .... ????? .... ?? for which there's a finite size such that the functional only depends on the restriction of the functions to ... ?? subsets below that size ???? ....
??mapping from spec(finite comm ring) to exponential spec(z^n)^[number ring] .... ????
spec(finite comm ring) X [number ring] -> spec(z^n) .... ??? ....
?? number of components that number ring splits into when tensored with finite comm ring .... ????? .....
?? for example number ring = gaussian integers ... ?? .... ?? and n = 3, say ... ?? ...
?? finite comm rings of size one ... ?? ....
?? zero components ??? ... ?? one map ?? ....
?? size two ?? ...
?? is it just one component ?? ... ??? because of ramification ??? ... so 3 maps ?? ....
?? size three ... ??? ...
?? one component .... ?? so 3 maps ??? .....
?? size 4 .... ??? 2 iso classes of semi-simple comm rings here .... ??? ... f_4 and f_2 X f_2 .... ???? .....
?? ...
spec(finite comm ring) X [number ring] -> spec(z^n) .... ??? ....
?? number of components that number ring splits into when tensored with finite comm ring .... ????? .....
?? for example number ring = gaussian integers ... ?? .... ?? and n = 3, say ... ?? ...
?? finite comm rings of size one ... ?? ....
?? zero components ??? ... ?? one map ?? ....
?? size two ?? ...
?? is it just one component ?? ... ??? because of ramification ??? ... so 3 maps ?? ....
?? size three ... ??? ...
?? one component .... ?? so 3 maps ??? .....
?? size 4 .... ??? 2 iso classes of semi-simple comm rings here .... ??? ... f_4 and f_2 X f_2 .... ???? .....
?? ...
Saturday, April 7, 2012
?? using lagrange extrapolation to construct alg ab gp structure on spec(xXz) for x quadratic number ring ..... .... .... ??? ...
?? exponentiableness and "flatness" ??? ... affine line ^ spec(r) .... ??? to what extent this gives good "algebraic model" of r ... ???? ..... ?? exponentiableness and geometric colimits (= algebraic limits ...) ... preservation of them by cartesian (tensor ...) product with exponent e .... ?? modification of colimits and getting more objects to be exponentiable .... ???? ...
?? power series ring .... ??? ....
?? doctrine in which concept "field" can be nicely expressed .... ???? ......
?? tag (??) theory for d-building .... ???? ....
?? spec(gaussian integers) as exponent ... ??? "for each original generator g, generators g_1 and g_i ... for each original relator r, relators r_1 and r_i ..." ...
?? bit about "underlying real variety of complex variety" .... ?????? ....
?? walking idempotent ... r = [gg=g] ... g_1=h, g_r=j ... (h+ji)(h+ji)=h+ji ...
hh-jj = h
jh+hj = j
????
?? measure vs function .... ????? .....
2hj = j
2hj-j = 0
(2h-1)j = 0 ...
?? .... 1/4 - jj = 1/2
jj - 1/4 = -1/2
jj = 1/4 - 1/2
jj = -1/4 ....
j =?= plus or minus i/2
?? h+ji = 1/2 + (i/2)i = 0 .....
?? what in the world is going on here ???? .....
?? spec(r^2) ^ spec(c) ..... ????? ?? over r ..... ????? .....
?? alg ab gp vs motive .... ???? .... ?? ab cat of alg ab gps ?? ... ???? ...
??? "imaginary numbers drag real numbers down so complex field has no real spectrum" .... ???? ....
?? some algebra/geometry contravariance confusion here, but ... ???? ...
?? mystical arrow / dominance / superscript confusion ??? ..... ....
hh-jj = h
2hj = j
(2h-1)j = 0
?? "h = 1/2 or [j=0 and consequently hh=h]" .... ????? ...
?? components of the exponential here ... ?? "real" vs ... ?? "unreal" (????) such .... ???? ...... actual morphism ... "occupied" .... ??? .... ???? ....
?? relationship to exponentiation in group action topos ?? ... ?? "preservation of exponentiation by forgetful functor" .... ???? ....
?? (1+1)^2 = 1+2+1 ..... ????? ....
?? exponentiableness and "flatness" ??? ... affine line ^ spec(r) .... ??? to what extent this gives good "algebraic model" of r ... ???? ..... ?? exponentiableness and geometric colimits (= algebraic limits ...) ... preservation of them by cartesian (tensor ...) product with exponent e .... ?? modification of colimits and getting more objects to be exponentiable .... ???? ...
?? power series ring .... ??? ....
?? doctrine in which concept "field" can be nicely expressed .... ???? ......
?? tag (??) theory for d-building .... ???? ....
?? spec(gaussian integers) as exponent ... ??? "for each original generator g, generators g_1 and g_i ... for each original relator r, relators r_1 and r_i ..." ...
?? bit about "underlying real variety of complex variety" .... ?????? ....
?? walking idempotent ... r = [gg=g] ... g_1=h, g_r=j ... (h+ji)(h+ji)=h+ji ...
hh-jj = h
jh+hj = j
????
?? measure vs function .... ????? .....
2hj = j
2hj-j = 0
(2h-1)j = 0 ...
?? .... 1/4 - jj = 1/2
jj - 1/4 = -1/2
jj = 1/4 - 1/2
jj = -1/4 ....
j =?= plus or minus i/2
?? h+ji = 1/2 + (i/2)i = 0 .....
?? what in the world is going on here ???? .....
?? spec(r^2) ^ spec(c) ..... ????? ?? over r ..... ????? .....
?? alg ab gp vs motive .... ???? .... ?? ab cat of alg ab gps ?? ... ???? ...
??? "imaginary numbers drag real numbers down so complex field has no real spectrum" .... ???? ....
?? some algebra/geometry contravariance confusion here, but ... ???? ...
?? mystical arrow / dominance / superscript confusion ??? ..... ....
hh-jj = h
2hj = j
(2h-1)j = 0
?? "h = 1/2 or [j=0 and consequently hh=h]" .... ????? ...
?? components of the exponential here ... ?? "real" vs ... ?? "unreal" (????) such .... ???? ...... actual morphism ... "occupied" .... ??? .... ???? ....
?? relationship to exponentiation in group action topos ?? ... ?? "preservation of exponentiation by forgetful functor" .... ???? ....
?? (1+1)^2 = 1+2+1 ..... ????? ....
Friday, April 6, 2012
?? gaussian integers .... homming into finite semi-simple comm ring .... ?? ....
0
1 1 - 1 1
2 f_2 1 1 1
3 0
4 f_2 X f_2 1,1 2 f_4 1 2 1
5 f_5 2 1 3 1 2
6 0
7 0
8 f_2 X f_2 X f_2 1,1,1 6 f_2 X f_4 1,1 2 f_8 1 3 1
??? counting points of spectrum of alg comm ring over finite fields .... ??? ...
_comm ring_ -> _comm ring_ .... ???? ... "tensoring with gaussian integers" ....
?? confusion ?? ... ?? level slip ??? .... ??? pretty big algebraic comm ring ... ??? polynomials in 2 variables .... inf-dim ... ??? vs pretty small hopf algs .... ???? .... ??? .... fin-dim .... ???? ...
?? 4th roots of 1 mod square roots of 1 ??? ..... ???? ....
?? 3rd roots of 1 ........ ?????? .....
0
1 1 - 1 1
2 f_2 1 1 1
3 0
4 f_2 X f_2 1,1 2 f_4 1 2 1
5 f_5 2 1 3 1 2
6 0
7 0
8 f_2 X f_2 X f_2 1,1,1 6 f_2 X f_4 1,1 2 f_8 1 3 1
??? counting points of spectrum of alg comm ring over finite fields .... ??? ...
_comm ring_ -> _comm ring_ .... ???? ... "tensoring with gaussian integers" ....
?? confusion ?? ... ?? level slip ??? .... ??? pretty big algebraic comm ring ... ??? polynomials in 2 variables .... inf-dim ... ??? vs pretty small hopf algs .... ???? .... ??? .... fin-dim .... ???? ...
?? 4th roots of 1 mod square roots of 1 ??? ..... ???? ....
?? 3rd roots of 1 ........ ?????? .....
?? reverse engineer hopf alg from galois rep ?? ... ???? ..... ???? ..... ?? generalize to case including context where "galois gp" is more naively (?? ...) "geometric" .... ??? ....
?? the (???) galois action on x (alg gp, or alg ring, or ... ?? ... ??? alg something ?? ....) depends only on the underlying scheme (?? ...) of x .... ??? ...
?? so for example, the (??? ...) galois action on "alg gaussian integers" as trivial .... ???? .... ??? ...
??? some other galois group / action here ??? .... ??? auts of "alg alg integers" ??? .... ??? ... ??? .... ??? ....
?? the (???) galois action on x (alg gp, or alg ring, or ... ?? ... ??? alg something ?? ....) depends only on the underlying scheme (?? ...) of x .... ??? ...
?? so for example, the (??? ...) galois action on "alg gaussian integers" as trivial .... ???? .... ??? ...
??? some other galois group / action here ??? .... ??? auts of "alg alg integers" ??? .... ??? ... ??? .... ??? ....
Thursday, April 5, 2012
?? role of flatness (?? ...) of k in .... ???? getting ... ??? i was going to distinguish between .... getting "underlying set of tensor product with k" to be representable, vs getting it to lift to _comm ring_ .... ?? but the latter seems completely tautological, right ??? ... ???? ....
?? consider ... ??? many different products here .... categorified and decategorified .... dircichlet series ... comm ring ... alg comm ring ... bistable hopf alg ... .... ??? .... .... cartesian, "tensor", "direct sum", ..... ???? .... correspondences / relationship among such ... ???? ....
??? confusion here (?? ...) ... ?? comm ring vs alg comm ring ... ??? ... ??? galois action .... ???? ....
?? consider ... ??? many different products here .... categorified and decategorified .... dircichlet series ... comm ring ... alg comm ring ... bistable hopf alg ... .... ??? .... .... cartesian, "tensor", "direct sum", ..... ???? .... correspondences / relationship among such ... ???? ....
??? confusion here (?? ...) ... ?? comm ring vs alg comm ring ... ??? ... ??? galois action .... ???? ....
?? so _does_ "direct sum" of bistable hopf algs correspond to multiplication of associated categorified zeta functions ??? .... ???? .... ?? work it out ...
?? even if it does, hard to see how could account for characteristically (...) weird coefficients of l-fn ... ??? unless ... ?? decategorification process used is somehow weirder than ... what it seems to be in the zeta function case .... ??? .... ??? .....
?? underlying additive algebraic (abelian) group of algebraic comm ring .... "direct sum" decomposition of that algebraic abelian group .... ??? ...
?? "algebraic direct sum" .... ???? tensor product .... ???? ......
?? hopf alg and fourier duality .... ???? .....
?? categorified zeta function with f_q-vsp coefficients .... ??? relating in some way to "green type" .... ???? .... ???? ..... ??? vague memories about thinking about stuff like ... ?? to what extent idea of substituting something (????) into dirichlet series makes (categorified ... ??? ...) sense .... ???? ....
?? even if it does, hard to see how could account for characteristically (...) weird coefficients of l-fn ... ??? unless ... ?? decategorification process used is somehow weirder than ... what it seems to be in the zeta function case .... ??? .... ??? .....
?? underlying additive algebraic (abelian) group of algebraic comm ring .... "direct sum" decomposition of that algebraic abelian group .... ??? ...
?? "algebraic direct sum" .... ???? tensor product .... ???? ......
?? hopf alg and fourier duality .... ???? .....
?? categorified zeta function with f_q-vsp coefficients .... ??? relating in some way to "green type" .... ???? .... ???? ..... ??? vague memories about thinking about stuff like ... ?? to what extent idea of substituting something (????) into dirichlet series makes (categorified ... ??? ...) sense .... ???? ....
?? ag morphism from theory of z/2-torsor to theory of line object ... ??? .... ?? pull z/2-graded vsp back along z -> z/2 ?? .... ??? ok to be sloppy about fourier dual of z/2 here because pretty galois-stable ??? ... ??? ... ??? ....
?? wait, does this make any sense ?? .... would be surprising if it did, since i think that it was based on a misconception .... ?? and seems like should give paradox .... ??? ... ??
?? go other way by pushing forward along z -> z/2 .... ???? ....
?? wait, does this make any sense ?? .... would be surprising if it did, since i think that it was based on a misconception .... ?? and seems like should give paradox .... ??? ... ??
?? go other way by pushing forward along z -> z/2 .... ???? ....
?? (?? affine ?? ...) algebraic ring whose underlying scheme is exponentiable (?? in "affine" sense ??? .... ???? ....)
?? affine algebraic ring like "the gaussian integers" .... ????? .... role of "flatness" (??? ... ????) of k in making functor _comm ring_ -> _comm ring_ -> _set_ given by [tensoring with k] followed by "underlying set" sufficiently good (=?= representable ... ??? ....) ... ????? .... ???? ....
?? "algebraic integers" here as trickier but maybe interesting ... ??? ... ?? not affine but ... ??? ....
?? had ideas about using this (...) sort of algebraic ring stuff to get nice twisted versions of finitary bistable hopf algs (and "distributive" such ...... ???? .....) ..... ??? not quite sure about that idea now ??? .... ????? ...... ?? level slip ??? .... coefficients vs .... ??? .... ??? .... ???????? "splitting" ...... ???? "trivial" .... ?? "locally trivial" .... ????? "flat" ????? ..... ????? ......
?? alg ideal in alg ring ... modding out by such .... ??? .... ??? ....
??? alg rep of alg gp .... ???? .... ?? alg hom to alg matrix (?? ...) ring ... ???? ....
??? vague memory about ... ?? "geometric (?? ....) version of frobenius ..." .... ???? categorified zeta function .... ????? .... ??? ....
rig .... ??? comm rig as between comm monoid and comm ring ... of course other things in between there too ... ???? ....
??? divisor concept on "reducible" variety ... ??? use product of meromorphic fields here ..... ????? .....
?? affine algebraic ring like "the gaussian integers" .... ????? .... role of "flatness" (??? ... ????) of k in making functor _comm ring_ -> _comm ring_ -> _set_ given by [tensoring with k] followed by "underlying set" sufficiently good (=?= representable ... ??? ....) ... ????? .... ???? ....
?? "algebraic integers" here as trickier but maybe interesting ... ??? ... ?? not affine but ... ??? ....
?? had ideas about using this (...) sort of algebraic ring stuff to get nice twisted versions of finitary bistable hopf algs (and "distributive" such ...... ???? .....) ..... ??? not quite sure about that idea now ??? .... ????? ...... ?? level slip ??? .... coefficients vs .... ??? .... ??? .... ???????? "splitting" ...... ???? "trivial" .... ?? "locally trivial" .... ????? "flat" ????? ..... ????? ......
?? alg ideal in alg ring ... modding out by such .... ??? .... ??? ....
??? alg rep of alg gp .... ???? .... ?? alg hom to alg matrix (?? ...) ring ... ???? ....
??? vague memory about ... ?? "geometric (?? ....) version of frobenius ..." .... ???? categorified zeta function .... ????? .... ??? ....
rig .... ??? comm rig as between comm monoid and comm ring ... of course other things in between there too ... ???? ....
??? divisor concept on "reducible" variety ... ??? use product of meromorphic fields here ..... ????? .....
?? "incestuous" nature of "l-function" ??? .... ??? coefficients may themselves undergo galois action ... ??? ... ???? .....
?? "fourier duality proof of quadratic reciprocity" .... ??? .... ????? .... ??? ....
?? lambda .... ... q .... .... ????? .... ???? ....
?? character of perm rep .... trace .... fixed points .... ???? ....
?? motive _as_ quasicoherent sheaf over some sort of "galois stack" ??? ... ???? ...
?? trying to set up "square" here .... ??? ....
?? "set-valued galois rep" .... ???? given by (for example) homming gaussian integers into q-bar .... ???? ..... ?? then decompose this (??2d ...) perm rep into irreps .... ?? perhaps over f_q for some q ??? ... ???? ... ??? hopf alg interpretation ??? .... of ... ??? ... ?? perm rep over f_q .... ???? ....
?? comm ring r st [r,q-bar] = (naturally ... ??? ...) 1 + [z[i],q-bar] + 1 ... ????? .... ??? z X z[i] X z ... ??? .... ?? making this into bistable hopf alg ?? ... ??? then decomposing it wrt tensor product of bistable hopf algs ?? ... ??? ..... ???? ....
?? fourier dual of z/2 ??? ....
?? z[x]/x^2-1 ...... ?????? ....
?? confusion ..... ???? ......
?? z X z X z X z[i] X z[i] X z[i] ..... ???? .....
?? z[x]/x^2+x+1 # ... ???? ....
?? z[x]/x^3-1 # (z X z X z)
?? z[x]/x^3+x # (z X z X z) ...... ???? .... hmmmmm ..... ????? ......
??? f_q as algebraic ring here ??? .... ??? free algebraic module of algebraic ring on algebraic set ??? .... ???? ....
??? possibility of "twisted" version of "f_q as algebraic ring" here ??? .... ???? .... ?? maybe many such ??? .... ??? abelian variety p-torsion .... ???? ..... ??? "complex multiplication" ????? ..... ????? .....
?? f_p^n .... ??? ....
?? algebraic ring "the ring" ..... ??? .... polynomials in one variable ..... ????? ....... ?????? ...... ??? dirichlet polynomial ...... ?????? ...... ?? co-ring .... ???? ....
?? homming irreducible tensor factor stable hopf algs here into finite fields and / or finite semi-simple comm rings .... ??? ..... ????? ..... ???? add / mult confusion ..... ????? ..... ????? .....
?? "fourier duality proof of quadratic reciprocity" .... ??? .... ????? .... ??? ....
?? lambda .... ... q .... .... ????? .... ???? ....
?? character of perm rep .... trace .... fixed points .... ???? ....
?? motive _as_ quasicoherent sheaf over some sort of "galois stack" ??? ... ???? ...
?? trying to set up "square" here .... ??? ....
?? "set-valued galois rep" .... ???? given by (for example) homming gaussian integers into q-bar .... ???? ..... ?? then decompose this (??2d ...) perm rep into irreps .... ?? perhaps over f_q for some q ??? ... ???? ... ??? hopf alg interpretation ??? .... of ... ??? ... ?? perm rep over f_q .... ???? ....
?? comm ring r st [r,q-bar] = (naturally ... ??? ...) 1 + [z[i],q-bar] + 1 ... ????? .... ??? z X z[i] X z ... ??? .... ?? making this into bistable hopf alg ?? ... ??? then decomposing it wrt tensor product of bistable hopf algs ?? ... ??? ..... ???? ....
?? fourier dual of z/2 ??? ....
?? z[x]/x^2-1 ...... ?????? ....
?? confusion ..... ???? ......
?? z X z X z X z[i] X z[i] X z[i] ..... ???? .....
?? z[x]/x^2+x+1 # ... ???? ....
?? z[x]/x^3-1 # (z X z X z)
?? z[x]/x^3+x # (z X z X z) ...... ???? .... hmmmmm ..... ????? ......
??? f_q as algebraic ring here ??? .... ??? free algebraic module of algebraic ring on algebraic set ??? .... ???? ....
??? possibility of "twisted" version of "f_q as algebraic ring" here ??? .... ???? .... ?? maybe many such ??? .... ??? abelian variety p-torsion .... ???? ..... ??? "complex multiplication" ????? ..... ????? .....
?? f_p^n .... ??? ....
?? algebraic ring "the ring" ..... ??? .... polynomials in one variable ..... ????? ....... ?????? ...... ??? dirichlet polynomial ...... ?????? ...... ?? co-ring .... ???? ....
?? homming irreducible tensor factor stable hopf algs here into finite fields and / or finite semi-simple comm rings .... ??? ..... ????? ..... ???? add / mult confusion ..... ????? ..... ????? .....
Wednesday, April 4, 2012
?? add / mult confusion with zeta / l-function ... ??? possibility of euler summand being key here ??? .... hard to believe will completely take care of it .... ???? characters .... ????? hmmmm, fourier duality here .... twisted vs untwisted .... integers vs roots of unity .... ????? .... ????? ..... ????? .....
Tuesday, April 3, 2012
?? some sort of limit where piston mass approaches zero ... ??? maybe should simultaneously re-scale velocity (?? ...) so that both chambers compress to same extent ... ?? visible from animation how "target" chamber compresses more if starting velocty stays fixed and piston mass goes to zero ... ??? .....
?? sym mon cocontinuous fr from [g,_set_] to _set_ .... ???? and / or to accidental topos (with tensor product ...) .... ??? cocomm comonoid adjunction bit here .... ???? ....
?? left-universal property of [g,_set_] as sym mon cocomplete cat .... ??? relationship to "double coset" / "orientation" logic .... ???? .... ?? "torsor" .... ???? ..... ?? "g-action x st x#x = g-fold sum of x, via certain cocone ..." .... ????? ....
?? adjunction confusion again .... ???? ..... ?? to accidental topos, vs to underlying sym mon cocomplete cat of ag th ... ?? possibly one coming from a tag theory ??? ..... ??? .... ??? revisiting more than one old confusion here ?? ... ?? relationship between them ?? ..... ????? ....... ?? "richer doctrine / poorer environment" (?? ...) confusion .... ??? vs .... ???? .... toric stack confusion ..... ????? ......
?? different concepts of "same theory but in different doctrine" ... ??? .... decategorified analog .... ???? ....... underlying cat / set changes, vs structure (...) on it changes .... ???? .... ???? ..... ????? sounds a bit like theory vs environment ?????? .... ????? .......
??? "toric rigidity" .... ?????? .....
?? left-universal property of [g,_set_] as sym mon cocomplete cat .... ??? relationship to "double coset" / "orientation" logic .... ???? .... ?? "torsor" .... ???? ..... ?? "g-action x st x#x = g-fold sum of x, via certain cocone ..." .... ????? ....
?? adjunction confusion again .... ???? ..... ?? to accidental topos, vs to underlying sym mon cocomplete cat of ag th ... ?? possibly one coming from a tag theory ??? ..... ??? .... ??? revisiting more than one old confusion here ?? ... ?? relationship between them ?? ..... ????? ....... ?? "richer doctrine / poorer environment" (?? ...) confusion .... ??? vs .... ???? .... toric stack confusion ..... ????? ......
?? different concepts of "same theory but in different doctrine" ... ??? .... decategorified analog .... ???? ....... underlying cat / set changes, vs structure (...) on it changes .... ???? .... ???? ..... ????? sounds a bit like theory vs environment ?????? .... ????? .......
??? "toric rigidity" .... ?????? .....
?? classification of symplectic manifolds ....
?? of poisson manifolds ??? .... ?? relationship to moduli stack of symplectic manifolds ..... ???? .....
?? of symplectic manifolds equipped with hamiltonian ....
?? and so forth .... ?? ...
?? vector field ... ??? ...
?? periodic orbits ... ??? of shortest periods ??? .... ?? ... ?? other periods .... ??? ... ??? ....
?? orbit closures .... ??? .....
?? sink , source ... ??? ....
?? somewhat canonical "reduction" associated with ... ?? taking as many conserved quantities as possible .... ??? .... ???? ....
?? of poisson manifolds ??? .... ?? relationship to moduli stack of symplectic manifolds ..... ???? .....
?? of symplectic manifolds equipped with hamiltonian ....
?? and so forth .... ?? ...
?? vector field ... ??? ...
?? periodic orbits ... ??? of shortest periods ??? .... ?? ... ?? other periods .... ??? ... ??? ....
?? orbit closures .... ??? .....
?? sink , source ... ??? ....
?? somewhat canonical "reduction" associated with ... ?? taking as many conserved quantities as possible .... ??? .... ???? ....
prasad asked me about the frontispiece drawing in dixmier's book, attributed to borho ... i said that it had to do with the canonical linear structure on the orbit space of the kaleidoscope group action on the (complexified ...) kaleidoscope .... ?? but to what extent do i really know how "canonical" it is ... ?? mere property of being a polynomial ring, vs structure of having the generators picked out .... ??? ....
?? the criticism that i saw of ash and gross tended to be of exactly the wrong kind, roughly "why bother trying to give a human conceptual account of this material ?" ... whereas i see it as a (somewhat) noble failure which should be criticized only for almost entirely failing to achieve it, rather than for attempting it ...
?? i think that i can tell that they fail at it, despite (or maybe because of ...) my not yet being able to understand a lot of what they're talking about ...
?? i think that i can tell that they fail at it, despite (or maybe because of ...) my not yet being able to understand a lot of what they're talking about ...
?? map from subtoposes of t to sublocales of localeization(t) .... ??? as weird maybe ??? .... ?? example of accidental toposes ??? .... ????? .... ??? non- / surjective, non- / injective .... ???? .... ?? _is_ map clear here ??? ....
?? vs open case .... ????? .... ?? bijective ??? ..... ???? .....
?? confusion about open subtopos of accidental topos .... ???? .... ??? subobject classifier .... ???? ....
?? stuff i sort of almost learned about open subtopos right before / around time of collapse of ideas about categorified bialgebras ... ??? did i already write this somewhere ?? ...
?? vs open case .... ????? .... ?? bijective ??? ..... ???? .....
?? confusion about open subtopos of accidental topos .... ???? .... ??? subobject classifier .... ???? ....
?? stuff i sort of almost learned about open subtopos right before / around time of collapse of ideas about categorified bialgebras ... ??? did i already write this somewhere ?? ...
Monday, April 2, 2012
?? sub-[toric variety] of toric variety as corresponding to arbitrary subtopos of accidental topos, vs open sub-topos ??? .... ??? or maybe for "makkai" topos (???? .... ????) there's no difference ???? ........ ????? ....... ??? what happens when you try to get accidental topos to have subtoposes corresponding to all those of "toric zariski locale" ??? ..... ????? ..... ????? .....
?? bringing out peculiarness of trying to get lattice of all subtoposes (vs just open such ... ??? ...) to form a frame .... ??? though consider idea (?? mentioned in later (...) post, about ... ?? also possible peculiarness of map from subtoposes of t to sublocales of localeization(t) .... ??? ....
?? bringing out peculiarness of trying to get lattice of all subtoposes (vs just open such ... ??? ...) to form a frame .... ??? though consider idea (?? mentioned in later (...) post, about ... ?? also possible peculiarness of map from subtoposes of t to sublocales of localeization(t) .... ??? ....
?? functor assigining to finite field x(?? or semi-simple comm ring ?? ... ???? ..... ??? ....) f_p-vsp (... ?? ... of pth roots of unity in x) ... ?????? ......
?? proof of cyclicness of finite (?? ...) subgp of mult gp of field .. ??? otherwise impossibly many nth roots of unity for some n ??? ... ???? ... ??? .... ?? generalizations ..... ???? .... ???? from 1d vsp to higher-dim ??? ... ????? ... ??? ....
?? proof of cyclicness of finite (?? ...) subgp of mult gp of field .. ??? otherwise impossibly many nth roots of unity for some n ??? ... ???? ... ??? .... ?? generalizations ..... ???? .... ???? from 1d vsp to higher-dim ??? ... ????? ... ??? ....
?? idea that ... ??? ... whole "constant group(oid ?? ...)" part of ag doctrine should survive in tag doctrine as well ??? ...... ???? .... ???? but ... ??? what about "toric rigidity of moduli stack" ??? ... ???? ....
?? "toric convolution" .... ?? confusion about whether that applies in _non_-constant group case as well ..... ??? .... ?? some fourier duality confusion here ??? .... bit about fourier dual of "constant" bistable hopf alg as very un-constant ??? .... ???? ..... ???? "quantum double" here ??? .... ????? .... ???? .... ?? anyway, trying to tie in whole "how fourier duality gets along (/ doesn't ?? ...) with galois action" bit with "how constant groupoids fit into (/don't ?? ...) tag doctrine" .... ????? ..... ???? ... ???? ...
?? lots of (other ?? ...) confusion here ... ??? .... ??? sym mon cat, vs arbitrary cat as "pro-cartesian" .... ???? .....
?? galois group as constant group ....... ???? .....
??? zeta type and l-type .... ??? ....relationship to "lambda operations" and their relationship to galois theory ???? ..... ????? .....
?? "twist by character of ..." .... ????? .....
?? "toric convolution" .... ?? confusion about whether that applies in _non_-constant group case as well ..... ??? .... ?? some fourier duality confusion here ??? .... bit about fourier dual of "constant" bistable hopf alg as very un-constant ??? .... ???? ..... ???? "quantum double" here ??? .... ????? .... ???? .... ?? anyway, trying to tie in whole "how fourier duality gets along (/ doesn't ?? ...) with galois action" bit with "how constant groupoids fit into (/don't ?? ...) tag doctrine" .... ????? ..... ???? ... ???? ...
?? lots of (other ?? ...) confusion here ... ??? .... ??? sym mon cat, vs arbitrary cat as "pro-cartesian" .... ???? .....
?? galois group as constant group ....... ???? .....
??? zeta type and l-type .... ??? ....relationship to "lambda operations" and their relationship to galois theory ???? ..... ????? .....
?? "twist by character of ..." .... ????? .....
?? "helix with longitudinal tails" .... ??? ... ?? farther extremes of tail can unwrap ... ??? ....
?? idea that we seem to be (with toric toposes ...) treating subtoposes as though they form a frame (?? ...) .... "glueing subtoposes together ..." ... ??? .... ?? vs rumors about subtoposes _not_ forming frame in certain (?? which ??) generality ... ??? possibility of nice lesser generality where it does work ?? ... ?? maybe "makkai" toposes ?? ... ??? .... filteredly cocontinuous set-valued functors ... ??? ....
?? localizing from nXn to zxn ... ?? second n just "along for the ride" ... "curry" ... ??? .... ?? hopefully makes it easy to prove glueing works nicely here ... ??? .... punctured plane example ... ??? ....
?? idea that we seem to be (with toric toposes ...) treating subtoposes as though they form a frame (?? ...) .... "glueing subtoposes together ..." ... ??? .... ?? vs rumors about subtoposes _not_ forming frame in certain (?? which ??) generality ... ??? possibility of nice lesser generality where it does work ?? ... ?? maybe "makkai" toposes ?? ... ??? .... filteredly cocontinuous set-valued functors ... ??? ....
?? localizing from nXn to zxn ... ?? second n just "along for the ride" ... "curry" ... ??? .... ?? hopefully makes it easy to prove glueing works nicely here ... ??? .... punctured plane example ... ??? ....
Sunday, April 1, 2012
?? theory of symmetric bimorphism vs of "nondegenerate" such ... ??? .... cups and caps vs bit about partition into pairs on image complement ... ??? .... ?? did baez really omit nondegeneracy condition in "composition alg" bit ?? ... ??? ...
?? morphism being injective map between family of t-structured set, vs morphism being map with t-structure on image complement ..... ???? .....
?? morphism being injective map between family of t-structured set, vs morphism being map with t-structure on image complement ..... ???? .....
?? idea of ... ?? emphasizing geometric doctrine and its "baby fragments" (products theories, lex theories, ...) as analogous to ag doctrine (?? also tag ?? ...) and its "baby fragments" (dimensional theories ... ?? ...) .... ???
but how good a parallel is it, really ? ... ?? particularly, is the right adjoint from _ag theory_ to _dimensional theory" really parallel in its uses (?? ...) to the right adjoints in the other case ??? .... ??? ....
but how good a parallel is it, really ? ... ?? particularly, is the right adjoint from _ag theory_ to _dimensional theory" really parallel in its uses (?? ...) to the right adjoints in the other case ??? .... ??? ....
?? blow-down followed by blow-up ..... ???? asf, os .... ????? .....
?? blow-up of curve in 3-space, vs "knot surgery" ... ???? .... ??? ... ??? also "cover of knot complement branched along knot" .... ???? .... ???? ....
?? "birational geometry as guide to glueing" .... ??? relationship to "divisor as holomorphic structure on standard meromorphic line bundle" .... ??? .... glueing vs dimensional doctrine (and related doctrines ...) .... ??? ...
?? (?? nice ...) structure types carried by meromorphic line bundles (= invertible vsps over field of meromorphic functions ...) ... ??? relationship to "non-toric fan" ??? .... ???? certain kind of dimensional theory over dimensional theory of meromorphic line bundles .... ???? _faithfully_ over ?? ... ??? ...
??? toric case ???? .... ??? sensible approach to "toric divisor" ?????? ... "toric holomorphic structure on standard toric meromorphic line bundle" ??? ..... ???? .....
m1 -> g1
| |
v v
m2 -> g2
m2 = another ab gp ...
g2 = trivial ....
??? simply _bi_-graded (by ab gps ...) comm monoid ??
?? faithfulness ??? .....
????? m1 -> m2 -> g1 ??? .... ?? m2 universal groupization of m1 ???? .... ???? ....
?? some confusion here ??? ... ???? kernel of groupization(m1)->g1 ??? ....
?? relationship between line bundle and toric structure ..... ????? .....
?? m2 = groupization(m1)
?? g2 = g1
.... ???? .....
?? blow-up of curve in 3-space, vs "knot surgery" ... ???? .... ??? ... ??? also "cover of knot complement branched along knot" .... ???? .... ???? ....
?? "birational geometry as guide to glueing" .... ??? relationship to "divisor as holomorphic structure on standard meromorphic line bundle" .... ??? .... glueing vs dimensional doctrine (and related doctrines ...) .... ??? ...
?? (?? nice ...) structure types carried by meromorphic line bundles (= invertible vsps over field of meromorphic functions ...) ... ??? relationship to "non-toric fan" ??? .... ???? certain kind of dimensional theory over dimensional theory of meromorphic line bundles .... ???? _faithfully_ over ?? ... ??? ...
??? toric case ???? .... ??? sensible approach to "toric divisor" ?????? ... "toric holomorphic structure on standard toric meromorphic line bundle" ??? ..... ???? .....
m1 -> g1
| |
v v
m2 -> g2
m2 = another ab gp ...
g2 = trivial ....
??? simply _bi_-graded (by ab gps ...) comm monoid ??
?? faithfulness ??? .....
????? m1 -> m2 -> g1 ??? .... ?? m2 universal groupization of m1 ???? .... ???? ....
?? some confusion here ??? ... ???? kernel of groupization(m1)->g1 ??? ....
?? relationship between line bundle and toric structure ..... ????? .....
?? m2 = groupization(m1)
?? g2 = g1
.... ???? .....
?? what happens to the piston system as the mass of the piston approaches zero ? ... ??? ....
p1 = h1/w12
p2 = h2/(1-w12)
w12'' = (p1-p2)/(piston mass)
h1' = -w12'*p1
h2' = w12'*p2
?? ....
?? pistons as giving interactions between chambers vs vice versa ... poincare duality ... ??? .... ??? property vs structure vs stuff here ?? ... ??? ... ?? "field" ... ?? ... ??? ....
?? constraint vs force vs field ??? ...... ???? ....
?? 1/r potential .... ??? .....
p1 = h1/w12
p2 = h2/(1-w12)
w12'' = (p1-p2)/(piston mass)
h1' = -w12'*p1
h2' = w12'*p2
?? ....
?? pistons as giving interactions between chambers vs vice versa ... poincare duality ... ??? .... ??? property vs structure vs stuff here ?? ... ??? ... ?? "field" ... ?? ... ??? ....
?? constraint vs force vs field ??? ...... ???? ....
?? 1/r potential .... ??? .....
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