blogger is malfunctioning badly enough at the moment that i'll try wordpress instead ...
notebook361.wordpress.com
Monday, May 21, 2012
Thursday, May 17, 2012
Tuesday, May 8, 2012
?? "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 .... ??? ...
?? 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 .... ????? ....
??? 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.
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 ... ??
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."
?? 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."
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
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11
31 3 3,6
21
20
12
10
22 2 2,4 ????????
01
20
211 2 4
1111 1 1,2
?? 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
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20
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02
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12
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1 1
2 2
3 1
4 2 ?
6 1
8 1
4 4 8
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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 ...
?? 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 .... ???? .....
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
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.
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" ... ??? .....
?? "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 ???? ..... ????? ..... ?????? ......
?? "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 ??? .... ???? .....
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?? "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" ...... ?????? ......
?? 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 .... ???? ......
?? 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" ..... ???? ......
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