[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 ... ??
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