User zoran škoda - MathOverflow

Recent fundamental new directions in PDEs

2013-05-04T10:44:11Z

My main interests are in modern geometry/topology, algebra and mathematical physics. I observe that there is a raising communication, language and social barrier between this community and the differential equations community, with the main exception of the study of geometrical PDEs. The areas of mathematics which are close to my knowledge have an impressive list of new appearing frameworks, which are changing the landscape and have the capacity of even redefining the basics of the subject. To mention just few which emerged in roughly last 15 years: derived algebraic geometry, tropical geometry, geometry over "the field of one element" $\mathbf{F}_1$, cluster algebras, higher categories, $(\infty,1)$-topoi, Kontsevich deformation quantization, homotopy type theory and univalent foundations, categorification in representation theory, derived categories of motives, $\mathbf{A}_1$-homotopy theory, geometric Langlands program, motivic integration, relation between motives and renormalization of QFT, geometric understanding of BV-formalism etc. It is hard to think of the subject without those, despite the fact that these framework appeared that recently.

I am interested of learning of similar recent landscape changes in PDEs (apart from the description, please take the subject in quite general sense, including variational calculus, stochastic PDEs etc. and give some pointers to the names or seminal references). It is hard to spot them from outside, and it s hard to believe that there is nothing vaguely comparable to the advances of the previous epoch (most notably, the work of Hoermander in 1960s on Fourier integral operators and linear and quasilinear PDEs; or, say, the work of Gromov on h-principle for partial differential relations in differential geometry).

Though, there is no need of repeating, as I am well aware of: the framework of polyfolds and general Fredholm theory due Hofer, and PDEs related to symplectic and contact geometry in general; then, of course, breakthrough in the study of Ricci flow after Perelman; the appearance of systematics methods of obtaining local index formulas by Connes and others; then advances related to integrable PDEs (including, most recently, the development of methods involving nonlinear Poisson vertex algebras by Victor Kac and collaborators), and those related to microlocal analysis and hyperfunctions; to cohomological analysis of PDEs related to secondary calculus. I am aware of the importance of the works of Villani and also of Tao, though I have somewhat too vague picture of which are the new general fundamental principles there. I am also aware that the numerical control and theoretical study of special cases of Navier-Stokes got to much higher level than before and that the homological algebra is recently systematically applied to the study of stabilty of finite element methods by Douglas Arnold and collaborators (see references). But I would like to learn of other new fundamental frameworks.

A related question is is if there are recent fundamentally new types of functional spaces which now promise to become central in the study of nonlinear PDEs.

monomials in the universal enveloping of a Lie algebra in terms of the symmetric basis

2011-12-12T11:52:14Z

Let $L$ be a finite-dimensional Lie algebra over a field $k$ of characteristic zero and $e_1,\ldots, e_n$ some basis of $L$. The formula $[e_i,e_j] = \sum_k C_{ij}^k e_k$ determines the structure coefficients $C_{ij}^k$. Given any ordered $k$-tuple $I = (i_1,\ldots,i_k)\in \lbrace 1,\ldots,n \rbrace^k$, define $e_I = e_{i_1}\cdots e_{i_k}\subset U(L)$ and $$ e^S_I = \frac{1}{n!}\sum_{\sigma\in\Sigma(k)} e_{\sigma(i_1)}\cdots e_{\sigma(i_k)}\in U(L). $$ As it is well known, from various forms of a PBW theorem, $e_I$, for all $I$ with $i_1\lt i_2\lt \ldots \lt i_k$, $k \geq 0$, form a basis and also $e^S_I$ (for the same set of $I$-s) form a basis. I need explicit formulas for $e_I$ in the linear basis of $e^S_J$-s where the coefficients are expressed in terms of the structure constants $C_{ij}^k$ (and combinatorial factors). In fact, for my present purposes, it would be enough to know explicitly the deepest, lowest order, linear term (but it is of course the hardest summand in the expansion). For example, for the easiest nontrivial case $k = 2$, $$e_{(i,j)} = e_i e_j = \frac{1}{2}(e_i e_j + e_j e_i) + \frac{1}{2} \sum_k C_{ij}^k e_k = e_{(i,j)}^S + \frac{1}{2} \sum_k C_{ij}^k e_k,$$ hence the linear term is $\frac{1}{2} \sum_k C_{ij}^k e_k$.

Answer by Zoran Škoda for References For Important Hopf Algebras

2011-08-29T08:51:07Z

Shahn Majid's book Foundations of quantum group theory (Cambridge Univ. Press 1995, 2000) has lots of examples and of classes of examples. These are not only examples of quantum groups in the narrow sense (cf. the $n$Lab page for other references on quantum groups in various senses).

Answer by Zoran Škoda for Plagiarism in the community

2011-08-06T13:18:58Z

While I know cases of stealing by people who heard the idea, or even unintentional incorporation into their other work (one often does not remember which idea helped her/him to come to certain point), this is possible in an unwanted way ONLY if this passing of idea is individual and in a way secretful. If you tell the idea to LOTS of people, at recordable places as well, then everybody knows it is your idea and you will get credit and be allowed to publish it even if some use the idea in various forms. So making it half-available is the only scenario which can possibly hurt you (though such cases are rare in pure mathematics and somewhat more common in areas with more money like physics, biomedical sciences and so on). There is no better protection from stealing a secret than making it no-secret at all.

Answer by Zoran Škoda for Basis of quantum SU(n)

2010-02-23T14:32:38Z

The question is slightly mistated, from the example for $n=2$ it is seen that the question is about $SL_q(n)$ and not $SU_q(n)$; the answer is of course those standard normally ordered monomials

$(t^1_1)^{a_{11}}(t^1_2)^{a_{12}}...(t^n_n)^{a_{nn}}$

satisfying the condition that at least one of the diagonal exponents $a_{ii}$ is zero. Unlike in $O_q(M_n)$ literal application of the Bergman's diamond lemma does not produce the algorithm, because the diagonal enetries are not one next to another so if one wants to exclude the diagonal extra occurences one needs to go against the semigroup law. This is possible to do with great effort, I have checked this in 1999 with lots of algorithmic combinatorics; namely the set of reductions used is infinite and given algorithmically rather than by explicit formulas. Unlike the general rule advised by Bergman, it is not wise in the straight diamond lemma approach to exclude the nested ambiguities. Some other Grober arguments not relying on standard diamond lemma can give easy answer though.

For generic $q$ it is of course enough to use the classical commutative case and deformation arguments (Edit: alluded in David's answer).

It is not true, what is stated above in the accepted answer that the simple technique for $O_q(M_n)$ via diamond lemma and with the relations taken as reductions works when setting $det_q =1$. Imagine you have expression $(x^1_1)^2 (x^2_2)^2 (x^3_3)^2$ in $SL_q(3)$. How will you use centrality of the quantum determinant to translate this into something what does not have all three diagonal generators ? You need first to rearrange thing to be able to complete to a quantum determinant to exclude a bad diagonal generator, but this is not very compatible with the ordering. It can be done systematically but by now means is trivial or implied by Klimyk-Schmuedgen book.

Riemann hypothesis via absolute geometry

2011-07-03T10:00:54Z

Several leading mathematicians (e.g. Yuri Manin) have written or said publicly that there is a known outline of a likely natural proof of the Riemann hypothesis using absolute algebraic geometry over the field of one element; some like Mochizuki and Durov are thinking of a possible application of $\mathbf{F}_1$-geometry to an even stronger abc conjecture. It seems that this is one of the driving forces for studying algebraic geometry over $\mathbf{F}_1$ and that the main obstacle to materializing this proof is that the geometry over $\mathbf{F}_1$ (cf. MO what is the field with one element, applications of algebaric geometry over a field with one element) is still not satisfactorily developed. Even a longer-term attacker of the Riemann hypothesis from outside the algebraic geometry community, Alain Connes, has concentrated recently in his collaboration with Katia Consani on the development of a version of geometry over $\mathbf{F}_1$.

Could somebody outline for us the ideas in the folklore sketch of the proof of the Riemann hypothesis via absolute geometry ? Is the proof analogous to the Deligne's proof (article) of the Riemann-Weil conjecture (see wikipedia and MathOverflow question equivalent-statements-of-riemann-hypothesis-in-the-weil-conjectures) ?

Grothendieck was not happy with Deligne's proof, since he expected that the proof would/should be based on substantial progress on motives and the standard conjectures on algebraic cycles. Is there any envisioned progress in the motivic picture based on $\mathbf{F}_1$-geometry, or even envisioned extensions of the motivic picture ?

Answer by Zoran Škoda for Cite articles or book where I first found the result?

2011-05-15T16:38:17Z

The one sided answers neglect the complexity of the issue. First of all most of the papers in the science are nowdays difficult to judge, survey and check for an average reader, not to mention the users like science agencies and general public. The production is huge, there is lots of repetition and formalism differences make it difficult to bring knowledge and fruitfulness to the public and we need to fight this not by messy inclusion of everything but by making honest and informative choices to the reader. So, the main criterion is to write primarily for the reader and the author should have the stand weather the article she cites are likely to be useful to reader or not at all; if one does not take reader as the main criterion but some agenda of pushing the agenda for agencies, why would then the reader respect author's agenda of misleading the reader which reference is good and readable just to fulfill the political agenda ? In mathematics, the pointers from the text to the bibliography, together with the basic standard knowledge in the field should make obvious path how to achieve the results in the paper. This is already a difficult goal and making references only by history and not lead by self-analysis of the paper and its predecessors will make it too difficult to reconstruct the preliminaries used. Of course, one needs to account for balance in making clear what the original resources are either by making clear pointers to other surveys and also citing directly relevant and used work with some emphasis on the "primary" sources, where the latter term is not fully well defined, as each idea has some predecessors.

I disagree with much of the answer and comments of Deane Young, who is on one hand saying cite generously much, cite when in doubt etc. as if we live in an ideal world where the size of bibliography is allowed to be as large as you want, (Example: One of my papers was delayed for 2 years because it was 19 pages and the limit in the journal was 17. Roughly, two pages where bibliography.) and on the other hand saying that we live in a bad world where people depend on our citations, so cite even if you did not read/check and where we need to adapt to "funding agencies". Well, if we all adapt to present treand in funding agencies we agree with a wrong attitude that the citation is proportional to research credit what is not true: often the most cited are often surveys and books, popular digests, and also some original papers which are often only famous but nobody reads, or can read (so many times they are more readable than the public thinks, by inertion). It is better to push the agencies to take this into account and fund according to the full description of the discoveries of the author and description of their impact by listing consequences and new directions opened, and not of numerical quasi-impact. Conforming to numerical quasi-impact as a main principle may exactly push down honest people who cite only papers they trust, they check, they believe the authors and so on, we may also cite references which are written in bad way and not recommendable to the reader. To extend this remark to extreme which appears in practice, I know many people who cite famous people just to show that their own work is relevant for the trends pushed by those. This is the worst kind of citation.

There is also a pressure from referees and editors to include the references of their choice. In such cases one has to follow common sense, and decide if the request is reasonable, even with expense of possibly changing the journal. We should not regret the referees' and editor's time if they abuse their position to make people cite papers of their friends and their own; however we should appreciate it in good cases when their knowledge is channeled to point to the sources we did not use or appreciate and to learn what is the better or more original reference in many cases.

Answer by Zoran Škoda for Publishing journals articles without transferring copyright.

2011-05-14T13:21:17Z

Most journals in math allow you to publish a version of the paper which was previously posted to the arxiv.org. They ask you often to take the copyright for the published version which just slightly differs from the arxiv version. So there is not much difference between having it public or having a slightly different version public. Some journals, on the other hand are free anyway and forever in their public versions, e.g. Theory and application of categories. If you choose a journal carefully you solve most of your concern. Some publishers are notorious of being nasty, expensive, proprietory, nonresponsive to author needs etc. You do not want to publish in expensive envelopes of crap, like Elsevier's Chaos, solitons and fractals used to be.

duality between universal enveloping and function algebra for GL(n)

2011-03-15T19:00:53Z

Motivation. Few years ago I constructed a family of internal Hopf algebras in the Loday-Pirashvili tensor category of linear maps which is in a sense a generalization of the algebra of regular functions on the group $GL(n)$. A student of mine (Matija Bašić) has worked out certain internal analogue of a Weyl algebra in that context and one could conceivably make much of the story on algebraic groups and invariant differential operators in that setup. However, we did not continue for a while and this project stalled in a way. Its motivation was to find a geometric theory of integration of Leibniz algebras in characteristic zero, and more generally Lie algebras in Loday-Pirashvili tensor category. There is an Ado theorem in this setup, so it is sufficient to do integration for Lie subalgebras in a Loday-Pirashvili analogue of $gl(n)$ (here $n$ generalizes to data of a linear map between two finite dimensional vector spaces). For the latter there is an internal version (as well as another version) of a universal enveloping Hopf algebra. On the other hand, there is my analogue of the function Hopf algebra on the $GL$ in Loday-Pirashvili category. I have a picture of certain modification of scheme theory to make it into an algebraic group in certain abstract context, but so far this is not fully done. Thus I would like to prove, not from abstract principles but by direct check that the two Hopf algebras are dual one to another.

But trouble: in classical case of the usual Lie groups and enveloping algebras I am aware only of the proofs using what a Lie group is and what its Lie algebra is, while my GL is given as a Hopf algebra somewhat alike usual functions on $GL(n)$, with just a little more exotic relations. So what I should generalize is down to earth explicit proof that $U(gl_n)$ and $\mathcal{O}(GL(n))$ are dual as Hopf algebras. Emphasis is on Hopf. Notice that it is not sufficient to write down the pairing for multiplicative generators, but for the whole vector space basis: the algebras are not free and to predict the pairing between higher order monomials does not follow from knowing pairing just between matrix elements $t^i_j$ and the generators of the Lie algebras $gl_n\subset U(gl_n)$. But it should not be that difficult.

Question: Do you know how to do algebraically prove that we have a Hopf pairing (possibly with some sort of formulas for the pairing) between classical $GL(n)$ and $U(gl_n)$ ? We should never use any knowledge on $GL(n)$ and $gl_n$ except their generators and relations, as most of other facts are nontrivial to generalize to my problem which motivates the question.

Answer by Zoran Škoda for Universal definition of tangent spaces (for schemes and manifolds)

2011-03-15T18:24:01Z

Steven already explained a bit about the key to the answer: Synthetic differential geometry (cf. nLab where the hints on a higher categorical analogue are also present), but I would like to put it in a much broader perspective, though more in words and references than really explaining, mainly due to space, time and expertise limits.

While there are mentions, in several of the answers above, of the (possibly relative) module of Kähler differentials leading to the algebraic version of cotangent space used in algebraic and analytic geometry; this partial notion predates a little bit the more fundamental work of Grothendieck, who invented an inherent geometrical way to found a differential calculus in geometry. Similarly to the differentiation in topological vector spaces, the basic idea is to approximate the maps with linear maps, but this time Grothendieck considered maps among sheaves of $\mathcal{O}$-modules over schemes; he described the linearization in the language of operations on sheaves in terms of infinitesimal neighborhoods of the diagonal $\Delta\subset X\times X$; these are described in terms of nilpotent elements in the structure sheaf; one can also define the related notion of infinitesimally close generalized points; the infinitesimal neighborhoods build up an increasing filtration, which induces a dual filtration on the hom-spaces, so called differential filtration. The union of the differential filtration is the differential part of the hom-bimodule, and its elements are regular differential operators. A "crystaline" variant of the picture related to divided powers leads to appropriate treatment of differential calculus in positive characteristics. The notion of crystal of quasicoherent sheaves is bases on the notion of infinitesimally closed generalized points; the geometric picture with pullbacks of sheaves, leads to a definition as sort of descent data, cf.

P. Berthelot, A. Ogus, Notes on crystalline cohomology, Princeton Univ. Press 1978. vi+243.
J. Lurie, Notes on crystals and algebraic $D$-modules, in Gaitsgory's seminar, pdf

These are a dual point of view on D-modules. The descent data in abelian context are equivalent to certain formally defined connection operator, called Grothendieck connection in this case. There are now abstract versions of the algebraic correspondence between descent data in abelian context and flat Kozsul connections for the associated "Amitsur" complexes (work of Roiter, T. Brzeziński and others, cf. connection for a coring). On the other hand, Grothendieck immediately came up with nonlinear version of crystals (crystals of schemes) which are dual point of view on what some now call D-schemes.

Grothendieck's point of view on differential calculus has been soon after the discovery at the end of 1950s, introduced in works of Malgrange, Kodaira and Spencer in the development of obstruction and deformation theory for differential equations. Both works together in late 1960s motivated Lawvere, Kock and Dubuc to extend that geoemtric approach to differential calculus into differential geometry. Dubuc introduced $C^\infty$-schemes as yet another approach to manifolds, in the spirit of the theory of schemes. Lawvere did not look only at then recent work of Grothendieck (and Malgrange, Spencer...), but also at classical work on "synthetic geometry". This is a terminology which requires caution: in 19th and early 20th century, synthetic was viewed as differing from coordinatized, analytic, and pertained to either work from axioms, not referring to coordinate and even metric aspects, and some people in axiomatic descriptive geometry refer to their geometry as synthetic even now in that "clean", but less powerful sense. Another sense is that it is close to the engineering point of view that the path of a particle can be considered either as a point in the space of paths or as a map from interval into the space, what implies that the infinite-dimensional spaces of paths should exist and one should have the exponential law, i.e. we need to embed our category of spaces into closed monoidal category; there are many such embeddings of the category of manifolds available now, and some models of them offer the model $D$ of infinitesimals, which represents the functor of taking the tangent space in particular. This model has been shaped with having in mind the Grothendieck's field of dual numbers in algebraic geometry, but the language and multiplicity of models made it very flexible in the approach of the synthetic differential geometry of Kock and Lawvere. First of all they had an independent axiomatic approach as well as study of the topos theoretic models; including the study of Cahiers topos which is even more faithful to the Grothendieck's point of view. In all these models, they had nilpotent infinitesimals, like in scheme theory, but not like infinitesimals in nonstandard analysis. More recent approach of Moerdijk-Reyes offered both nilpotent and non-nilpotent infinitesimals (though possible variant related to nonstandard analysis is not known to me). For a differential geometer there are many attractive tools in synthetic differential geometry like infinitesimal simplices, enabling intuitive and effective of many quantities involving differential forms and geometry.

On the other hand, the usual differential geometry is faithfully embedded into synthetic models, so one is bound to be conservative, i.e. not to get results about usual notions in manifolds theory which are inconsistent with the usual definitions. One just gets more intuitive and technical power.

I should also mention that the Grothendieck's picture with the infinitesimal thickenings, aka resolutions of diagonals, leading to differential calculus, can be extended to noncommutative spaces rerpesented by abelian categories "of quasicoherent modules". This has been done in 1996 preprints

V. A. Lunts, A. L. Rosenberg, Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings, MPI 1996-53 pdf, II. D-Calculus in the braided case. The localization of quantized enveloping algebras, MPI 1996-76 pdf

The resulting definition of the rings of regular differential operators on noncommutative rings has been used in the study toward the Beilinson-Bernstein correspondence for quantum groups in later published two articles, which however skip the geometric derivation of the definition of the differential operators used:

V. A. Lunts, A. L. Rosenberg, Differential operators on noncommutative rings, Selecta Math. (N.S.) 3 (1997), no. 3, 335--359 (doi); sequel: Localization for quantum groups, Selecta Math. (N.S.) 5 (1999), no. 1, pp. 123--159 (doi).

Somewhat similar analysis in infinite-categorical setup of a $(\infty,1)$-version of the Cahiers Topos is in recent master diploma

Herman Stel, ∞-Stacks and their Function Algebras – with applications to ∞-Lie theory, Utrecht 2010, webpage, pdf

under the guidance of Urs Schreiber. This work leads to a correct theory of higher Lie algebroids.

Answer by Zoran Škoda for Grothendieck and Non-commutative Geometry?

2011-03-15T16:52:55Z

No and yes, depending on the level of understanding. The consideration of noncommutative rings telling about geometry is almost nonexistent in Grothendieck's published opus. One of the exceptions is that he considered cohomologies for the possibly noncommutative sheaves of $\mathcal{O}$-algebras for commutative $\mathcal{O}$ (the latter is used in Semiquantum geometry). On the other hand, Grothendieck has been pioneer on abandoning the points of spaces as primary objects and promoting the category of sheaves over the space as defining the space. This is the point of view of topos theory which he invented; he noticed that the topological properties do not depend on a site but only on the associated topos of sheaves, and proposed a topos as a natural generalization of a topological space. Manin took Grothendieck's advice that one should consider the topos of sheaves as replacing the space, together with Serre's theorem that the category of quasicoherent modules determines a projective variety, as a mogivation to his approach to noncommutative geometry and quantum groups. The modern view of noncommutative geometry is that it is about the presentation of space via the structures consisting of all possible objects of some kind living on a space (algebra of functions, some structures consisting of cocycles, like category of vector bundles, category of sheaves, higher category of higher stacks).

In late 1960s W. Lawvere, with help from Tierney, extended the Grothendieck topoi to the theory of elementary topoi. This was not his only contribution of Lawvere in 1960s. Lawvere promoted also the duality between spaces and dual objects which he calls quantity (cf. space and quantity). While Lawvere's impact has been deep, I object to the terminology: in physics a quantity is normally a single observable; physicist do not consider the algebra of all observables a quantity, but rather a field of quantities, or algebra of quantities. But never mind the terminology, Lawvere went on very deeply in presenting this point of view, which is really generalized noncommutative geometry. Of course, neither Grothendieck nor Lawvere did not pay that particular attention to reconstructing the differential geometry and measure theory from the study of operator algebras, what is the huge contribution of Connes, or from the study of noncommutative rings, which was implicit in Gabriel 1961 and more explicit with works of J. S. Golan, van Oystaeyen (and P. M. Cohn with his affine spectrum) and others in mid 1970s, working with spectra of noncommutative rings and noncommutative localization theory as a noncommutative analogue of Zariski topology. One should mention that sporadic appearance of operator algebras from the noncommutative geometry point of view is present to some extent in 1970s book of Semadeni on Banach spaces of continuous functions (MR296671), where he studies, among other topics, the noncommutative analogues of many topological properties of topological spaces; in less explicit form there are also works of Irving Segal which had a similar motivation.

Grothendieck says in his memoirs that the concept of abelian category as he promoted it in Tohoku is part of the same philosophy -- abelian categories, possibly with additional axioms like AB5 are sort of categories of sheaves of modules, and should be viewed as an idea which is sort of abelian/stable version of Grothendieck topoi. More precisely, in this line, there is a recent Nikolai Durov's concept of a vectoid. Pierre Gabriel, who was close to Grothendieck's school in his early days, had in his prophetic work of 1961 reconstruction theorem for schemes and study of subcategories and localizations in abelian categories which represent open or closed subschemes and so on. Gabriel's work is in fact the first big work in noncommutative algebraic geometry and his reconstruction theorem is really the basic motivation in algebraic flavour of the theorem. In a sense, Gabriel's work is an abelian version of some Grothendieck's basic ideas of topos theory (cf. noncommutative scheme for one of the modern ideas along that line of thought) and Grothendieck was well aware of the abelian direction of this thinking from the Tohoku times.

For a general vista, I recommend

Pierre Cartier, A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry, Bull. Amer. Math. Soc. 38 (2001), 389-408, pdf.

Answer by Zoran Škoda for Affine morphisms in different settings coincide?

2011-03-14T22:31:21Z

For case 2) $f: X\to Y$ is affine if its direct image functor $f_*:Qcoh_X\to Qcoh_Y$ is faithful and admits not only a left adjoint (inverse image) $f^{*}$, but also a right adjoint, say $f^{!}$.

If $f$ is quasi-compact and $X$ separated, then $f$ is affine iff it is cohomologically affine, that is, $f_*$ is exact (Serre's criterium of affiness, cf. EGA II 5.2.2, EGA IV 1.7.17).

Answer by Zoran Škoda for Never appeared forthcoming papers

2011-03-10T00:05:08Z

S. Gel'fand, Yu. Manin, Methods of homological algebra, first appeared in Russian as Методы гомологической алгебры. Введение в теорию когомологий и производные категории. Т. 1 (that is VOLUME 1). Volume 2 has been given up and the Springer Western edition does not cite Russian original, has many typing errors in formulas which Russian original does not have and it scraped off the tome 1 from the title.
M. Demazure, P. Gabriel, Groupes algebriques, tome 1, Mason and Cie, Paris 1970 -- later volumes never appeared
Z. Semadeni, Banach Spaces of Continuous Functions, Polish Scientific Publishers, Warzawa, 1971, never appeared from the Polish Sci. Publ. There is however a different book with a similar title in Springer in 1982, Schauder bases in Banach spaces of continuous functions. Lecture Notes in Mathematics 918. Springer 1982. v+136 pp. MR83g:46023.
John W. Gray, Formal category theory: adjointness for 2-categories, Lecture Notes in Mathematics 391, Springer-Verlag 1974. xii+282 pp. has been envisioned as a m3 volume project on formal category theory, some material is mentioned in volume 1 and never appeared. The monograph is very innovative and some of the material from the latter volumes was undoubtfully sketched by the author in some detail. The author later drifted to theoretical computer science.
John Duskin started a paper in several parts "Nerves of bicategories", part I appeared with great delay, partly due serious health problem the author experienced few years ago. Second and third part did not appear, although the contents description looks very promising. We wish the author good health and more to be seen!

Grothendieck planned not only later EGAs but also later SGA (e.g. some Berthelot's works in SGA 8). Bourbaki Elements are of course never finished as well (an now are very slow, asymptotically stalling) as the German encyclopedic work by Klein's students at the beginning of the 20th century. M. M. Postnikov wrote two volumes of a course on algebraic topology in Russian about basics of homotopy theory and promised the homology in "next semester" but no books appeared on that.

Answer by Zoran Škoda for Ore Extensions and the Construction of the Quantum General Linear Group

2011-03-03T19:17:49Z

I have hard time understanding what is your notion of usual; most references indeed define the quantum linear group $M_q(N)$ by generators and relations (or by the universal property as defined by Manin) and then localize at $det_q$. So far as the construction goes. But it is of course very useful to know that one can start with the polynomial ring and then do the iterated Ore extension $N^2-1$ times to obtain $M_q(N)$. This is useful to infer many useful ring-theoretic properties. For example, it follows that the matrix bialgebra $M_q(N)$ and its Hopf envelope $GL_q(N)$ are Ore domains (the same O. Ore!), hence they are contained in the Ore quotient ring in which we can do many useful computations. Various intermediate Ore localizations are also useful.

Answer by Zoran Škoda for History of classifying spaces

2011-01-17T15:20:33Z

Historically the first version is the nerve of a covering, which has been used in the works of P. S. Aleksandrov in late 1920-s. The nerve of a covering in that version was treated as a simplicial complex had the elements (which are some open sets) of the covering as vertices, and an $n$-simplex is corresponding to an $(n+1)$-tuple of elements of the covering which have a common nonempty intersection; in particular one gets a finite combplex for a finite covering. This version was soon later used in Čech theory. I emphasize this as often nowdays the Vietoris complex which is a bigger complex whose vertices are pairs $(U,x)$ where $U$ is an open set and $x\in U$ is nowdays often called Čech complex as well, as the finite, original, version is now more rarely used. Simplicial sets replaced old-fashioned simplicial complexes a couple of decades later.

Grothendieck generalized the nerve to the case of categories. The simplicial complexes in their combinatorial and topological reincarnation were from the beginning taken interchangeably. However, for simplicial sets, the nice categorical treatment is from Milnor, who formally introduced a notion of geometric realization in modern context; the concept was essentially known but not its properties at the time. Classifying spaces for group case, were of course studied first in the context of group cohomology, so MacLane is probably among the first ones using it. Segal in late 1960s, not only studied the concept in depth but also introduced more complicated version for simplicial categories.

Answer by Zoran Škoda for Free, high quality mathematical writing online?

2010-11-21T11:26:40Z

In nlab we keep a list of main links of archives and free book collections in our main areas of interests (we were intentionally selective there):

For top level directory for math resources see http://ncatlab.org/nlab/show/math+resources, from where you can go to archives, individual author collections, blogs and institutions.

Answer by Zoran Škoda for Which math paper maximizes the ratio (importance)/(length)?

2010-11-02T11:18:22Z

Instead of answering directly about which paper (I don't know), I think that a journal with amazing importance/page ratio was Funktsional. Anal. i ego Prilozhen./Functional analysis and its applications at the time when Gel'fand was the main editor (or Kirillov at some point). Typical paper in 1970-s was of much importance, recognizable names and results nowdays, while being usually something like 4 pages. If one looks at all the volumes in 1970-s together it is just a short interval at a bookshelf, amazing compression of thousands of important results, especially in view of many junk commercial journals nowdays which flag with impact factors like the notorious Chaos, solitons and fractals...

Answer by Zoran Škoda for Derived categories of coherent sheaves: suggested references?

2010-10-18T12:24:54Z

Kapustin-Orlov'a survey of derived categories of coherent sheaves is pretty good,

A. N. Kapustin, D. O. Orlov, Lectures on mirror symmetry, derived categories, and D-branes, Uspehi Mat. Nauk 59 (2004), no. 5(359), 101--134; translation in Russian Math. Surveys 59 (2004), no. 5, 907--940, math.AG/0308173

but more slow/elementary exposition starting with fundamentals of derived categories is in an earlier survey of Orlov

D. O. Orlov, Derived categories of coherent sheaves and equivalences between them, Uspekhi Mat. Nauk, 2003, Vol. 58, issue 3(351), pp. 89–172, Russian pdf, English transl. in Russian Mathematical Surveys (2003),58(3):511, doi link, pdf at Orlov's webpage (not on arXiv!)

There are also Orlov's handwritten slides in djvu from a 5-lecture course in Bonn

djvu, but the link is temporary

For derived categories per se, apart from Gelfand-Manin methods book and Weibel's hoological algebra remember that a really good expositor is Bernhard Keller. E.g. his text

Bernhard Keller, Introduction to abelian and derived categories, pdf

...and also his Handbook of Algebra entry on derived categories: pdf

Answer by Zoran Škoda for Generalized notions of solutions in various areas of mathematics

2010-10-07T14:01:02Z

Moduli problem: find a good parametrization of geometric objects of some type; parametrization should form a collection equipped with some natural geometric structure, therefore being a geometric object in its own right. While naive "parameter space" is a set, in structured formulation it is replaced by a moduli space which classifies the geometric objects we started with. In the simplest case, the moduli problem is representable by a space in a usual sense, an object in more or less the same category in which the original geometric object was. For example a manifold or a scheme where the original objects were manifolds or schemes. With harder problems the moduli lead to more and more general kinds of objects. This motivated new types of spaces as stacks, higher stacks, derived stacks and so on.

It appears that starting with original geometric category, most of the generalized objects needed to solve the moduli problem live in some nice geometric subcategory (e.g. algebraic stacks) of the category of (possibly categorified) presheaves or sheaves on the original category, including higher versions like simplicial presheaves and so on. The original category embeds by the corresponding version of Yoneda embedding into the category of (pre)sheaves. The new ambient category of presheaves not only more generically has a solution to the moduli problem, but also has many other improved natural properties like closedness under limits.

Cohomology theories, various generalized cocycles and so on, generalized smoothness notions and so on, can also be accomodated after Yoneda embedding into a homotopy correct version of presheaf category, like in the emerging subject of derived geometry. In the original terms of non-generalized spaces, one would need to use all kinds of difficult and dirty technique to define and study the generalized notions, for example introducing various piecewise-continuous cocycles, multivalued or infinite-dimensional models and so on. Methods depending on Yoneda philosophy give rather universal setting to attack moduli problems and many other problems (like deformation theory), allowing to often eliminate construction of very elaborate but ad hoc modifications of original concepts. Inside the bigger category it may be easier to cut out some nice geometric subcategory of geometric spaces which include the solutions to the moduli problem than constructing some similar category in terms of original geometry. Of course, sometimes the difficult elementary models have their own specific strengths, which do not follow from the application of general methods.

Answer by Zoran Škoda for Most helpful math resources on the web

2010-10-05T21:01:01Z

Proceedings of all past ICM-s can be found here: http://www.mathunion.org/ICM

The following nlab pages list some of the main resources

http://www.ncatlab.org/nlab/show/Online+Resources -- a long list of math blogs and forums
http://www.ncatlab.org/nlab/show/math+institutions -- main institutions
http://www.ncatlab.org/nlab/show/math+archives -- preprint/journal/book/review archives

http://numdam.org is a collection of old issues of many mainly French math journals. http://www.mathnet.ru site has links to free old issues of most of the Russian math journals (and even some video lectures) in Russian and links to some non-free English versions. There is also an English mode of the site: http://www.mathnet.ru/index.phtml?&option_lang=eng. A smaller free depository of old issues of Polish math journals is http://matwbn.icm.edu.pl (click on the flag for English).

Max Planck maintains links to a very long list of journals, most of which are proprietary and usable only from their site, but the list is still useful because a sizeable fractions of links are also to free journals or some volumes of journals which are free, and those are mainly usable from all locations. The current URL is http://rzblx1.uni-regensburg.de/ezeit/fl.phtml?bibid=MPIMA&colors=3&lang=en&notation=SA-SP

Many resources can be found at the sites of main world math institutes like ihes, mpim-bonn, Oberwolfach, msri, kitp, ictp, rims, ias, Steklov, Clay, crm Barcelona, Mittag-Leffler, Banff, Fields, Newton, ihp Paris

AMS keeps a long list of math societies throughout the world with links to their sites, which are often useful. One should also recommend more general AMS directory of links Math on the Web http://www.ams.org/mathweb/index.html.

Answer by Zoran Škoda for Understand Cech Cohomology

2010-10-01T14:56:47Z

On 3rd question: Čech homology is not a homology theory in the sense of Eilenberg-Steenrod: the exactness axiom (long exact sequence in homology) does not hold, exactly because of the problem with limits. There is however a sophisticated method, discovered by Sibe Mardešić, to correct this, by modifying slighly the Čech definition. The resulting "strong homology theory" agrees with singular homology on the spaces having homotopy type of CW complexes, and does give long exact sequence of pairs $(X,A)$ where $X$ is paracompact and $A$ closed; moreover for metric compacta it satisfies not only all the axioms of Eilenberg-Steenrod, but also the relative homeomorphism axiom and the wedge axiom. The only homology theory on the metric compacta satisfying not only the Eilenberg-Steenrod but also the wedge axiom is the Steenrod-Sitnikov homology theory, hence the strong homology agrees with it.

Answer by Zoran Škoda for Did Durov's work give an example of noncommutative schemes?

2010-07-27T14:16:06Z

No, there is no need for the Rosenberg noncommutative scheme that the categories be abelian in general; whatever being said in his 1998 paper he does not mean so. He defines a relative scheme over a base category given by exactness properties of direct and inverse image functors describing covers used for gluing and the affinity property. In general one has to be careful, with formulating correctly the exactness properties for nonabelian context.

Of course, to justify this one needs to say that the Durov's schemes are determined by the categories of quasicoherent modules. I do not know if the reconstruction theorem a la Gabriel-Rosenberg holds for the generalizes schemes of Durov. Durov glues schems along categorical localizations which he calls for some reason "pseudolocalizations". They just have the correct exactness properties. On the other hand, for commutative monads, Durov has two nice versions of prime spectra; now one should compare those with a version of Rosenberg's spectrum for nonabelian context. Now one version if the spectrum for right exact categories of Rosenberg. What is a right exact structure for the case of categories like the ones in Durov's work ? Well Durov has spent some time to develop a theory of vectoids which generalize topoi, but also the categories of modules over finitary monads in Set and, more generally, the categories of quasicoherent sheaves of $\mathcal{O}$-modules over generalized schemes. Durov wrote a draft text in Russian about vectoids (which I have seen but is not released yet), and a video of a talk at Steklov. Is there a canonical right exact structure on a vectoid for which the spectrum of the right exact category in the sense of Rosenberg gives a sensible reconstruction theorem, it would be very interesting to investigate.

Remark: I believe that that the quantum flag variety of Lunts-Rosenberg is isomorphic in the case of $SL_n$ to the flag variety studied in my thesis from the dual point of view and with explicit Ore localizations. I never had time to check and publish all the details.

Answer by Zoran Škoda for Functorial point of view for formal schemes

2010-07-23T10:57:25Z

There are many nonequivalent generalities in which one can define a formal scheme, for example the definitions in Hartshorne and in EGA are not quite the same. (I use in this answer some parts of my own editing in nlab's entry where more references can be found). In my understanding, whatever the definition is, the category of formal schemes is a realization of certain subcategory of Ind-schemes. Typically one requires at least that the [[ind-object]] in the subcategory may be represented by a diagram whose connecting morphisms are closed immersions of schemes. A pretty modern treatment is in

A. Beilinson, V. Drinfel'd, Quantization of Hitchin's integrable system and Hecke eigensheaves on Hitchin system, preliminary version (pdf)

Some subcategories of Ind-objects in many algebraic categories can be described by putting the topology on algebraic objects. Thus the complete local rings, or more general the pseudocompact case, in the Grothendieck's approach to local schemes. One can use a topological version of Yoneda on rings to get a nice theory of formal schemes, over an arbitrary ring:

B. Pareigis, R. A. Morris, Formal groups and Hopf algebras over discrete rings, Trans. Amer. Math. Soc. 197 (1974), 113--129 (doi).

Nikolai Durov suggests to use directly the Gabriel-Demazure approach but not over Aff but over the opposite to the category of pairs (commutative ring, nilpotent ideal). Formal schemes should be an appropriate subcategory of that category of presheaves. That larger category (but without singling out there the smaller subcategory which would correspond more precisely to Grothendieck's formal schemes) is sketched in ch. 7-9 of

N. Durov, S. Meljanac, A. Samsarov, Z. Škoda, A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, Journal of Algebra 309, n. 1, 318--359 (2007) (doi:jalgebra) (math.RT/0604096).

Answer by Zoran Škoda for What is a noncommutative fiber bundle?

2010-06-29T11:42:29Z

One can consider torsors (principal bundles) within any sufficiently nice category and with respect to descent with respect to given Grothendieck topology and given fibered category over the ground site. See for example

Tomasz Brzeziński, On synthetic interpretation of quantum principal bundles, AJSE D - Mathematics 35(1D): 13-27, 2010 arxiv:0912.0213.

where in the main motivational part the codomain fibration and regular epimorphism topology are implicitly used. For the fibered category of modules (viewed as quasicoherent sheaves) over noncommutative rings, the faithfully flat Hopf-Galois extensions are the answer provided we accept Hopf coactions as dual representations of group actions. The tensor product is not a monoidal product in the category of associative algebras so this is a bit of a problem. Next thing is that one needs to consider nonaffine objects, if one is in algebraic framework, what can allow for more general concept of noncommutative principal bundles over the covers by noncommutative localizations which are analogues of covers in a Grothendieck topology. This kind of noncommutative principal bundles were skecthed in my articles

Z. Škoda, Localizations for construction of quantum coset spaces, math.QA/0301090, Banach Center Publ. 61, pp. 265--298, Warszawa 2003;
Z. Škoda, Coherent states for Hopf algebras, Letters in Mathematical Physics 81, N.1, pp. 1-17, July 2007. (earlier arXiv version: math.QA/0303357),

based on general picture of actions in noncommutative algebraic geometry as explained in the newer survey

Z. Škoda, Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183--202, arXiv:0811.4770.

See also the nlab:noncommutative principal bundle. I will hopefully release 2 more articles in this direction within next month or two.

Yes, for the spaces of sections of associated bundles with structure Hopf algebra one uses the cotensor product construction in the affine case; this is well known in the literature; the recipe can also be globalized by gluing along localizations. However this does not give the total spaces of associated bundles (in the category of noncommutative spaces) in satisfactory way in general, but only the spaces of sections.

Answer by Zoran Škoda for Major mathematical advances past age fifty

2010-05-24T14:34:02Z

This is not really an answer but an objection to most of the answers at this pages and in particular to not so well formed question (it does not do justice to Hardy's book in my opinion).

If you read the whole chapter of Hardy's book where the excerpt is from, Hardy explains somewhere that he does not know a highest class mathematician whose best discoveries came after 50. I recall after reading the whole chapter that I was convinced with the bulk of text that Hardy meant that there are no major advances by a mathematician after 50, unless they had major discoveries also before 50. So Euler and Poincare are not counterexamples to Hardy's experience, and some other answers in this column are not as well! Of course some people completed earlier work after 50, or continued with major advances while they already became major mathematicians before, but do you really a know a mathematician who done no major research before 50 and done such world class research after 50 ?? Also do not look the publication dates but the creation dates.

Answer by Zoran Škoda for Mathematical Physics? (Particularly computational)

2010-05-18T14:31:55Z

There is a problem with this kind of question, namely for many mathematicians the most interesting mathematical physics is a new vast area on the interface of quantum field theory and geometry/topology emerging from about late 1960s till now. You will find no word on this new mathematical physics in the classical books like Reed-Simon, Morse-Feshbach (Methods of mathematical physics, 1953 and later ed.), Vladimirov (Equations of mathematical physics) and even older Courant-Hilbert which focus on the integral and differential equations of mathematical physics, special functions, generalized functions (distributions), representations of classical groups and functional analysis. For your classical hydrodynamics indeed the classical textbooks and reference books suffice, but for people interested in a bit more modern mathematical physics we could add (in various level of exposition and specialization)

Yvonne Choquet-Bruhat, Cecile Dewitt-Morette, Analysis, manifolds and physics, 1982 and 2001
Albert Schwartz, Quantum field theory and topology, Grundlehren der Math. Wissen. 307, Springer 1993. (translated from Russian original)
Bernard F. Schutz, Geometrical methods of mathematical physics (elementary intro)
Eberhard Zeidler, Quantum field theory. A bridge between mathematicians and physicists. I: Basics in mathematics and physics. , II: Quantum electrodynamics
Charles Nash, Differential topology and quantum field theory, Acad. Press 1991.
P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and E. Witten, eds. Quantum fields and strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)
Gregory L. Naber, Topology, geometry, and gauge fields: interactions
Mikio Nakahara, Geometry, topology and physics
Peter Olver, Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, UK, 1995.
James Glimm, Arthur Jaffe, Quantum physics: a functional integral point of view, Springer
Sternberg, Shlomo (1994), Group theory and physics, Cambridge University Press.
V. I. Arnold, Mathematical methods of classical mechanics, Springer (1989).
V. Guillemin, S. Sternberg, Symplectic techniques in physics, Cambridge University Press (1990)
Leon A. Takhtajan, Quantum mechanics for mathematicians, Graduate Studies in Mathematics 95, Amer. Math. Soc. 2008.
Marian Fecko, Differential geometry and Lie groups for physicists
V. S. Varadarajan, Supersymmetry for mathematicians: an introduction, AMS and Courant Institute, 2004.
R. E. Borcherds, A. Barnard, Lectures on QFT, arxiv:math-ph/0204014
Paul Aspinwall, Tom Bridgeland, Alastair Craw, Michael R. Douglas, Mark Gross, Dirichlet branes and mirror symmetry, Amer. Math. Soc. Clay Math. Institute 2009.
R. S. Ward, R. O. Wells, Twistor geometry and field theory (CUP, 1990)
N. N. Bogoliubov, A. A. Logunov, I. T. Todorov, Introduction to axiomatic quantum field theory, 1975
O. Babelon, D. Bernard, M. Talon, Introduction to classical integrable systems, Cambridge Univ. Press 2003.
Martin Schottenloher, A mathematical introduction to conformal field theory
Philippe Di Francesco,Pierre Mathieu,David Sénéchal, Conformal field theory, Springer 1997
T. Miwa, M. Jimbo, E. Date, Solitons: Differential equations, symmetries and infinite dimensional algebras, Cambridge Tracts in Mathematics 135, translated from Japanese by Miles Reid
V. Kac, Vertex algebras for beginners, Amer. Math. Soc.
Ludwig D. Faddeev, Leon Takhtajan, Hamiltonian methods in the theory of solitons, Springer
V.E. Korepin, N. M. Bogoliubov, A. G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge Univ. Press 1997.
N. P. Landsman, Mathematical topics between classical and quantum mechanics, Springer Monographs in Mathematics 1998. xx+529 pp.
Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, pdf
A. Cannas da Silva, A. Weinstein, Geometric models for noncommutative algebras, 1999, pdf

I have placed these references in new nlab entry books and reviews in mathematical physics which will be updated at times wuth more specialized references.

Answer by Zoran Škoda for What is descent theory?

2010-04-21T18:47:03Z

Suppose we are given some category (or higher category) of "spaces" in which each space $X$ is equipped with a fiber, i.e. a category $C_X$ of objects of some type over it. For example, a space can be a smooth manifold and the fiber is the category of vector bundles over it; or a space is an object of the category dual to the category of rings and the fiber is its category of left modules. Given a map $f: Y\to X$, one often has an induced functor $f^* : C_X\to C_Y$ (pullback, inverse image functor, extension of scalars). The basic questions of classical descent theory are:

When an object $G$ in $C_Y$ is in the image via $f^*$ of some object in $C_X$ ?
Classify all forms of object $G\in C_Y$, that is find all $E\in C_X$ for which $f^*(E)\cong G$.

Grothendieck introduced pseudofunctors and fibred categories to formalize an ingenious method to deal with descent questions. He introduces additional data on an object $G$ in $C_Y$ to have a chance of determining an isomorphism class of an object in $C_X$. Such an enriched object over $X$ is called a ``descent datum''. $f$ is an effective descent morphism if the morphism $f$ induces a canonical equivalence of the category of the descent data (for $f$ over $X$) with $C_X$. It is a nontrivial result that in the case of rings and modules, the effective descent morphisms are preciselly pure morphisms of rings. Grothendieck's flat descent theory tells a weaker result that faithfully flat morphisms are of effective descent. In algebraic situations one often introduces a (co)monad $T_f : C_X\to C_X$ (say with the multiplication $\mu: T_f \circ T_f \to T_f$) induced by the morphism $f$. The category of descent data is then nothing else than the Eilenberg-Moore category $T_f-\mathrm{Mod}$ of (co)modules (also called (co)algebras) over $T_f$. Then, by the definition, $f$ is of an effective descent if and only if the comparison map (defined in the (co)monad theory) between $C_X$ and $T_f-\mathrm{Mod}$ is an equivalence. Several variants of Barr-Beck theorem give conditions which are equivalent or (in some variants) sufficient to the comparison map for a monad induced by a pair of adjoint functors being an equivalence. Generically such theorems are called monadicity (or tripleability) theorems. One can describe most of (but not all) situations of 1-categorical descent theory via the monadic approach.

There are numerous generalizations of monadicity theorems, higher cocycles and descent, both in monadic and in fibered category setup in higher categorical context (Giraud, Breen, Street, K. Brown, Hermida, Marmolejo, Mauri-Tierney, Jardine, Joyal, Simpson, Rosenberg-Kontsevich, Lurie...); the theory of stacks, gerbes and of general cohomology is almost the same as the general descent theory, in a point of view.

For examples, it is better to consult the literature. It takes a while to treat them.

Answer by Zoran Škoda for Definition of an algebra over a noncommutative ring

2010-04-20T03:51:08Z

The commutative notion of an (associative or not) algebra $A$ over a commutative ring $R$ has two natural generalization to the noncommutative setup, but the one you list with defined left $R$-linearity in both arguments is neither of them; in particular your multiplication does not necessarily induce a map from the tensor product, unless the image of $R$ is in the center. Most useful is the notion of an $R$-ring $A$ (or a ring $A$ over $R$), which is just a monoid in the monoidal category of $R$-bimodules: in other words the multiplication is a map $A\otimes A\to A$ which is left linear in first and right linear in the second factor. If we drop the associativity for the multiplication all works the same way, but I do not know if there is a common name (maybe descriptive like magma internal to the monoidal category of $R$-bimodules; or one may try a rare term nonassociative $R$-ring).

In the commutative case, the mutliplication is both left and right linear in each factor, what is here possible only if $R$ maps into the center of $A$. (Edit: I erased here one additional nonsense sentence clearly written when tired ;) ). Thus the two useful concepts in the noncommutative case are $R$-rings (possibly nonassociative!) and, well, the subclass with that property: $R$ maps into $Z(A)$, deserving the full name of "algebra". There is also a notion of $R$-coring, which is a comonoid in the monoidal category of $R$-bimodules, generalizing the notion of an $R$-coalgebra to a noncommutative ground ring.

Edit: I suggest also this link.

Answer by Zoran Škoda for Connections on principal bundles via stacks?

2010-04-20T14:55:26Z

The only thing which has to be replaced in the representation of a principal bundle as a suitable class of a maps $M\to [*/G]$ in order to introduce a flat connection is to replace $M$ by its fundamental 1-groupoid $\Pi_1(M)$, or, if the connection is not flat, by the thin homotopy version $P_1(M)$ of it, cf. nlab:path groupoid. The same way it works for higher categorical generalizations, see Schreiber:differential nonabelian cohomology and for details also Sec. 7.4 (from page 27 on in version 1) in arxiv/1004.2472.

Answer by Zoran Škoda for What to expect from attending an ICM?

2010-04-08T12:41:51Z

I was attending ICM 2002 in Beijing, in my first postdoc year, which was wonderfully organized. I expected that as a general mathematics meeting, the talks should be very accessible, but even plenary talks (with few exceptions, like the talks by Mumford and Hopkins) were accessible (for nonexperts) only about first 15 minutes or so, depending on your attention stamina in the midst of flood of data and the background. My background is reasonably wide, and I am used to find myself well at conferences in a number of area, but most of the talks at ICM were too detailed and fast. Some plenary talks had of the order of 45 transparencies. During the breaks people run in other lecture rooms, and in the mass of people people get lost around. I would recommend knowing and contacting in advance some people you like to hang out, improvising is more difficult at such big conferences than in small ones, which are mathematically good for me. It also depends on how far is the hotel from the site.

India is a wonderful country, which I visited in 2007; the summer is a bit too hot though Hyderabad is slightly in the hilly and drier area than big cities like Madras and Kolkatta; in India hygiene is often a problem (and much worse than in China), especially with vegetables and water (UV-filtered water which they use is OK, you do not need bottled water though). I brought a bottle of home made brandy and was taking a sip few times a day for precaution and never had a stomach problem, unlike most westerners.

I have math/physics friends in India, and if Croatian science funding were not in crisis (and still declining) I would go. I do not know your background, but if you come unprepared, without good acquaintances and so on, be aware of hectic atmosphere of big congresses and the needs of the climate, hygienic and other adaptation in subtropics.

Comment by Zoran Škoda

2013-02-15T15:55:35Z

Andy, I do not think his request is ridiculous. In old times, people like Morse-Feshbach wrote massive books which covered all math needed to do theoretical physics at that time. Regarding that much of those topics (excessive details on special functions and special solutions of PDEs etc.) can be taken away, or have easier replacements, those can now substitute with more modern content. In Russia the separation between math and physics in education did not grow that big as in the west where mathematicians lost feeling any more what to give to physicists, while the latter do not feel the call.

Comment by Zoran Škoda

2013-02-15T15:49:13Z

Princeton Companion is more focused on more classical areas like analysis, number theory, logic, combinatorics and graph theory, differential equations, while it largely neglects modern categorical and sheaf theoretical foundations and viewpoint and has very little on cohomological methods, virtually none on abstract homotopy, higher categories, model categories and derived categories.

Comment by Zoran Škoda

2013-02-15T15:44:45Z

Many of the topics listed above are to some extent covered in Yvonne Choquet-Bruhat, Cecile Dewitt-Morette, Analysis, manifolds and physics, 1982 and 2001. See also ncatlab.org/nlab/show/…

Comment by Zoran Škoda

2013-02-01T16:39:17Z

By the way, taking a commutative monoid in a symmetric monoidal category as an affine object in AG is usually more properly attributed (say, in Gabber's book and other follow ups) to Deligne, Grothendieck's Festschrift article on Tannakian formalism and not to much later work of Toen-Vaquie.

Comment by Zoran Škoda

2013-02-01T16:36:41Z

Saša, understanding the interrelations in the world we kind of know is worthy by itself.

Comment by Zoran Škoda

2013-01-07T14:17:48Z

Quoted is a very old version of Igor's manuscript, from 2009 or so. Igor had written in the meantime several much longer and more detailed manuscripts on the subject; hopefully he will be soon happy enough with them to post to the arXiv. There are also available slides from his Cambridge talk on a related topic of fibrations within tricategories <dpmms.cam.ac.uk/~jg352/pdf/pssl93talks/…;. Igor this December gave a talk on the present state of the subject at Paris VII.

Comment by Zoran Škoda

2012-07-03T11:14:17Z

Neeraj, justification of any path in intellectual pursuit is inner, and the typical requirements of some sponsor institution who get hysteric about something they do not understand about are not invalidating the inner turns. I do not understand "does not make sense". If somebody gets interested in something new, the interest itself generates large momentum. Instead I think that a restriction to what someone will live for does not make sense, and needs why ?

Comment by Zoran Škoda

2012-07-03T11:09:37Z

The question is very unclear. "Acceptable" presupposes some sort of value system of authorities who accept or not. Is it about your parents, your ex-advisor, your ex-sponsor, your beer pals, colleagues in your thesis area, or a prospective committee for a tenure track hiring in USA or China ?

Comment by Zoran Škoda

2012-02-28T10:23:11Z

Martin, this is a useful comment, but not an answer to his question: is it possible to axiomatize the theory of local rings as a coherent theory or at least a geometric theory in the technical sense of the word.

Comment by Zoran Škoda

2012-02-14T14:42:35Z

Johaness and Martin: what about Dubuc's C-inf schemes and Moerdijk-Reyes models of smooth infinitesimal analysis (synthetic differential geometry) -- they are based on $C^\infty$-rings. ncatlab.org/nlab/show/smooth%20algebra

Comment by Zoran Škoda

2012-02-14T14:39:03Z

lierre: it is not a choice of coordinates but an existence of coordinates. Coordinate free means that there is some low level language which refers to coordinates but does not depend on choices, and then one builds up on this low level language rarely or almost never needing in general considerations to unpack into the lowest level language.

Comment by Zoran Škoda

2012-02-14T14:32:17Z

One thing is "easiest" for the beginning of the theory, and another is what a power may be gained when going to subtleties and extensions of the theory. By no means introduction of sheaves means one should leave other tools like coordinates. One often needs to extend some constructions from manifolds to categories closed under interesting operations like quotients etc. It is easier to pass to e.g. supermanifolds, stacks and derived geometry, understanding the sheaf aspect of the story first, not only about sheaves on spaces, but also understanding of spaces as sheaves on site of local models.

Comment by Zoran Škoda

2012-01-20T21:01:39Z

See Parshall, Wang book/memoirs volume.

Comment by Zoran Škoda

2012-01-20T20:22:28Z

vote to close...

Comment by Zoran Škoda

2012-01-10T12:04:19Z

There is some hesitation between terminology continuous and cocontinuous. Bass used right and left continuous and right continuous was more important in module theory, so some retained continuous for what pure category theorists say cocontinuous. See also Lurie's book which also has it that way (and also Rosenberg who follows Bass).