Monday, May 21, 2012

blogger is malfunctioning badly enough at the moment that i'll try wordpress instead ...

notebook361.wordpress.com

Thursday, May 17, 2012

?? set with lagrange extrapolation structure on it .... ???? .....

???? .......

??? "affine" ??? ......

Tuesday, May 8, 2012

?? lagrange extrapolation as making human god-like ... ??? ...

?? things even god can't do .... ??? ...

?? "galois-equivariant parabolic induction" ...

?? group whose reps re being considered as algebraic .... ?? ... reps as ... ???? ....

??? vs ... ??? "stable hopf alg as galois rep" .... ????? ..... ??? rep as .... ??? gp as ... ??? ....

?? alg vs abstract ... ???? .....

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

?? duality .... ??? ...

Monday, May 7, 2012

?? "q-deformed schur-weyl duality" .... ?? lots of confusion here ... ?? ?? relationship to categorification of quantum gp, but maybe funny level slip vs non-q-deformed schur-weyl duality ??? .... ??? "green subsitution" ...

??? does cuspidal rep give anything interesting in connection with q-deformed quantum gp ..... ????? ....

??? a,b,c table .... ????? ....

[email to jacob lurie]

hi. my name is james dolan; you may have heard of me in connection with the so-called "baez-dolan conjectures". martin brandenburg has told me how helpful and encouraging you've been to him on occasions.

i consider myself more of a teacher than a researcher but these activities are for me more in harmony than in conflict. in this however i find myself at odds against the mathematical community at large, and combined with the fact that writing is for me an absurdly inefficient form of communication (i've never written a math paper), these are the primary reasons for my failure to obtain academic degrees or academic jobs up till now.

(i did obtain a master's degree as a side-effect of an abortive attempt to obtain a phd at suny buffalo, but i never got a bachelor's degree or phd despite some years of trying.)

in moderate financial desperation, last year i wrote a sketch for a phd thesis and contacted david yetter at the math department at kansas state university about the possibility of enrolling as a grad student there to get a phd. yetter was very helpful to me but my negotiations with ksu hit a terminal snag due to bureaucratic requirements that i perform as a teaching assistant under excessively heavy-handed supervision.

(i'd be ok with not teaching at all, and i'd be even happier teaching independently, but as i have very strong ideas about the right way to teach i'd be very unhappy (and likely make others unhappy too) as an over-regulated teaching assistant).

yetter suggested to me that harvard might have more flexibility than ksu with regard to grad student teaching duties and length of residency needed to obtain a phd, and that your familiarity (to say the least) with some of my work might help my chances of getting admitted to your department as a grad student. that was enough to convince me to explore the possibility.

i'm writing to you to see whether you have any opinions/ideas/advice about how practical it might be for me to apply to the harvard math department as a grad student, or about whether there might be any particular more practical alternatives.

the thesis sketch that i wrote is here. it is intentionally very modest and unambitious, in contrast to most of my research which is insanely ambitious beyond my own ability to bring to fruition. i believe that, padded out with verbiage and proofs of the theorems, the sketch would be approximately sufficient for a phd, at least at most math departments.

also here is a brief dicussion of my recent mathematical activities and interests, including how the thesis sketch fits into the picture.

my recent mathematical activities and interests

[some ideas here owe much to collaborators and friends, especially todd trimble, alex hoffnung, and martin brandenburg ... however i won't make any particular effort to give more detail about such intellectual debts here...]

much of my work in recent years has centered around algebraic geometry, a subject that i was largely ignorant of until a certain realization struck me a few years ago.

previously i'd had the rough impression that algebraic geometry is more or less the study of the spectrums of commutative rings, except that for some reason these objects, the affine schemes, are not "complete" enough, and so more general schemes are glued together from the affine ones, analogous to how manifolds are glued together from coordinate patches. i didn't find this story convincing or interesting. (i may not have been exposed to v i arnold's rants against the "glueing" approach to manifolds and schemes by that time, but if i had then i likely would have sympathized with them to a certain extent.)

then in some elementary introduction to algebraic geometry i encountered the slogan that "projective n-space is the classifying space for line bundles with n+1 linearly independent holomorphic sections".

this puzzled me at first, not so much because i couldn't understand it as because it seemed like a warped version of something that i _could_ understand, but from a different branch of mathematics, namely algebraic topology (where the language of "classifying spaces" is a way of talking about the universal properties of objects in the (infinity,1)-category of (convenient) spaces).

i put the puzzle aside for a while, but as i learned more about algebraic geometry the true meaning of the slogan suddenly struck me: the category qcoh(p(v)) of quasicoherent sheaves over the projective space p(v) of a vector space v is the free "good tensor category" on a "line object" l equipped with a "good embedding" e : l -> v.

i didn't have precise meanings for the concepts inside quotation marks yet, but the basic conceptual picture was clear: p(v) is intuitively the space of 1-dimensional subspaces of v, and the universal property of qcoh(p(v)) is a direct translation of this intuitive characterization of p(v) into the language of good tensor categories. thus a point of p(v) "over" a good tensor category x is a homomorphism of good tensor categories from qcoh(p(v)) into x, which amounts to "a line object l in x equipped with a good embedding e : l -> [1_x]#v", or in other words "a 1-dimensional subspace of v, internal to x".

(here "[1_x]#v" is the unit object 1_x of x tensored with the external vector space v, or in other words "the x-internalization of v".)

the philosophy that suggests itself here is that categories possessing an appropriate formal algebraic structure (in this case the "good tensor categories" such as the category of quasicoherent sheaves over a nice scheme, or of representations of a nice group) give us a language in which the intuitive geometric ideas that we wish to study should be clearly and directly expressible. we write down a "theory" (that is, a good tensor category or "categorified presentation" thereof) , and voila, the moduli stack of models of that theory is brought into existence as an algebraico-geometric object. no laborious construction is involved, you just say what you want and you get it, because the formalism is good. conversely, any reasonable "scheme" or "stack" is seen as the moduli stack of _something_, namely of models of the theory given by the good tensor category of quasicoherent sheaves over it.

i see this philosophy as a merging of two big research programs:

1 the program of "categorical logic" (initially developed especially by bill lawvere with influences from jon beck), where categories possessing a particular sort of formal algebraic structure are said to form a "doctrine", and a particular such category t is said to be a "theory" of the doctrine, the structure-preserving functors from t to some other such category t' being considered as "models" of t in the "universe" provided by t'.

2 the "tannakian" program of algebraic geometry, where the primary objects of study are stacks over some sort of algebraico-geometric site, but an equivalence (a "gabriel-rosenberg theorem") is sought with tensor categories of some sort.

it seemed to me that the synergy implicit in the merging of these two programs was an intriguing potential that hadn't been sufficiently exploited, and i set out to exploit it. the rest of this discussion is primarily a description of my attempts in this direction.

1 concerning my original motivating example of the projective space p(v) of a vector space v as a moduli stack, i found out that martin brandenburg was also interested in clarifying the left-universal property of
the tensor category qcoh(p(v)), though from a somewhat different viewpoint than mine.

(brandenburg's interest in the question made it more plausible to me that it was not already completely resolved, and might thus be of interest beyond just purposes of my own education.)

i was naive enough at that point to suggest that in fleshing out the statement "qcoh(p(v)) is the free good tensor category on a line object l equipped with a good embedding e : l -> v", it would be sufficient to use the follwing definitions:

1 "good tensor category" = symmetric monoidal cocomplete category enriched in vector spaces over a ground field k;

2 "line object" = invertible object with trivial self-braiding;

3 "good embedding" between dualizable objects = morphism whose mate is epi.

brandenburg, however, suggested reasons why this conjecture of mine was unlikely to be correct, which i was able to confirm by showing that ...

in response to brandenburg's demolition of my first conjecture i formulated a second one: to take a "good embedding" between dualizable objects to be a morphism which is the mate of the cokernel of the mate of its cokernel.

(i see this property as morally a sort of "regular epi" making sense in tensor context ... ??? ... ??? ....)

brandenburg told me that after much work he proved a version of this conjecture. however ... ??? .... ??? still interested in original conjecture .... ???? shouldn't be too hard to resolve but i didn't get too much of a chance to talk it over with anyone .... ??? ....


?? tension between brandenburg pursuing true statements and me pursuing good ones .... ????? ...

?? weak pushout of ag theories asf .... ???? ....

?? ag theories that are "non-abelian" ... ??? asf .... ??? ....


2 ?? zariski topos ??

i thought about trying to understand the "zariski topos" of a scheme or stack x (?? as well as other grothendieck toposes associated to x ...) using my philosophy of algebraic geometry ... my conjecture is that zt(x) can nicely be thought of as _the same theory_ embodied by x, but expressed in a different linguistic formalism ...

?? go into lawvere's ideas to significant extent here ....

?? "boilerplate" idea .... ??? ....

3 ?? dimensional theories ??

an interesting "subdoctrine" of the doctrine of ag theories is given by the "dimensional theories". a dimensional theory is a symmetric monoidal category enriched in vector spaces over a ground field k, where every object is a line object. i call them "dimensional theories" because they can be thought of as rudimentary scientific theories expressed according to the rules of "dimensional analysis"; an object being a "dimension" ("line object" = "1-dimensional object" = "dimension") and a morpism x->y being a quantity in dimesiion y/x. this relates interestingly to lawvere's ideas about "theories" and in fact i have a vague memory of lawvere talking about dimensional theories (not under that name) in a lecture long ago. i've written to lawvere trying to obtain information about this but didn't recieve a response.

on the other hand, dimensional theories are essentially the "multi-homogeneous coordinate algebras" of projective algebraic geometry. the correspondence between "generalized projective varieties" and the dimensional theories obtained by taking line bundles over them and sections of those line bundles can be thought of as a baby version of the "tannakian correspondence" of algebraic geometry.

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

?? relationship between 1 and 3 .... ??? ....


4 ?? toric geometry ??? ...

one way to think about toric varieties is that they are algebraic varieties built using the category of sets as a foundation instead of the category of abelian groups (or vector spaces ...); thus for example where an ordinary affine variety corresponds to a commutative monoid in the category of abelian gorups an affine toric variety corresponds to a commutative monoid in the category of sets. many aspects of algebraic geometry become especially simple in the toric context because problems of finite-dimensional linear algebra are replaced by problems of finite combinatorics; thus toric geometry serves sometimes as an experimental toy universe for exploring aspects of algebraic geometry which might otherwise be annoyingly difficult of intractable.

i decided to apply this "experimental toy universe" status of toric geometry to the understanding of the "tannakian correspondence" between algebraic schemes and/or stacks and tensor categories of a certain kind. besides the advantage described above, some other pleasantly amusing things happen here, in that abelian categories get replaced by grothendieck toposes, in which i have an independent (?? ....) interest. the amusement value here may be compensated to some extent by potential confusions ... but that's part of the amusement ... ?? and/or, maybe there _is_ something deeper and interesting going here ... ??? ....

this is what i wrote a sketch of a phd thesis about ....

?? infinity-topos ????? ...... ???? .... derived ..... ???? ?? as achieving some sort of perfection not available at lower levels .... ???? ..... and/or as mysterious .... ???? ....


5 ?? modular forms ...

the graded commutative algebra of modular forms (freely generated by a generator x in grade 2 and
another generator y in grade 3) can be thought of as an example of a dimensional algebra or dimensional theory. (see section 3 above.) evidently a model of this theory is "an invertible object equipped with a polynomial function with just quadratic and cubic terms". because of the connection of modular forms to elliptic curves, we should expect that such models are (roughly) equivalent to elliptic curves, and in fact it's clear how this works, roughly: the projective completion of the 1d vector space l has the point at infinity as a special point, plus the special points arising as the zeros of the cubic polynomial function; the double cover with these 4 branch points is the elliptic curve. from the elliptic curve with its identity element marked, l can be recovered as ... ??? ....

(?? ... connections with dimensional theories here ... ?? ....)

6 woolf ... thom spectrum ... ??? ...

i'm interested in the work of jonathan woolf on formalizing the idea suggested by baez-dolan of obtaining an infinity-category from a stratified space by starting with the fundamental infinity-groupoid and taking a certain sub-infinity-category the n-morphisms in which are characterized by a certain transversality property relative to the stratification ...

baez-dolan saw this as a generalization of baez-dolan-lurie idea .... to more general thom spectrums .... i'm interested in pursuing this .... ??? ag ideas ??????????? ..... thom .... ????? .....

?? though also how this relates to other ideas about ... stratification ... transversality .... ????
cohomology ??? ??? buzzword i'm looking for ??? ....

?? perverse ??? .... ?? not quite, but .... ????? .....

?? middle perversity ..... ??? ......

?? two cultures ...

?? overlap between the two programs, but ... not as fully merged as might be optimal for further progress ... so this became a big personal program of my own .... ?? exploitation and evangelization of merging of these two programs .... ??? .....

??? link bit about [?? my personal emphasis on "syntactic" side ... ??? ...] to bit about [?? phoniness of pretense to "geometric"ness ... generational ... ??? ...] ?? ....


?? homomorphism of good tensor categories ... ?? ...

?? familiarity of this philosophy .... lawvere, beck, doctrines ..... ????? ....

?? but also ... my intro to "tannakian" philosophy .... ??? relationship between these ... two cultures .... ??? ....

?? conceptual level slips here ??? .... ?? one person's "intuitive geometric ideas" ... ?? another's "appropriate formal algebraic structure" .... ??? ....

?? propositional vs predicate logic here .... affine vs non-affine .... ????? .... ??? "moore-postnikov factorization" ....


?? each generation of algebraic geometers as going through motions of claiming current version of algebraic geometric as "geometric" by taking advantage of formal (?? ...) semantics determined by syntax ..... ????? ......


?? generalization, abstraction, exhaustive systematization .... ??? vs (?? ...) fitting of interesting stuff into picture .... ????? .... ??? working form interesting stuff end vs from exhaustive picture end .... ?????? ....


?? good for me to have collaborators who read outside stuff .... ??? ...

?? even though (??? .....) i myself experience alienation .... ?? may be able to help others become less alienated ..... ???? .... ????? .....

?? doctrines ...

?? dimensional theories ... renormalization ...

?? toric quasicoherent sheaves ...

?? kummer's chemistry analogy .... ???? "downward decorrelation" .... ??? ....

?? "stacks vs infinitesimal stacks" .... ??? .... ?? rational htpy theory ....

?? woolf ... thom ..... ??? .....

?? zariski topos .... ?? "boilerplate" idea .... ???? .... ?? divorce ... ??? ... ?? software ... ??? ....

?? g2 ....

?? langlands ...

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

?? "moduli stack" ... modular forms .... ??? .... ??? hilbert scheme ... ?? other moduli stacks ... higher genus curves asf ... ???? "hodge structure" ??? .... ??? ...

?? flag geometry ... singularities of schubert varieties ... invariant distributions .... ???? ....

?? "logic" .... ??? .....

?? kaleidoscope fan .... ???? ....

?? physics ..... ???? ..... ?? string theory ... ??? mirror symmetry ?? ....

??? teaching ???? .....

?? i hope that this discussion ... ??? sample of some of my recent work .... ???? ... ... ?? ... gives some sense of unifying theme ... ?? even though haven't given much detail ... ?? ... ?? and / or connecting thread(s?? ...) .... ???? .....

?? lots of other stuff ... ??? ...

?? amateurishness ... naive ... isolated ??? .... ??? ... teaching ... ??? to other mathematicians ... ?? especially category-theorists .... ??? ....

?? teaching .... ??? baez papers as distorted / corrupted lesson plans for course of instruction unfortunately never taught yet ..... ??? ....

?? my interest in teaching continues ... the papers written by john baez under my influence (including ones where i'm listed as co-author but others as well) are, roughly, distorted and corrupted fragments of lesson plans for a vast course of instruction in areas such as mathematics, computer science, physics, ... which unfortunately i've had only very little chance to try out on actual students. last summer however i taught an informal course on galois theory and related ideas for a small group of advanced high school students ....

?? intersection co-/homology ...

?? locally presentable .... ??? ....

?? lex .... ???? .... topos .... ???? .... categorify ??? .....

?? string ... ??? ....

?? ultimate outsider ... ??

Sunday, May 6, 2012

?? is "whittaker model" idea somehow how something like "automorphic rep" and "automorphic form" get conflated ??? .... ??? .... ?? well, but if so then phenomenon mentioned of "degenerate" rep without whittaker model seems very annoying .... ??? ...

?? vaguely reminding me of how multihomogeneous coordinate algebra of flag variety can be thought of as direct sum of the irreps ..... ???? ....


"The motivation for this local conjecture comes largely from global considerations. We review the theory of Artin L-functions from a more sophisticated standpoint than in Section 1.8. In that section, we considered the theory of Artin L-functions attached to Galois representations. There is another theory of L-functions, namely, the Hecke-Tate theory, which we considered in Section 3.1, and there is some overlap between the theories of these two classes. Let F be a global field and let A be its adele ring, and let p : Gal(F-bar/F) -> GL(1, C) = C^* be a one-dimensional representation. Then p factors through Gal(F^ab/F), where F^ab is the maximal Abelian extension, and the composition of p with the reciprocity law homomorphism A^*/F^* -> Gal(F^ab/F) gives us a Hecke character whose L-function agrees with the Artin L-function of p.

Thus L-functions of some Hecke characters are also Artin L-functions. Not all are, however, because it follows from the "no small subgroups" argument (Exercise 3.1.1(a)) that the image of any continuous homomorphism p : Gal(F-bar/F) -> C^* is necessarily finite. Thus any Hecke character of infinite order, such as the ones we employed in Section 1.9, does not arise in this way. We see that although the two theories overlap, neither theory is contained in the other.

In order to obtain a theory that subsumes both these overlapping theories in a unified framework, Weil (1951) introduced a topological group W_F called the (absolute) Weil group of F, which is a substitute for the Galois group."

???? ....

"The result of Langlands is stated precisely in Tate (1979) Theorem 3.4.1, Deligne (1973b) Theorem 4.1, or Langlands (1970a) Theorem 1. If Fv is a local field, and if pv : WFv ->� GL(n, C) is a representation, then there are defined local L- and e-factors L(s, pv) and e(s, pv, fa), the latter depending on the choice of an additive character fa. These must satisfy certain axioms, the most important of which is a compatibility with induction. If pv is one dimensional, then in view of the isomorphism Wp? = F*, pv is essentially a quasicharacter of Fvx, and L(s, pv) and �(s, pV9fa) agree with the Tate factors that were defined in Section 3.1. The consistency of the conditions that �(s, pv,fa) must satisfy is by no means obvious because some representations might be expressible as linear combinations of representations induced from one-dimensional characters in more than one way, and when this occurs, an identity between Gauss sums must be satisfied. This consistency is the content of the result of Langlands."

???? ....


"The local Langlands conjecture asserts that if F is a local field and p : WF -> GL(n,C) is an irreducible representation, then there exists a supercuspidal representation pi of GL(n, F) whose L- and e-factors agree with those of p. (If n = 2, we defined these L- and e-factors in Section 3.5 and Section 4.7.) It is assumed that if x is a quasicharacter of F^* , then L(s, p) = L{s, x) and e(s, x#p, pitchfork) = e(s, pi, x, pitchfork). (We are suppressing the subscripts v from the notation, of course.) It is a consequence of Proposition 4.7.6 that at most one representation n can have this characterization. This representation is denoted pi(p).

The local Langlands conjecture thus asserts that the supercuspidal representations of GL(n, F), where F is anon-Archimedean local field, are in bijection with the irreducible n-dimensional representations of W_F- We can expand the scope of the conjecture to include all irreducible admissible representations of GL(n, F), which are in rough bijection with all semisimple representations of W_F, including the reducible ones - for example, if n = 2, the principal series representations are parametrized by the pairs of quasicharacters of F^* = C_F, which correspond to representations of W_F that are the direct sum of a pair of quasicharacters. One must be somewhat careful because this scheme does not account for the special representations. To obtain a precise bijection, one employs not the Weil group,but a slightly larger group, the Weil-Deligne group (Deligne (1973b), Tate (1979) and Borel (1979)). We avoid this issue by considering only supercuspidal representations."

?? our third term as "principal series" ??? ... ??? ....

"Analogous intertwining operators occur in the theory of GL(2, F) when F is local; their introduction may be motivated by the theory of the constant terms of Eisenstein series." .... ????? ....

Saturday, May 5, 2012

?? so ... ?? vague idea now is that what bump more or less means is ... ?? related to how joyal and street describe the rep theory of gl(n,f_q) ... in that ... the "cuspidal content" of an irrep in the joyal-street picture corresponds to the "maximal torus together with character" in bump's picture ... ?? ...

?? issue of ... ?? what bump means by "maximal torus" ... ?? extent to which intended concept can be / is / should be seen as purely "abstract group"-theoretic ... vs "alg gp"-theoretic in some sense .... ???? .... ?? seems more like the alg alternative, though in some ways that seems surprising / weird ... ??? elements contained in no maximal torus ... ??? elt of order 3 in gl(2,f_3), for example .... ?? 8 conjugacy classes here ... ??? 3 conjugacy classes in split maximal torus .... 11,21,22 ...

4, 31, 22, 211, 1111

?? 1111 splits in 2 ??? ....

?? 31 also splits in 2 ?? ...

?? what else ?? .... ?? maybe the even ones split in 2 ??? ...

01
21

21
20

20
02

02
12

12
10

10
01


1 1

2 2

3 1

4 2 ?

6 1

8 1


4 4 8

12
11

31 3 3,6

21
20

12
10


22 2 2,4 ????????

01
20

211 2 4
1111 1 1,2
?? "galois-equivariant parabolic induction" ...

?? real / complex case .... ???? ...

?? line bundle over complex projective line (aka riemann sphere) ... real projective line as equator of riemann sphere ... pole-reversal galois action ... ???? equivariant sections ???? .... ???? ... ??? ....

?? vs .... sections over equator ... ??? .....

?? numerology of f_3 case ???? .... ???? .....

??? gl(n,c) as limiting (????? ....) case of gl(n,f_q) .... ????? ..... a,b,c, table ..... ???? .... ???? ....

?? .... categorified gram-schmidt ... kazhdan-lusztig .... ??? .... ???? ......

?? ... verma module ...

?? unification between :

1 grinding out construction of given irrep of simple (?? ...) alg gp (?? ...) via "schur functor" .... ???? ..... ???? .... ag doctrine .... ???? ....

2 constructing that irrep as holomorphic (?? ...) sections of hopefully somewhat evident line bundle over flag variety .... ??? .... taking very conceptually slick "algebraic" approach to "holomorphic section" here .... ????? ......

?? example ....

??irrep of gl(2) .... functor assigning to 2d vsp v new vsp v' , obtained by ... ???.... p(v) as (more or less ...) dimensional theory ... certain line object in that theory ..... ???? .... sections thereof ..... ????? .... ??? ....


?? idea that "getting hold of not-quite-irrep instead of irrep is sometimes almost as good" (?? ...) vs ... ?? idea that it's too vacuous, because for example "cayley rep contains everything so you could just declare victory and quit right there" ... ...??? so, first idea here needs some refinement ... ?? getting hold of irrep, up to it being "categorified leading term" in non-irrep ... ??? various ways of trying to formalize "leading" here .... ???? ....

?? applying schur fr to cayley rep of ... ??? finite gp ??? ... other sort of gp ??? .... ???? ....

?? hmm, maybe the unification above is really just that of "holomorphic sections of line bundle over flag variety" idea with "grade (?? wrt "highest weight" grading rather than wrt "weight" grading ?? ...) of multihomogeneous coordinate algebra of flag variety" idea ..... ??? .... ?? not sure how close to thinking this outloud we've come before .... ??? ...

??? bott ... ?? riemann-roch .... ???? .....

Friday, May 4, 2012

?? baez mentions theorem "every projective variety is a quiver grassmanian" ... ?? vague "mckay correspondence" flavor to proof ?? ...

?? not sure .... ??? ....

Thursday, May 3, 2012

[email to david yetter]

hi. i'd like to thank you for trying to help me get admitted to ksu, but it looks like it's not going to work out.

i hope that you understand why it makes no sense for me to accept the conditions that i've been offered: i know that teaching in a coordinated setting is a guarantee of failure for me, and i also know that under more reasonable conditions i can prove that i can be a competent and successful teacher. it's no good for me to be promised to be allowed to teach independently contingent upon first demonstrating that i can teach competently working under a coordinator, when i know that i can't teach competently working under a coordinator.

the really absurd thing for me here is this:

when i was gathering references from people as part of the application process, a number of people mentioned to me that some citation index recorded a comparatively impressive number of citations of "my" papers. i barely know what a citation index is, and i presume that the papers in question are mainly ones actually written by john baez, for which i somehow got listed as co-author. without having carefully read those papers, i imagine that they give only a distorted impression of my actual ideas. nevertheless, i believe that those papers were often cited in significant part because of their expository qualities, and that those expository qualities reflect my contribution to a significant extent. so in a way, my expository abilities have been recognized by the mathematics community- and yet i've been in effect blacklisted from teaching for the last twenty-five years, when the primary business of teaching is exposition, the explanation of ideas.

could it be that it actually makes sense to give someone who's demonstrated an unusual expository ability a chance to teach in the classroom, when that's what they really want ???

it would be funny if it weren't tragic. hell, it's funny even though it _is_ tragic.

of course i don't mean to deny baez's own contributions to the exposition in those papers; in fact he wrote those papers single-handedly, usually rejecting any explicit attempt that i made to shape the narrative. but what he wrote about was often ideas that he learned from me. often he gave people the impression that i was the inventor of those ideas and that he was the expositor, but in those cases i was often actually telling him about ideas that he'd already been exposed to without understanding them, explaining them to him in a way that he could understand, and that he could then adapt to explain to other people via his natural medium of exposition which is writing. my own natural medium of exposition, in contrast, is standing up in front of a bunch of people and talking to them, which coincidentally happens to be what classroom teaching is like.

if you have any suggestions about how to convince gpac to change their minds and to agree to a fairer compromise, then i'd be interested to hear them. i expect that it's too late for fall 2012, but i may still be available after that and i'm still interested in ksu as one of my last best hopes.

i've arranged for teaching evaluation forms to be sent to the five students from the course on galois theory and related ideas that i taught last summer. i don't know whether any of the forms will actually be turned in, or what they might say. it was the first time in about 25 years that i had anything close to a real teaching job, and i felt very stiff and out of practice from decades of involuntary unemployment. it's unrealistic for me to expect to be able to return to the classroom and immediately be as good as i was 25 years ago, but with regular practice i think that i can get back to that level.

as for teaching evaluations from earlier teaching experiences, i've asked the schools involved for copies of such evaluations if they exist, but no evidence has turned up that any such evaluations still exist.

i would be open to the possibility of enrolling at ksu without having to teach at all, if such an arrangement could be made practical. i would even be open to the possibility of teaching under a coordinator if i wouldn't be penalized for my inevitable poor performance in such a circumstance; such a possibility however would make even less sense than what's actually been offered to me.

the only reasonable and fair possibility is for me to be allowed to teach independently, perhaps after first demonstrating my teaching ability by means of a "summer tryout" as i've suggested. there's even less excuse for being unable to arrange such a tryout now that the time frame for it is shifting from summer 2012 to summer 2013.

i am not open to the possibility of teaching under a coordinator if i will be penalized for my inevitable poor performance under such a circumstance.

Tuesday, May 1, 2012

??? "galois-equivariant parabolic induction" ??? ... ??? to what extent does this make sense, and to what extent does it realize vague idea that i was trying to get at with "split, parabolically induct, de-split" ?? .... ??? ....

?? "de-split" by taking galois-invariant part ... ??? ... ???? .....

?? real/complex case .... ??? .... "cusp ..." .... ???? .... ???? .....

?? "cohomological induction" ... ??? .....

?? "finite fields from aperiodic necklaces" .... ???? finite semi-simple commutative rings from just plain necklaces ??? .....

?? "fixing the frobenius automorphism" here .... ???? ....

?? zeta vs l here ..... ?????? ..... ?? "frobenius" .... mystical ..... ???? ..... ?? "the frobenius" ... ?? level slip .... ?? "formal ..." .... ????? .....

?? natural dual pairing vs natural bijection ??? .... ????? .....

??? ag theory given by modules of group z ... (and / or quotient groups thereof ... ?? ...) .... ??? vs (?? ...) one given by modules of number ring ..... ???? .... ??? interaction ???? ..... ????? ..... ?????? ......

Monday, April 30, 2012

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" ...... ?????? ......
?? 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 .... ???? ......

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" ..... ???? ......

?? "split case" =?= "parabolic induction" ??? .... ???? ....

?? extend, then induce ... ?? ....

?? unsplit case ... ???? .....

?? structure preserved by unsplit maximal torus .... ???? .....
?? when unsplit maximal torus splits, keeping track of what happens (??? under induction ??? .... ??? ....) to those irreps which corresponded to characters of it .... ???? .....

?? "algebraic representation of algebraic group" ??? .... ???? .... .... ???? ......

?? what is given maximal torus doing while particular irrep is being parameterized by character of _other_ kind of maximal torus ??? .....

?? long axises of cube as forming f_3 projective line .... ??? some (?? ...) axises of dodecahedron as forming projective line over f_5 ?? ...

?? but gl(2,f_5) has 10+8+6 = 24 irreps, whereas e8 mckay group has 8 ?? .... ???? .....

?? interpretation of paths in ade graphs under mckay correspondence as "clebsch-gordan terms" ?? ... ??? .....
?? "t-genic ag theory" .... ?? ....in mckay correspondence situation ... ??? ... ?? in situation of trying to prove nice universal property of ag theory given syntactically by quasicoherent sheaves over projective space of v .... ??? .... ???? ....

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

?? seems like there really should be some sort of "kaleidoscope-folding" here because ... ?? bump says "maximal toruses" but seeming to mean conjugacy (???? ....) classes thereof ..... ???? ....

gl(2,f_3) .... ???? .....

?? maximal toruses ....

?? might help if group here has some semi-familiar secret identity .... ???? .....

?? 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" .... ??? .....

?? "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" ... ??? used to know something about something like that in lie case ... ?? ....

?? left-universal property of _fd f_q-vsp_ as braided monoidal finitely cocomplete algebroid .... ???? ..... ????? .....

?? "green substitution" .... ????? .....



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 ....

?? "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 .... ???? ......

?? "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 .... ????? ......

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" .... ???? .....

?? 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" ... ??? .....

Friday, April 27, 2012

?? galois action on cuspidal reps ..... ???? ..... ???? generalizing gl(1) case .... ???? .... ???? ..... ???? ....

?? with numerology of gl(n,f_q) reps apparently working out to some extent now, perhaps good to try more categorified approach in n=2 case .... ???? .....

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

?? "categorification" ...

?? "quantization" ...

?? "integration" ... ?? ...

?? 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 ... ???? ....
?? testing to try to get some sense of how badly broken new editor options might be ... 1 2 3 4 a b c d xxxxx xxxxx xxx xxxxxxx

Tuesday, April 24, 2012

?? exploiting coincidences (?? ...) in setting up "a,b,c" table .... ??? ...

?? bump ... "split vs unsplit torus" .... ???? 2-box vs 1-box young diagram ... parabolic induction from cuspidals .... ???? ....
?? e c comics ... quantum suicide ... anthropic principle ... stephen hawking .... ???? .....

Sunday, April 22, 2012

?? "scratch" ??? ...

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

???? ......

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 ..." .... ??? .....
?? "c-deformed gl(a,f_b)" .... ??? which portions of a,b,c space make sense ?? .... ?? what portions does "schur-weyl duality" connect to each other ??? .... ?? b,c as both "q" variables ??? .... ????? ......

q prime power vs q root of unity ..... ????? .....

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 .... ???? ......
?? "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 ..." ??? ...

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 ....
?? "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 .... ??? .....

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" .... ???? ....
?? "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 ??? .... ???? .... ?? ....
?? 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" ??? ... ???? ... ???? .....

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 .... ???? ....
?? "testing" conjecture about geometric morphisms between accidental toposes using case of 0d toric variety and ... ??? ideas about cofan cocones as classical models here ... ??? .... ?? to what extent did we already od this ??? .... ??? seems like maybe it works out ok ??? ....
?? notebook-y-4 .... ?? galois gp associated with "moduli field" of ec ... ??? as "related to sl(2,z) .... ??? ... ??? was going to try to suggest something here about "adele", but .... ???? confusion ??? ....

?? many ec's ... ???? .....
?? "langlands correspondence" as motivated by role in stating generalized version of "langlands conjecture" ?? ... ???? ....
?? "symbol" .... ?? "fake" ... ??? "fake higher-order tangent" ... ??? ....
?? 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 .... ??? ...

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 .... ??? ...
??? 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 ?? ... ??? .....
?? 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 ..... ???? .....
?? 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 .... ??? ....
?? 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" .... ??? ......
?? "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 ???? .... ????? ..... ???
y^2 = x^3 + g2*x + g3


?? 0 = x^3 + g2*x + g3 ..... ???? ....

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

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

?? shafarevich .... ??? .....

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 ... ??? ....

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 .... ???? ....
?? "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

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

Thursday, April 12, 2012

?? in ag theory of (...) graded modules of gca of modular forms, is there nice gcm for "the" ec ??? .... ??? .... ??? .....

?? aut gp of gca for particular ec ??? ....

?? some confusion about .... ??? "graded ideal" here ... ?? modding out by such, vs also inverting ... ??? ...... ??? ....
?? 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 ?? ... ??? ....

Wednesday, April 11, 2012

?? dimensional analysis interpretation of riemann-roch ?? ...
?? instead of trying to translate nice structure on ec into nice structure on cqa, try going other way ??? .... ??? ....

?? where structure may degenerate ... branch points .... ??? ....

?? pure structure ... no property part .... ???? ...

?? "congruence subgp" .... ??? .....

?? ....
?? 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 .... ?? ....
?? 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" .... ???? .....

???? .......