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Today I corrected the quantifiers in coherent sheaf, the very definition, thanks Qing Liu (the author of a great book on algebraic geometry, in many aspects better than Hartshorne's) who spoke loudly at Math Overflow that nlab has it wrong.
Thanks, Zoran.
By the way, concerning the entry five lemma: it looks like no other entry links to this? We should try to ensure that new entries are connected to the web of other entries. Here may a link from homological algebra is indicated.
Do you have a link to that MO discussion?
Hey, Zoran, I know that this isn't directly relevant to this thread, but I appreciate what you did for me. Thanks.
http://mathoverflow.net/questions/15223/what-are-the-relationship-between-various-definitions-for-quasi-coherent-sheaves
Liu has put the 3rd comment to the very question (not to the answers).
Two additions at Jacobson ideal.
At D-module, Grothendieck connection and some other places I put additional references (including one book at perverse sheaf). Special attention to very neat notes pdf from Gaitsgory on crystals and D-modules.
I have added yesterday some references and/or links at derivator, characteristic class, Morse lemma, D-module, regular differential operator. New entries algebraic Lefschetz formula, jet space.
References (corrections and additions) at Lie-Rinehart pair.
I am now at IHES for a month hopefully (some administrative problems occured at home and I may be forced to make an emergency trip back: cross your fingers for me not to have to do it!).
Gerd Faltings, stable bundle and moduli space of bundles. Please do not contaminate it with tables of contents as long as it does not have much material to be lost in.
New entry Quot scheme.
Typing query at the bottom of quantum mechanics.
Albert Schwarz of AKSZ model and of Morse theory (I moved the sentence on founders from concept page Morse function to the subject page Morse theory).
PROBLEM: trying to edit BV theory (BTW Urs and I agreed before that BV formalism is a better title with full names redirects like Batalin-Vilkovisky formalism) redirects to HomePage ! However we have a small separate entry Batalin-Vilkovisky quantization with a redirect BV quantization.
I wanted to add reference
to BV theory and to rename the entry.
Now I see we have also a third entry, much bigger (though less cited within the nlab!) BV-BRST formalism which is I guess the central entry on the subject. I DISAGREE with the title. BRST formalism is much older than BV formalism and amount to a special case when the singularities are not that bad. Most of the specialists still do distinguish BRST as a simpler older formalism and BV as more general and later formalism. BRST has ghost fields, BV has in addition antifields and antighosts. BRST is sort of a resolution, BV is a resolution which in addition has doubling a dimension to simulate complex dimension and residue computations of the original problem, so it is inherently complex geometric. BRST is sort of Koszul, BV is lind of a Koszul with supersymmetry aspect.
PROBLEM: trying to edit BV theory (BTW Urs and I agreed before that BV formalism is a better title
Yes, we had already changed this to BV-BRST formalism. I don't know why the old page "BV-theory" still displays. That's the cache bug, I think. That page shouldn't really exist anymore.
BRST formalism is much older than BV formalism and amount to a special case when the singularities are not that bad.
BRST is concerned with dg-algebras just in non-negative degrees, while BV-BRST add the negative degrees. Geometrically speaking, BRST is about oo-Lie algebroids (parameterized on ordinary algebras), while BV-BRST is about derived oo-Lie algebroids (parameterized over simplicial algebras/ negatively graded cochain dg-algebras).
So I think BRST should be discussed in the context of the more general BV.
It could be, one can always do a notion as a special case of more general notion, but the clear distinction should be made explicit, the entry seems to consider the two formalisms synonyms. It is like making an entry
and never pointing out that a scheme and algebraic stack are not the same level of generality.
BRST algebra, BRST identities etc. are much simpler than the general case and have some representations which could not be done for the BV case. There are people who never need the far more complicated case of BV. Our entry mainly concentrates on easier 0-dim case so the sharp disproportion in complexity is not that manifest.
It would be nice also to understand also the relations between Hodge theory and the mathematics of BV in works of Sergei Barannikov, partly based on his collaboration with Kontsevich.
Right, I agree. So we should create an entry on BRST formalism. But I am too busy right now with oo-adjunctions...
No need to hurry, we should just be coherent together into plans of future nlab architecture :)
Jim had some remark on BFV versus BV and promised to expand later on this but never did (in the BRST-BV quantization entry). Could somebody remind him ?
quadratic differential on the to-do list; frame created though
Created Christophe Soulé wanted at contributors to algebraic geometry with a redirect Christophe Soule.
Only one ’r’ in ’Terence’.
I corrected by changing the name to Terence Tao and leaving redirect Terrence Tao. But depite the redirect in good page, the rr-link opens the old page with rr in the text. There is some bug here…
It's the Dreaded Batter Pudding Hurler.
I mean, it's the Infamous Cache bug. I've deleted the cached page so it redirects as expected.
Oh Cache only ? Than it is a matter of waiting few days and clicking fingers and reciting mantras (with some good ceremonial wine at hand) to disappear… If so I drop my charges, and go for a bottle :)
The mantra to recite is:
Lab elf, lab elf, please take my cache.
The serious point is that, at the moment, when there's a page that didn't get deleted from the cache but should have been, then some needs to go in an manually delete it. To do that, we need to know what they are so please keep mentioning them.
It may be off-topic but I was surprised how the list of references and external links to the entry “category theory” in wikipedia is not up-to-date. But with their formats of citing it was too time consuming for me last night to fully update it so I added just the books of Lurie (HTT), Simpson, Kashiwara-Schapira and Bucur-Deleanu among the category books, and of course the links to nlab and Joyal”s Catlab in the external links.
Federigo Enriques, required at contributors to algebraic geometry.
Paul Horja, Duiliu-Emanuel Diaconescu (the mathematical physicist, not the D. of the famous theorem on topoi). Added redirect algebraic geometer at algebraic geometry; got sock of typing the profession link.
New references and links at Higgs bundle and Carlos Simpson. New entry Martin Olsson and additions to contributors to algebraic geometry. Reference updates at deformation theory including some MR links.
Peter Bantay (as he is now visiting Zagreb and gave a nice talk on vector modular forms and their connections to physics).
Guenter Harder and David Kazhdan. Updates to modular form, conformal field theory, vertex operator algebra.
There existed also an extremely stubby (almost empty) entry topological modular form. Currently this seems to still have priority over the redirect to “modular form” that you put. We should think about if we want to have separate entries on this, eventually. Otherwise we should move “topological modular form” to History.
It was not me who put the redirect “topological modular form” to the entry modular form, and I have now erased that redirect which makes no sense to me; we clearly want to keep a separate topic about the latter as it is one of the foci of our interest; though the latter entry is still a microstub.
I added some of my today’s algebraic geometry entries (rational map, birational map, image of a rational map, rational variety, unirational variety) into the geometry list where I also appended another sentence on cohomology at the end. Added the Joyce-Song 2008 arXiv reference do Donaldson-Thomas invariant and to Dominic Joyce; there is a new version out today.
Lev Landau (stress on the second syllable) and Landau-Lifschitz
Stub protomodular category, required at the entry nonabelian homological algebra.
I made some changes at homological algebra, not that much in the main text which is somewhat obsolete but more to refercnes, removed a query discussion and landed part of it into nForum discussion. A related question on linearity of homological algebra as opposed to the theory of resultants for which I just created an entry. Updates to algebra.
I have added today's arxiv paper of Vistoli on deformation theory via fibered categories (rather than Artin functors) into the bibliography in deformation theory and the work of Bruguières and Natale into new created entry Hopf monad and into exact sequence of Hopf algebras. While the variant of nonabelian cohomology for Hopf algebras (in the sense of Schreier's theory) is not satisfactorily developed, in some cases (say over an algebraically closed field of characteristics zero, what is an unpleasant restruction), one can use Tannaka formalism and replace the Hopf algebra by its category of modules or of comodules. Then one can see the generalization of Schneider's conditions in this Tannaka-categorifed context and the things work a bit better. Still more systematic nonabelian cohomology is desired for the extensions here.
New person-entry Mark Rieffel.
Jean-Pierre Jouanolou and stub linear Poisson structure. Related page to linear Poisson in my personal pages is the last week's new entry symmetric ordering for Lie algebras (zoranskoda).
Started quasinormal mode, oscillation, Ehud Hrushovski. By the way there is no page model theory ! Despite so much of foundations in nlab.
New references at Hall algebra, Donaldson-Thomas invariant, wall crossing. Notice the funny diagram proof of associativity in the new Kontsevich-Soibelman paper from this month, at page 11 and the 9-fold limit construction at page 107...amazing compression of new mathematics.
I added sheaf of ideals. The defining sheaf of ideals is an example. This continues a mini-series from today including subscheme of an abelian category, conormal bundle and conormal sheaf, prompted by Urs’s interest in infinitesimal thickenings.
Alexei Bondal. I have chosen spelling Alexei as it is used mostly at western conference sites. His employee Russian Academy of Sciences has rather transliteration Aleksei. Ideal would be Aleksej, as in Slavic linguistics circles, but it is not used here often.
Some updates today at Dmitri Orlov, mirror symmetry, and new entries Floer homology, symplectic topology and Dusa McDuff. In minutes I will add Joel Robbin and Dietmar Salamon.
New page Donu Arapura, needed at (also new) entry cohomological descent. Donu is now also listed under contributors to algebraic geometry. Very deep and humble mathematician, occasionally contributing to MO. I forgot to say, I updated with links etc. Croatian Black Hole School.
New entry Francis Borceux.
New references, links and redirects at Alexander Rosenberg, topologizing subcategory etc. to support today’s 3 new entries spectrum of an abelian category, local abelian category and Gabriel-Rosenberg theorem.
Expanded Albert Schwarz.
New entry Gaston Giribet required at Croatian Black Hole School and today’s addition AdS-CFT. More references at black hole.
New entry Rehren duality.
New stub Bekenstein-Hawking entropy. Earlier today diamond lemma (zoranskoda), diamond, George Bergman, Anatolij Širšov.
New entry Teimuraz Pirashvili and links to n-cafe and Jibladze-Pirashvili paper at theory.
I corrected the incorrect link to Durov’s thesis at Nikolai Durov. I thank Todd and others to follow my advice to enrich nlab with new entries like hypermonoid on the basis of interesting discussion and answers in nForum and external material. I have some vague further thoughts on the examples and digest till better understanding.
New entry Shahn Majid and some additions to Chern-Weil theory.
More redirects and a note at Sweedler’s coring that is a grouplike element in each Sweedler’s coring. This is important: it enables to describe the descent data (comodules over the coring) as flat connections. I would be interested to know what is the relation of this to the correspondence between transport functors and descent data from the paper of Urs and Waldorf. More references at grouplike element and new page Tomasz Brzezinski with redirects like T. Brzeziński.
MO link at BV-BRST formalism, MO links at path integral etc.
Just in the case it is lost in conversation with Urs: in addition to changing and adding some in Chevalley-Eilenberg algebra, Chevalley-Eilenberg cochain complex, Lie algebra cohomology I created new entries Chevalley-Eilenberg chain complex, Lie algebra homology. New quick entry which needs check is resolution with redirects projective resolution, injective resolution and the plurals; Urs added fibrant resolution and cofibrant resolution. Alternative approximation terminology like in Hirschhorn is mentioned. New entries contramodule and semiinfinite cohomology and few minor changes in Koszul duality, Boris Feigin, Edward Witten, Weil algebra, vertex operator algebra, algebra, connection for a coring, George Bergman and so on. Yesterday new name-entry Gabriella Böhm.
Stubs coderived category, Robert Wisbauer, separable functor. A new reference at coring (relating separability), a link at Bernhard Keller. John has new post at cafe on separability, post 2.
New entry Fred van Oystaeyen and his historical reference under separable functor. New entry Alain Verschoren. Updates and redirects at various personal entries, e.g. Ludmil Katzarkov.
More references at quantum dilogarithm and link to a workshop on the topic in Aarhus 9-13 Aug. Also a link to a master class in Aarhus under wall crossing and a dedicated entry wall crossing at Aarhus.
New person entries for the operad practioners Jean-Louis Loday and Benoit Fresse. This should accompany changes at Loday-Pirashvili tensor category and Leibniz algebra.
No new entries, but I did not log a number of recent minor changes. E.g. some additions to OnlineResources like http://math.stackexchange.com, http://tex.stackexchange.com, http://www.equalis.com; link to Lie theory added in mathematics under Geometry and Topology section, new reference from arXiv on cotangent complex, on generalized complex geometry, to renormalization (Borcherds) and so on. I noted that some write arxiv or other references without a link (e.g. references to Lurie’s papers); well, while it is known to some to find the references anyway for some newcomer it is not obvious that one needs to go to Lurie’s webpage to find those of his papers which are not at arxiv. So I correct here or there some missing thing.
People may notice that I have contributed less in last about a week. It is not because of the vacation, that is still to come. I had problems with laptop which consumed my time: going back to factory image does not work with HP-backup and recovery utility as many other people had teh same problem: the factory .iso image has problems, when recovering the microsoftblotware can not decipher some of its own keys (who came up with the idea of the registry in XP ? just a virus and error attractor…). So I was forced finally to abandon the XP and installed ubuntu 10.04, it took just few minutes, less than one bloody service-pack from Seattle. But now I should arrange all other nice software and files on it, to work nice. And my stuff is in hundreds of directories at two desktops which need careful merging what will take most of my time before vacation.
New stub cyclic cohomology with redirect (for now) for cyclic homology. Related changes and additions at Alain Connes, J-L. Loday etc.
New person entry Fernando Muro and update to books and reviews in mathematical physics, namely the part II and III of the reference from Hisham Sati.
Entry Feynman proof of the Lorentz force equations (zoranskoda) (for now just the main references with links) as yet another approach to the gauge theories and classical gravity. Did anybody look at the categorification ?
Person entry Peter Olver and references by Olver (and also by Krupková) at variational calculus. Olver's Applications of Lie groups... is very good introduction into the variational bicomplex, among other topics. Unfortunately, a friend lost my own copy (of Russian translation) from which I learned most of what I knew about Poisson manifolds as undergraduate.
I added two original references from Vinogradov to variational bicomplex, the new entry which I split off from variational calculus entry.
I was extremely busy recently, and last week on vacation, where I had expensive and slow internet, so I was not logging many changes; I also had made some contributions to some math articles in wikipedia recently, in few cases which were staring in my eyes, mainly in Croatian wikipedia section, but also additions to English entry en.wikipedia:Beck’s monadicity theorem where Mike was one of past contributors and the entries on noncommutative algebraic geometry, quasi-category, weak n-category and higher category theory and rewrite a section on modern approach in Wikipedia entry on algebraic geometry; and Croatian entries on nonstandard analysis, mathematical analysis, category theory (here some of the formulas from English page were used), Alain Connes and few others; I mention this here just if somebody from lab community might be interested to check; and I along the way cited lab in some most relevant English and Croatian Wikipedia pages (only in the case when I added also something else).
Now latest changes. In lab I started entry comonadic functor different from monadic functor, stubs for Arne Ostvaer and noncommutative motive, added a proof to skeleton, new references and links to several entries, including asymptotic isometry which has a new redirect, namely asymptotic symmetry (this was prompted by a new preprint by Henneaux on arXiv and my recent memories from Croatian Black Hole School). I translated roughly most of what I wrote in Croatian Wikipedia about Alain Connes to enhance the lab entry Alain Connes, and Urs made some fine formatting.
As noted in the discussion in Atrium/blog, I opened a new entry compatible localization in lab, while working on the related entry distributive law for idempotent monad (zoranskoda).
A proof to the equivalences (and new variants) added to idempotent monad. Yesterday I wrote a stub for affine localization. On the other hand, distributive law for idempotent monad (zoranskoda) has some errors, which I will repair later, so now it is under (re)construction.
New person entries Andrey Lazarev and Jonathan Rosenberg. Regarding that I often use page for Alexander Rosenberg I put the redirect Rosenberg to Alexander Rosenberg while the link to Jonathan Rosenberg will be placed there as well.
Lazard’s analyser and Alastair Hamilton.
More references at Joel Robbin and Dietmar Salamon. I looked a bit at entry renormalization and I am pretty unhappy with the bias it has: it identifies the general concept of renormalization with its very special case. Renormalization is not confined to relativistic QFT (it exist e.g. in statistical field theory), it exists even when the quantities are not infinite, and one should not identify BPHZ procedure with general renormalization scheme as there are many others. There are also renormalization procedures in perturbative calculations of spectra of various operators in mathematics, not necessarily related to physics and so on. For gauge theories one should not identify the standard covariant case with renormalization for gauge theories in general. For example, there is no known renormalizability in some noncovariant gauges, including physically important case of Coulomb gauge.
I have no time to fix all that now but I would like that people are aware of these things.
New stub operator product expansion with redirect OPE; also enriched multicategory to complement elsewhere reported suitable monad and relaxed multicategory. P.S. the OPEs are the most well known in CFT, I would like however a more general treatment here as it was used in physics literature before CFT became so important for mathematical physics.
New stub chiral differential operator having links to the main references.
I am pretty unhappy with the bias it has: it identifies the general concept of renormalization with its very special case. Renormalization is not confined to relativistic QFT
The entry does not seem to mention the worrd “relativistic”. (?)
(it exist e.g. in statistical field theory), it exists even when the quantities are not infinite, and one should not identify BPHZ procedure with general renormalization
There is a section titled “Hopf algebraic renormalization” that discusses Hopf-algebraic renormalization. That’s not a bias. It’s just an indication that somebody took the time to write about that aspect and nobody else took the time to write about any other aspect.
The entry does not seem to mention the worrd “relativistic”.
Urs, you clearly do not see the point. The relativistic theories have that special feature that the renormalization comes up only when the divergences appear. If you read the article it is all the time defining renormalization with the reference to the divergences. However in general in QFT and in statistical field theory there are many cases of finite renormalization interpreted as self-consistent screening and other effects. The correspondence between removing analytic singularities and the cases requiring renormalization which is implicit in the whole article comes up by theorems of axiomatic field theory in relativistic setup.
Now you defend the sentence saying that adding counterterms and extracting the renormalized amplitude is BPHZ by saying that the section is about Hopf algebras. So what Hopf has to do with BPHZ ? One has a Hopf algebra of functions on the adequate torsor of homotopic resolutions in any renormalization scheme not only BPHZ. I agree that BPHZ is here most well understood, but one should certainly not DEFINE extraction of amplitudes as BPHZ. The article says
A prescription – called renormalization scheme – for adding to a given amplitude in a certain recursive fashion further terms – called counterterms – such that the resulting modified amplitude – called the renormalized amplitude – is finite at z=0 was once given by physicists and is called the BPHZ-procedure .
Not only BPHZ procedure; there are other (less well known) renormalization schemes ewhich give correct and finite answer. Besides, BPHZ is for relativistic QFT.
Urs, you clearly do not see the point.
Right.
But go ahead and expand the entry. If there is actually anything wrong with my summary of the Connes-Kreimer formalism, let me know.
Once I will have time to do it I will. The Hopf combinatorics is quite big so eventually most of it should be transferred to a separate entry. Most of nlab is about general point of view on mathematical physics, so we should not mislead here by a too particular point of view here regardless current fashion in some circles.
However in general in QFT and in statistical field theory there are many cases of finite renormalization
If I may have been understood here. By finite I mean that you never have divergences but the cases in which renormalization changes from a finite quantity to another finite quantity. One easy example is in the book on QFT in condensed matter physics by Abrikosov, Gorkov Dyzaloshinski. In relativistic setup such things do not exist. Only renormalization of infinite things to finite. If something is already finite it does not need renormalization in the relativistic case (as explained in classical QFT textbook by Albert Schwarz).
Added a list of chapters at Categories for the Working Mathematician.
Updates to BV-BRST formalism, references.
Gelfond’s theorem from number theory.
Slight extension of the rudimentary article socle.
Various new stubs in Lab, including KK-theory, E-theory (=asymptotic morphism), Laplace operator, Jean Dieudonné, and a large extension of the article Bourbaki. In my “private lab” I started some new entries including today’s general math books and literature. Unfortunately I clearly do not have time to finish these things soon in the way I planned. By the way I asked once (Toby?) in nForum and did not get or notice an answer about Paul Taylor’s Practical Foundations, if anybody has a readable copy. The everywhere available online copy is, as noted, in largely unreadable format.
thanks for starting a stub on Laplace operators. I think the MO answers fail to mention what I think is the really good answer: Alain Connes’ answer. That also subsumes the answer that currently has the most MO-reputation points.
Well, many relevant features of Laplace are shared by a whole class of elliptic differential operators. On the other hand, there are also various features related to invariance. For example, there are analogues of Laplace operator in discrete setting, in representation theory (basically in the form of the quadratic Casimir operator) etc. Having and a formal adjoint much of the homological algebra of Hodge decompositions for holds at the formal level, as used in old representation theory papers by Kostant. Not all is about analysis here, I suspect…
Minor changes, mainly references at D-module, intersection cohomology, Masaki Kashiwara etc. New entry Beilinson-Bernstein localization.
Stub hyperset. By the way, how does one get the official list of all current wiki-categories of articles (like meta, axiom etc.) ? I knew that but now I don’t.
New person pages Brian Parshall, Alexandre Vinogradov. A new book listed at Hall algebra and changes at diffiety, new links and references in particular.
New person entry Rick Jardine (could somebody explicit the full bame corresponding to J F initials). More references at homological algebra and redirects and minor updates at some related entries.
could somebody explicit the full bame corresponding to J F initials
Harry recently reported here that the “F” is for (whence the “Rick”) No guanatee on spelling (“Frederick”?)
see http://www.genealogy.math.ndsu.nodak.edu/id.php?id=17633
The J seems to be John. see Rick Jardine for the updated version!