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https://mathoverflow.net/questions/5209 | 5 | There are a few standard notions of matrix derivatives, e.g.
* If *f* is a function defined on the entries of a matrix *A*, then one can talk about the matrix of partial derivatives of *f*.
* If the entries of a matrix are all functions of a scalar *x*, then it makes sense to talk about the derivative of the matrix a... | https://mathoverflow.net/users/498 | Notions of Matrix Differentiation | There is another interpretation of Elisha's question that I think has not yet been addressed: How, and to what extent, can you do differential calculus with functional expressions of square matrices? For instance, how do you differentiate $\exp(A)$, which is defined for all square matrices $A$?
There is a good answer... | 9 | https://mathoverflow.net/users/1450 | 5318 | 3,579 |
https://mathoverflow.net/questions/5321 | 25 | In order get a minimal model for a given a variety $X$, we can carry out a sequence of contractions $X\rightarrow X\_1\ldots \rightarrow X\_n$ in such a way that that every map contracts some curves on which the canonical divisor $K\_{X\_j}$ is negative.
Here we have, at least, the following technical problem: in con... | https://mathoverflow.net/users/1547 | Flips in the Minimal Model Program | I am also just learning this stuff, and I'm partly writing this out for my own benefit. Experts, please correct and up/down vote as appropriate!
The goal of the minimal model program is to give a standard, nonsingular, representative for each birational class of algebraic variety. As stated, this goal is too ambitiou... | 16 | https://mathoverflow.net/users/297 | 5341 | 3,594 |
https://mathoverflow.net/questions/5342 | 3 | I wanted to learn prediction, forecasting etc. I also have time series data on millions of online videos. I would like to test out prediction algorithms etc on this data set, for eg. Linear Prediction, Kalman filter.
Are there any good resources out there to get me started on those?
Edit: Let me rephrase the quest... | https://mathoverflow.net/users/1761 | Where can i find 'getting started' resources for statistical prediction | The following might be of some help
[linear regression](http://en.wikipedia.org/wiki/Linear_regression)
[Bayesian inference](http://en.wikipedia.org/wiki/Bayesian_inference)
(possibly).
| 2 | https://mathoverflow.net/users/1536 | 5350 | 3,601 |
https://mathoverflow.net/questions/5329 | 9 | I think I'm a bit confused about the relationship between some concepts in mathematical logic, namely constructions that require the axiom of choice and "explicit" results.
For example, let's take the existence of well-orderings on $\mathbb{R}$. As we all know after reading [this answer by Ori Gurel-Gurevich](https:... | https://mathoverflow.net/users/302 | "Requires axiom of choice" vs. "explicitly constructible" | In Goedel's proof of consistency of AC, we in fact get much more. There is an explicit relation defined, which is (provably in ZF) a well-ordering of a certain subset of the reals. It is consistent (and follows from the axiom V=L) that the subset is all of the reals.
| 11 | https://mathoverflow.net/users/454 | 5356 | 3,604 |
https://mathoverflow.net/questions/5323 | 29 | At the time of writing, question 5191 is closed with the accusation of homework. But I don't have a clue about what is going on in that question (other than part 3) [Edit: Anton's comments at 5191 clarify at least some of the things going on and are well worth reading] [Edit: FC's excellent answers shows that my lack o... | https://mathoverflow.net/users/1384 | Infinitely many primes of the form $2^n+c$ as $n$ varies? | Buzzard is correct to be skeptical of the most naive arguments: Erdos observed that $2^n + 9262111$ is never prime.
[**edit** Jan 2017 by Buzzard: the 9262111 has sat here for 7 years but there's a slip in Pomerance's slides where he calculates the CRT solution. The correct conclusion from Pomerance's arguments is th... | 34 | https://mathoverflow.net/users/nan | 5367 | 3,612 |
https://mathoverflow.net/questions/5344 | 25 | Given a compact smooth manifold $M$, it's relatively well known that $C^\infty(M)$ determines $M$ up to diffeomorphism. That is, if $M$ and $N$ are two smooth manifolds and there is an $\mathbb{R}$-algebra isomorphism between $C^\infty(M)$ and $C^\infty(N)$, then $M$ and $N$ are diffeomorphic.
(See, for example, Miln... | https://mathoverflow.net/users/1708 | Algebraic description of compact smooth manifolds? | There is a very cool answer to your question, and it goes by the name [well-adapted models](http://ncatlab.org/nlab/show/Models+for+Smooth+Infinitesimal+Analysis) for [synthetic differential geometry](http://ncatlab.org/nlab/show/synthetic+differential+geometry). Andrew Stacey already indicated it in his reply, but may... | 25 | https://mathoverflow.net/users/381 | 5368 | 3,613 |
https://mathoverflow.net/questions/5364 | 39 | I have several naive and possibly stupid questions about deformations of categories. I hope that someone can at least point me to some appropriate references.
What is a deformation of a (linear, dg, A-infinity) category? Is it a "bundle of categories" over a scheme? How can you make such a notion rigorous? Maybe via ... | https://mathoverflow.net/users/83 | What is a deformation of a category? | Certainly not my field, but you might want to check the paper by Lowen and Van den Bergh [Deformation theory of abelian categories](https://arxiv.org/abs/math/0405226). I think that's where the first notion of deformation of a category appeared.
| 17 | https://mathoverflow.net/users/914 | 5371 | 3,616 |
https://mathoverflow.net/questions/2339 | 18 | Suppose that $N$ is prime, and consider the normalized cuspidal Hecke eigenforms of weight 2 and level $\Gamma\_0(N)$.
For such an eigenform $f$, the coefficients generate (an order in) the ring of integers of a totally real field $E$. The corresponding simple abelian variety quotient $A\_f$ of $J\_0(N)$ has dimensio... | https://mathoverflow.net/users/nan | 2-adic Coefficients of Modular Hecke Eigenforms | A natural way to approach this question is to ask for an eigenform $f$ with coefficients in a local field $K$ with residue field degree $\ge 2$. Or, asking for slightly more, with coefficient ring $\mathcal{O}$ admitting a surjective map to a field $\mathbf{F}$ of order divisible by $4$ (this is slighly more since the ... | 12 | https://mathoverflow.net/users/nan | 5407 | 3,646 |
https://mathoverflow.net/questions/4665 | 36 | Fix an integer n. Can you find two finite CW-complexes X and Y which
\* are both n connected,
\* are not homotopy equivalent, yet
\* $\pi\_q X \approx \pi\_q Y$ for all $q$.
In [Are there two non-homotopy equivalent spaces with equal homotopy groups?](https://mathoverflow.net/questions/3540/) some solutions ar... | https://mathoverflow.net/users/437 | Are there pairs of highly connected finite CW-complexes with the same homotopy groups? | Here is a method for constructing examples. If a fiber bundle $F \to E \to B$ has a section, the associated long exact sequence of homotopy groups splits, so the homotopy groups of $E$ are the same as for the product $F \times B$, at least when these spaces are simply-connected so the homotopy groups are all abelian. T... | 34 | https://mathoverflow.net/users/23571 | 5431 | 3,665 |
https://mathoverflow.net/questions/5340 | 11 | Given that $Q\_d$ is the hypercube graph of dimension $d$ then it is a known fact (not so trivial to prove though) that given a perfect matching $M$ of $Q\_d$ ($d\geq 2$) it is possible to find another perfect matching $N$ of $Q\_d$ such that $M \cup N$ is a Hamiltonian cycle in $Q\_d$.
The question now is - given a ... | https://mathoverflow.net/users/1737 | Is every matching of the hypercube graph extensible to a Hamiltonian cycle | This is a known open problem. See "[Matchings extend to Hamiltonian cycles in hypercubes](http://garden.irmacs.sfu.ca/?q=op/matchings_extends_to_hamilton_cycles_in_hypercubes)" over at the Open Problem Garden.
| 15 | https://mathoverflow.net/users/1079 | 5433 | 3,667 |
https://mathoverflow.net/questions/5427 | 9 | I am wondering if there are analytic tools to find asymptotic formulae for the coefficients of a complex power series of a function with branch singularities. For example, it is possible to show using elementary means that, for $q>1$, the coefficients of $\frac{1}{1-z} log(\frac{1}{1-qz})$ are asymptotic to $\frac{1}{1... | https://mathoverflow.net/users/1770 | Asymptotics of Power Series With Branch Singularities | The answer is "Quite often yes, but the error terms are seldom as good as in the meromorphic case". The reason the asymptotics for the meromorphic functions works is that we know the exact coefficients for $c\_k(z-z\_0)^{-k}$ and can always remove the singularity by subtracting a few terms of this kind thus increasing ... | 8 | https://mathoverflow.net/users/1131 | 5434 | 3,668 |
https://mathoverflow.net/questions/5436 | 3 | Let X be a (finite dimensional) manifold. Consider smooth mapping space $$PX = C^\infty(I, X)$$ where I = [0,1] is the closed interval. Is this space paracompact? What if we fix a point x in X and consider the *pointed* path space, is this space paracompact?
| https://mathoverflow.net/users/184 | Are mapping spaces paracompact? | What topology do you want on $C^\infty(I,X)$? One natural topology, the one of uniform convergence of all derivatives, is metrizable, so paracompact.
| 5 | https://mathoverflow.net/users/1409 | 5438 | 3,671 |
https://mathoverflow.net/questions/5457 | 10 | I've been reading through Lurie's book on higher topos theory, where he develops the theory of $(\infty,1)$-toposes, which leads me to the following question: Is there any sort of higher topos theory on the more general $\omega$-categories, where we don't require all higher morphisms to be invertible?
| https://mathoverflow.net/users/1353 | $\omega$-topos theory? | The short answer is no. Even 2-toposes are poorly understood -- we don't know what the right definition is. For higher dimensions, including $\infty$, we *definitely* don't have the answers.
Just as the primordial example of a (1-)topos is $\mathbf{Set}$ (the 1-category of sets and functions), the primordial example ... | 9 | https://mathoverflow.net/users/586 | 5461 | 3,688 |
https://mathoverflow.net/questions/5378 | 47 | This question is being asked on behalf of a colleague of mine.
Let $X$ be a topological space. It is well known that the abelian category of sheaves on $X$ has enough injectives: that is, every sheaf can be monomorphically mapped to an injective sheaf. The proof is similarly well known: one uses the concept of "gener... | https://mathoverflow.net/users/1149 | When are there enough projective sheaves on a space X? | About [Jon Woolf's answer](https://mathoverflow.net/a/5401/64073), it seems to me that the condition that "$x$ is a closed point" was implicitly used: the extension by zero $Z\_A$ is only defined for a locally closed subset $A$ (see e.g. Tennison "*Sheaf theory,*" 3.8.6). So $X-x$ must be locally closed. How about the ... | 20 | https://mathoverflow.net/users/1784 | 5470 | 3,695 |
https://mathoverflow.net/questions/3344 | 6 | The other day I came across the statement that in the triangulated category $\mathfrak{KK}$ (of C\*-algebras with KK-groups as morphism sets) "there are many other sources of exact triangles besides extensions". Except for mapping cone triangles I don't know what is meant. What can you come up with?
I would also appr... | https://mathoverflow.net/users/1291 | Sources for exact triangles in triangulated categories. | This question has kind of been bothering me since I started thinking about it - I am far from an expert on KK-theory but I thought I'd throw something out there and maybe someone else will see it and come along and agree with me or correct me.
I think this statement is a way of thinking rather than something precise.... | 4 | https://mathoverflow.net/users/310 | 5472 | 3,697 |
https://mathoverflow.net/questions/5485 | 86 | Although we are not so numerous as other respected professionals, like for example lawyers, I wonder if we could come up with a reasonable estimate of our population.
Needless to say, the question more or less amounts to the definition of"mathematician".
Since I should like to count only research mathematicians (an... | https://mathoverflow.net/users/450 | How many mathematicians are there? | Current count of [Mathematics Genealogy Project](http://genealogy.math.ndsu.nodak.edu/) is 137672 (I am assuming that the PhD students that graduated are ranked as "research mathematicians"). But the problem is.. Mathematics Genealogy is mostly for universities of developed countries. There could be some really good un... | 25 | https://mathoverflow.net/users/1245 | 5487 | 3,709 |
https://mathoverflow.net/questions/5518 | 26 | The standard approach to proving that $H^n(X; G)$ is represented by $K(G, n)$ seems to be to prove that $\text{Hom}(X, K(G, n))$ defines a cohomology theory and then use Eilenberg-Steenrod uniqueness. This is utterly spiffing, but as far as I can see gives little geometric intuition. In his treatment, Hatcher mentions ... | https://mathoverflow.net/users/1202 | "Dirty" proof that Eilenberg-MacLane spaces represent cohomology? | I'd suggest looking up some basic material on obstruction theory. There, you generally find classification of maps $X \to Y$ with domain a CW-complex in terms of cohomology groups $H^s(X;\pi\_t(Y))$. The proofs are often very cellular indeed.
In the case where the range is an Eilenberg-Maclane space (for an abelian g... | 31 | https://mathoverflow.net/users/360 | 5521 | 3,739 |
https://mathoverflow.net/questions/5532 | 17 | It seems as though many consider ZF to be the foundational set of axioms for all of mathematics (or at least, a crucial part of the foundations); when a theorem is found to be independent of ZF, it's generally accepted that there will never be a proof of the theorem one way or another. My question is, why is this? It s... | https://mathoverflow.net/users/1455 | The Importance of ZF | Very few mathematicions these days wish to base their mathematics on ZF without the axiom of choice, as your question seems to imply. Yes, there is an intuitionist school about which I don't know a whole lot, but they seem to be quite the minority. So let's consider ZFC. I think the main philosophical argument for it i... | 11 | https://mathoverflow.net/users/802 | 5540 | 3,750 |
https://mathoverflow.net/questions/5522 | 10 | I recall the following question from Ulam's book "Unsolved math problems": show that the ring of Dirichlet series with integer coefficients is a factorial ring. I believe that soon after Ulam wrote his book, this problem was solved; essentially, this ring is isomorphic to the ring of formal power series in infinitely m... | https://mathoverflow.net/users/1306 | Dirichlet series with integer coefficients as a UFD | First, I want to nail down a reference to the solution to the problem alluded to above: the Dirichlet ring of functions f: Z+ -> Z with pointwise addition and convolution product is a UFD. This was proved by L. Durst in his 1961 master's thesis at Rice University and independently by Cashwell and Everett. References:
... | 4 | https://mathoverflow.net/users/1149 | 5549 | 3,756 |
https://mathoverflow.net/questions/5547 | 18 | I work in planar algebras and subfactors, where the idea of path algebras on a graph (alternately known as graph algebras, graph planar algebras, etc.) is quite useful. The particular result I'm thinking of is a forthcoming result of Jones and some others; it says that any subfactor planar algebra can be found inside t... | https://mathoverflow.net/users/699 | ubiquity, importance of path algebras | *As far as I can tell, "quiver" is a fancy word for a directed finite graph*
Yes. It doesn't even have to be finite.
*Or is there some philosophical reason path algebras are important?*
A huge application of path algebras lately is the path algebra of a quiver of Dynkin type. Following the ideas of Lusztig and Ri... | 10 | https://mathoverflow.net/users/1450 | 5556 | 3,762 |
https://mathoverflow.net/questions/5558 | 7 | What is the impact of Mathematics in social science today?. That is to say, what are the mathematics that a social scientist is using and, from the point of view of a mathematician, what are the mathematics they should use more or start out using.
Let me give a particular example to start with: Social Inequality. Eve... | https://mathoverflow.net/users/1547 | Math Vs Social Science | Right now, the idea of actually applying the results of Social Choice theory (aka voting theory) is gaining steam. There are plenty of mathematical results that show that plurality voting method (what US has) is one of the least effective ways to measure the will of the electorate, and something like Borda count or App... | 17 | https://mathoverflow.net/users/619 | 5575 | 3,778 |
https://mathoverflow.net/questions/5568 | 3 | I've heard about this construction on the lecture about **higher representation theory**:
>
> Given a Lie algebra $g$, one constructs $\mathcal A$, a category whose $K\_0$ is the universal enveloping algebra of $g$. Conjecture: any $\mathcal A$-acted triangulated category $\mathcal V$ (with its $K$ locally finite) ... | https://mathoverflow.net/users/65 | Categorifying the group representations | The $sl\_n$ version of this shouldn't so bad. I think it's just self-dual objects in parabolic category O and shuffling functors, though I'll admit, I haven't checked this myself, and doubt it's written properly somewhere. Probably the best reference is the papers of Brundan and Kleshchev (for example "Schur-Weyl duali... | 4 | https://mathoverflow.net/users/66 | 5592 | 3,793 |
https://mathoverflow.net/questions/5305 | 14 | If you have two Möbius transformations represented as:
$f(z) = \frac{az + b}{cz + d}$
$g(z) = \frac{pz + q}{rz + s}$
where $a, b, c, d, p, q, r, s, z \in \mathbb{C}$
Is it possible to derive a third function $h(z, t)$, where $t \in \mathbb{R}$ and $0 \leq t \leq 1$, which "smoothly" interpolates between the tra... | https://mathoverflow.net/users/1747 | How to smootly interpolate between möbius transformations? | I'd like to address two issues about how to implement these interpolations and what they mean. The group that you ask about is actually equivalent to the group of real M\"obius transformations. If you were in the upper half plane model, the coefficients would be real numbers. You actually don't need to explicitly use t... | 11 | https://mathoverflow.net/users/1450 | 5608 | 3,807 |
https://mathoverflow.net/questions/5597 | 6 | ... are all isomorphic to $l^1$ on some other index set. At least, that much I "know" from 2nd-hand sources, since the original proof is apparently in a paper of Köthe from the 1930s 1960s (in German) that I can't get hold of have had trouble digesting. Since there are some Banach space specialists reading MO, I wonder... | https://mathoverflow.net/users/763 | Closed, complemented subspaces of $l^1(X)$ when $X$ is uncountable | A proof in English (modulo some details involving the Pelczynski decomposition method) can be found in the article 'On relatively disjoint families of measures' (Studia Math, 37, p.28-29) by Haskell Rosenthal.
Regarding the analogous result for $\ell\_p (X)$ ($p\in (1,\infty)$) and $c\_0 (X)$, I seem to recall readin... | 7 | https://mathoverflow.net/users/848 | 5610 | 3,809 |
https://mathoverflow.net/questions/5553 | 16 | This is a follow-up to [a previous question](https://mathoverflow.net/questions/4912/ "Graphs with incidence matrices whose pseudoinverses are proportional to their transposes"). What graphs have incidence matrices of full rank?
Obvious members of the class: complete graphs.
Obvious counterexamples: Graph with more... | https://mathoverflow.net/users/1674 | Which graphs have incidence matrices of full rank? | The first answer identifies "incidence matrix" with "adjacency matrix". The latter is the
vertices-by-vertices matrix that Sciriha writes about. But the original question
appears to concern the incidence matrix, which is vertices-by-edges. The precise
answer is as follows.
Theorem: The rows of the incidence matrix of... | 27 | https://mathoverflow.net/users/1266 | 5621 | 3,814 |
https://mathoverflow.net/questions/5618 | 5 | This question has been inspired by [covering 3-torus post](https://mathoverflow.net/questions/5546/ramified-covers-of-3-torus).
>
> Is it true that any good (smooth, compact, oriented) $n$-manifold can be mapped to $S^n$ in such a way that the map is true covering away from codimension 2?
>
>
>
| https://mathoverflow.net/users/65 | Ramified covers of S^n | Yes. See Feighn's short [note](http://www.raco.cat/index.php/CollectaneaMathematica/article/view/56965) "Branched covers according to J.W. Alexander".
| 11 | https://mathoverflow.net/users/1650 | 5626 | 3,819 |
https://mathoverflow.net/questions/5179 | 12 | I'm reading the classical Brown-Gersten's paper "Algebraic K-theory as generalized sheaf cohomology" and I'm stuck with their choose of global fibrations. Namely, a morphism of simplicial sheaves $p : E \longrightarrow B$ is a global fibration if for every inclusion of open sets $U\subset V$ the natural map $E(V) \long... | https://mathoverflow.net/users/1246 | global fibrations of simplicial sheaves | For model structures on simplicial sheaves, there is a difference between the Joyal-Jardine approach and the Brown-Gersten approach. This is well explained in Voevodsky's preprint: *Homotopy theory of simplicial presheaves in completely decomposable topologies*, available [here](https://arxiv.org/abs/0805.4578). Briefl... | 5 | https://mathoverflow.net/users/349 | 5637 | 3,827 |
https://mathoverflow.net/questions/5635 | 170 | Purely for fun, I was playing around with iteratively applying $\DeclareMathOperator{\Aut}{Aut}\Aut$ to a group $G$; that is, studying groups of the form
$$ {\Aut}^n(G):= \Aut(\Aut(\dots\Aut(G)\dots)). $$
Some quick results:
* For finitely-generated abelian groups, it isn't hard to see that this sequence eventual... | https://mathoverflow.net/users/750 | Does $\DeclareMathOperator\Aut{Aut}\Aut(\Aut(\dots\Aut(G)\dots))$ stabilize? | I remember that my old grad classmate from Berkeley, Joel David Hamkins, worked on the transfinite version of this problem. [The Automorphism Tower Problem](http://www.math.rutgers.edu/%7Esthomas/book.ps), by Simon Thomas, is an entire book on this subject. The beginning of the book gives the example of the infinite di... | 134 | https://mathoverflow.net/users/1450 | 5638 | 3,828 |
https://mathoverflow.net/questions/5546 | 10 | It is known that every orientable 3-manfiold can be obtained as a ramified cover of S3 with a ramification (of some order) at a link in S3. I am curious if there is a reasonable characterization of 3-manifolds that cover 3-torus?
Added. Notice that such a manifold is enlargeble, so it does not admit a metric of posi... | https://mathoverflow.net/users/943 | Ramified covers of 3-torus | Note that a branched covering induces an injection of rational cohomology rings, by transfer considerations. Therefore the cohomology of a manifold that is a branched covering of $T^3$ must contain three classes of degree 1 whose triple cup product is nontrivial.
This condition on a manifold $M^3$ implies the existe... | 13 | https://mathoverflow.net/users/1822 | 5644 | 3,834 |
https://mathoverflow.net/questions/5650 | 2 | The [splitting lemma](http://en.wikipedia.org/wiki/Splitting_lemma) says:
>
> Given a short exact sequence with maps $q$ and $r$:
>
>
> $0 \rightarrow A \overset{q}{\rightarrow} B \overset{r}{\rightarrow} C \rightarrow 0$
>
>
> then the following are equivalent:
>
>
> 1. ...
> 2. there exists a map $u : C \ri... | https://mathoverflow.net/users/1825 | Splitting lemma under assumption of the axiom of choice | I assume you are working in some fixed abelian category $\mathcal{A}$.
It is not true in general that every short exact sequence in $\mathcal{A}$ will split. The problem is that although you can pick a preimage for every 'element' $c\in C$ there is no guarantee that you can assemble this into a morphism in $\mathcal{... | 7 | https://mathoverflow.net/users/310 | 5653 | 3,841 |
https://mathoverflow.net/questions/5611 | 24 | If K is a number field then the Dedekind zeta function Zeta\_K(s) can be written as a sum over ideal classes A of Zeta\_K(s, A) = sum over ideals I in A of 1/N(I)^s. The class number formula follows from calculation of the residue of the (simple) pole of Zeta\_K(s, A) at s = 1 (which turns out to be independent of A).
... | https://mathoverflow.net/users/683 | The class number formula, the BSD conjecture, and the Kronecker limit formula | I apologize for in advance for making just a few superificial remarks. These are:
1. The question is not uninteresting. Just because Sha doesn't appear in the definition of L easily, there's no reason one shouldn't ask about manifestations more fundamental than the usual one.
2. An approach might be to think about th... | 12 | https://mathoverflow.net/users/1826 | 5655 | 3,843 |
https://mathoverflow.net/questions/5666 | 2 | For the question, everything is over an algebraically closed field and by a scheme we mean a scheme of finite type. The theorem of the cube is the following:
Let $X$ be a complete variety, $Y$ a scheme, and $L$ a line bundle on $X\times Y$. Then there is a unique closed subscheme $Y\_1$ of $Y$ satisfying the following ... | https://mathoverflow.net/users/1468 | An application of the Künneth formula in the proof of the theorem of the cube | One uses the following trick.
By the projection formula we have
$${p\_2}\_\*(\mathcal{O}\_{X\times Y\_1}) \cong {p\_2}\_\*(p\_2^\*M\_1^{-1} \otimes L\_1) \cong M\_1^{-1} \otimes {p\_2}\_\*(L\_1)$$
and since $X$ is complete (and $k$ is algebraically closed) it follows from the Künneth formula that
$${p\_2}\_\*\mathca... | 3 | https://mathoverflow.net/users/310 | 5669 | 3,853 |
https://mathoverflow.net/questions/5299 | 5 | How do you define unbounded measurable functions for a general von Neumann algebra?
For the commutative algebra $L^\infty(X,\mu)$, we can consider the space of all measurable functions that are almost everywhere finite. This set has certain nice properties: it is closed under multiplication and there is the notion of... | https://mathoverflow.net/users/1704 | Measurable functions and unbounded operators in von Neumann algebras | I think your question should be as follows:
Given a von Neumann algebra $M$, can we define a canonical set $S$ such that
* if $M=L^\infty(X)$ acting on $L^2(X)$, then $S$ is (isomorphic to) the set of a.e. defined measurable functions, and
* if $M=B(H)$ acting on $H$, then $S$ is the **densely defined, closed** unb... | 6 | https://mathoverflow.net/users/351 | 5679 | 3,857 |
https://mathoverflow.net/questions/5658 | 3 | $l\_1$ minimization / compressed sensing comes to mind. Does anyone have any concrete examples? Or is such a construct completely useless?
| https://mathoverflow.net/users/1745 | Is there a use for a Hilbert space that uses a different norm than the one induced by the inner product? | One place where this is used is with Hilbert bundles. Taking $L^2$-functions on a space generally works very badly, so one instead takes $L^{2,1}$-functions - functions which are differentiable (almost everywhere) with square-integrable first derivative. However, the transition functions aren't isometries with respect ... | 7 | https://mathoverflow.net/users/45 | 5683 | 3,860 |
https://mathoverflow.net/questions/5684 | 11 | If I have the periods $$\pi\_1(\lambda)=\int\_0^1\frac{dx}{\sqrt{x(x-1)(x-\lambda)}}$$ and $\pi\_2(\lambda)$ similarly defined of the cubic curve $$y^2z=z(x-z)(x-\lambda z)$$ Such functions will be holomorphic on $\lambda \in \mathbb{C}-\{0,1\}$. Then the sum $\pi(\lambda)=\pi\_1(\lambda)+\pi\_2(\lambda)$, as well as e... | https://mathoverflow.net/users/1547 | Picard-Fuchs equations | Here is a very rough outline:
Take your family $E$ of elliptic curves over $B := \mathbb{C} - \{0,1\}$. Then take the associated "(co)homology bundle" over $B$, whose fibre over $\lambda$ is the (singular) (co)homology of the elliptic curve $E\_\lambda$. To be rigorous, the $i$-th cohomology bundle is $R^i \pi\_\ast\... | 17 | https://mathoverflow.net/users/83 | 5695 | 3,868 |
https://mathoverflow.net/questions/5691 | 5 | The exercise is the following:
>
> Let $A$ be a valuation ring, $K$ its field of fractions. Show that every subring of $K$ which contains $A$ is a local ring of $A$.
>
>
>
Does anyone know what is meant by "to be a local ring of a valuation ring"?
| https://mathoverflow.net/users/1841 | Atiyah-MacDonald: exercise 5.29 - "local ring of a valuation ring" | Let B be the subring of K containing A. I am going to prove that B is the localization of A at a prime ideal $P\subset A$, which seems a reasonable interpretation of the statement that B is a local ring of A.
First B is a local ring [by Atiyah-MacDonald Prop 5.18 i)] ; let $M\_B$ be its maximal ideal.
Similarly let $... | 4 | https://mathoverflow.net/users/450 | 5703 | 3,875 |
https://mathoverflow.net/questions/5085 | 3 | I have a 2D rectangular domain. The governing equation on this domain is Laplace equation:
$\nabla^2 f = 0$
In the left edge there is [Neumann boundary conditon](http://en.wikipedia.org/wiki/Neumann_boundary_condition) :
$\frac{\partial f}{\partial n} = -a$
n is the normal vector to the domain's boundary(here o... | https://mathoverflow.net/users/1591 | How to apply Neuman boundary condition to Finite-Element-Method problems? | You need to modify right-hand vector b of an equation Kx = b, where K is your stiffness matrix.
Here's how to do it, depending on which edge is on von Neumann boundary:
* edge 1-2 (i.e. connecting local nodes 1 and 2),
```
l = sqrt(x21*x21 + y21*y21);
b[node1] += a * l / 2.0f;
b[node2] += a * l / 2.0f;
b[node3]... | 1 | https://mathoverflow.net/users/1846 | 5718 | 3,885 |
https://mathoverflow.net/questions/5723 | 3 | I'm trying to understand the Hodge-Index Theorem at the moment. What does it say explicitly for the case of $\mathbb{CP}^2$?
| https://mathoverflow.net/users/1095 | Hodge-Index Theorem for $\mathbb{C}P^2$ | That the inner product on $H^2(\mathbb{CP}^2)$ has signature $(1,0)$.
| 4 | https://mathoverflow.net/users/297 | 5725 | 3,891 |
https://mathoverflow.net/questions/5711 | 5 | The Fell-Doran problem is a problem in functional analysis. It goes as follows: Let $A$ be a complex unital algebra, $X$ a locally convex space, and $L(X)$ the algebra of all continuous endomorphisms of $X$. Suppose that we have a representation of $A$ on $X$, by which we simply mean an algebra homomorphism
$$
T : A \... | https://mathoverflow.net/users/401 | Is the Fell-Doran problem trivial in a topological setting? | I don't agree with Andrew: more specifically, if $A$ is the algebra of compact operators on a Hilbert space $H$, then let $T\in L(H)$. If, say, $H$ is separable, then let $(e\_n)$ be an orthonormal sequence, and let $P\_n$ be the orthogonal projection on the span of $e\_1,\cdots,e\_n$. Then $P\_n$ is compact, and $P\_n... | 2 | https://mathoverflow.net/users/406 | 5729 | 3,894 |
https://mathoverflow.net/questions/5733 | 9 | Let $X$ be a random variable with mean $0$ and variance $1$, and let $X\_1, X\_2, X\_3, \dots$ be iid copies of $X$. Under what conditions can we say that the density of $\frac{X\_1+\dots+X\_n}{\sqrt{n}}$ converges pointwise to $N(0,1)$?
In particular, when can I say that for any sequence $\epsilon\_n \rightarrow 0$ ... | https://mathoverflow.net/users/405 | When does a pointwise CLT hold? | Bounded density will suffice, I think. Basically what one needs is for the Fourier transforms (aka characteristic functions) of the $X\_1 + \ldots + X\_n / \sqrt{n}$ to converge pointwise to the Fourier transform of normal distribution while being dominated by something integrable plus something whose L^1 norm goes to ... | 15 | https://mathoverflow.net/users/766 | 5738 | 3,901 |
https://mathoverflow.net/questions/5740 | 9 | If $d(x,y)$ and $e(x,y)$ are metrics then $d(x,y)+e(x,y)$ and $\frac{d(x,y)}{1+d(x,y)}$ are metrics.
If $d\_i(x,y)$ for $i=1,\dots,n$ are metrics then so is $\sqrt{\sum\_{i=1}^n{d\_i^2(x,y)}}$
Are there other interesting ways of constructing new metrics from old metrics?
| https://mathoverflow.net/users/812 | What are some interesting ways of making new metrics out of old metrics? | Personally, I prefer $\min(d(x,y),\epsilon)$ over the standard $d(x,y)/(1+d(x,y))$ trick if the goal is to turn a metric into an equivalent metric – it's a lot easier to prove the metric axioms for this one.
From that follows one way to turn a countably infinite number of metrics into a new metric: $d(x,y)=\sum\\_{n=... | 8 | https://mathoverflow.net/users/802 | 5748 | 3,907 |
https://mathoverflow.net/questions/5745 | 5 |
>
> For a fixed $d$, is there a relationship between the homotopy groups of smooth $d$-manifolds?
>
>
>
The $d=1$ case is trivial, but I already don't know how to approach $d=3$ (I should have said that the case of $d=2$ is simple as well, since there is only a sphere to consider, but I don't know how to formula... | https://mathoverflow.net/users/65 | Homotopy groups of smooth manifolds? | For $d=3$ the homotopy groups can be pretty elaborate. Consider the connect-sum of some lens spaces. The universal cover embeds in $S^3$ as the complement of a cantor set (except for a few degenerate cases where you have $\mathbb RP^3$ summands). So the homotopy-groups are pretty complicated ($\pi\_2$ is finitely gener... | 9 | https://mathoverflow.net/users/1465 | 5749 | 3,908 |
https://mathoverflow.net/questions/5196 | 0 | Let $L:K$ be a field extension. Let $A$ be a set of elements in $L$, all of which are algebraic over $K$. Construct the field extension $M=K(A)$. I have two questions:
[1] Is $M:K$ an algebraic field extension?
[2] Take $\beta\in M$ where $\beta$ is algebraic over $K$. Then $K(\beta):K$ is a finite extension. Can I... | https://mathoverflow.net/users/801 | On algebraic field extensions | The question was a bit confusing because it was not clear what was meant as the definition of "the field extension $K(A)$". Nick says that "the smallest field containing" is the definition that he wanted, to make the question non-trivial. (But still routine, in my opinion.)
Let's agree then that both (1) and (2) are ... | 3 | https://mathoverflow.net/users/1450 | 5763 | 3,917 |
https://mathoverflow.net/questions/5760 | 27 | In particular, suppose I have an affine algebraic variety over $\mathbb{R}^n$ described by generators of a radical ideal $I$ and I want to find (perhaps not all of the) points in the variety. Several important questions come up in practice:
1. are there versions of Buchberger's algorithm that work with inexact data? ... | https://mathoverflow.net/users/1557 | Can Gröbner bases be used to compute solutions to large, real-world problems? | First, the Gröbner basis is not sparse. I am speaking a little off-the-cuff, but empirically when I ask SAGE for the Gröbner basis of $(y^n-1,xy+x+1)$ in the ring $\mathbb{Q}[x,y]$, it gets worse and worse as $n$ increases. Any bound would have to be in terms of the degrees of the original generators as well as their s... | 23 | https://mathoverflow.net/users/1450 | 5767 | 3,921 |
https://mathoverflow.net/questions/5772 | 22 | What is the exact relationship between principal bundles, representations, and vector bundles?
| https://mathoverflow.net/users/1648 | Principal bundles, representations, and vector bundles | Let G be an algebraic group (or, since the question was tagged as differential geometry, a Lie group). Then if we're given a principal G-bundle $E\_G$ and a representation V of G, we get a vector bundle out of this through the associated bundle construction: $(E\_G \times V)/G$ is a vector bundle with generic fiber V. ... | 28 | https://mathoverflow.net/users/916 | 5776 | 3,927 |
https://mathoverflow.net/questions/5769 | 8 | I'm trying to understand the Hodge-Index Theorem at the moment. What does it say explicitly for the case of the complex Grassmannian Gr($n,k$), and can this be established without recourse to the theorem?
| https://mathoverflow.net/users/1095 | Hodge Index Theorem for Gr(n,k) | The answer is a happy surprise for me: The Hodge index theorem for a Grassmannian matches a special case of John Stembridge's $q=-1$ phenomenon, that I also studied in an old paper.
First, some generalities about what is going on, and about when you do or don't "need" the Hodge Index Theorem. I am following the Hodge... | 14 | https://mathoverflow.net/users/1450 | 5780 | 3,929 |
https://mathoverflow.net/questions/5786 | 89 | Because adjoint functors are just cool, and knowing that a pair of functors is an adjoint pair gives you a bunch of information from generalized abstract nonsense, I often find myself saying, "Hey, cool functor. Wonder if it has an adjoint?" The problem is, I don't know enough category theory to be able to check this f... | https://mathoverflow.net/users/382 | How do I check if a functor has a (left/right) adjoint? | The adjoint functor theorem [as stated here](http://en.wikipedia.org/wiki/Adjoint_functor_theorem#Existence) and the special adjoint functor theorem (which can also both be found in Mac Lane) are both very handy for showing the existence of adjoint functors.
First here is the statement of the special adjoint functor... | 79 | https://mathoverflow.net/users/310 | 5788 | 3,935 |
https://mathoverflow.net/questions/5800 | 14 | The classic puzzle goes something like this: "You are standing in front of a lake with a 3 gallon bucket and a 5 gallon bucket, how can you get 4 gallons of water?"
Is there an easy way to generate the triple (A,B,C) where you can get C gallons of water using buckets of size A and B?
| https://mathoverflow.net/users/1857 | Generalization of the two bucket puzzle | Simon's answer points out that the Euclidean algorithm shows that gcd(A,B) divides C is necessary, but the lack of large container makes the problem more difficult, because obviously you can't get $C$ if $C \gt A+B$. However, the following modification of the algorithm seems to work.
Let's assume $A \lt B$ and gcd$(A... | 23 | https://mathoverflow.net/users/1198 | 5807 | 3,949 |
https://mathoverflow.net/questions/5808 | 5 | It is well known that left [Bousfield localizations](http://ncatlab.org/nlab/show/Bousfield+localization) of the [global functor model category](http://ncatlab.org/nlab/show/global+model+structure+on+functors) $Func(C^{op}, SSet\_{standard})$ of functors with values in simplicial sets equipped with the [standard model ... | https://mathoverflow.net/users/381 | Local Joyal-simplicial presheaves? | If $V$ is a reasonnable model category (i.e. combinatorial, etc), and $C$ a small category endowed with a Grothendieck topology $\tau$, there are $\tau$-local model structures on the category $Fun(C^{op},V)$ (a projective version as well as an injective version). You may have a look at this [paper](http://www.intlpress... | 11 | https://mathoverflow.net/users/1017 | 5818 | 3,957 |
https://mathoverflow.net/questions/5823 | 2 | I'm looking for a list of examples of random variables to use in teaching a measure-theoretic probability course. For example, the Rademacher functions are an explicit construction of independent Bernoulli random variables.
If you were teaching such a course and had a list of canonical examples to illustrate definiti... | https://mathoverflow.net/users/136 | Examples of random variables | That will be not quite an answer for your question. Anyway it may be helpful.
1. If you have a sequence of independent Bernoulli r.v. $(B\_i)$ then you can define a uniform variable by $U = \sum 2^{-i} B\_i$ and further you can obtain an infinite sequence of independent uniform r.v $U\_i$. (just by splitting $B\_i$ ... | 5 | https://mathoverflow.net/users/1302 | 5829 | 3,965 |
https://mathoverflow.net/questions/5761 | 6 | Let $\psi(x):=\sum\_{n\leq x}\Lambda(n)$ denotes the 2nd Chebyshev function, where $\Lambda$ stands for the von Mangoldt function. Are there any known (and 'nice') estimates for the change rates $\psi(x+h)-\psi(x)$ for general or special $x$ and $h$?
Thanks in advance,
efq
| https://mathoverflow.net/users/1849 | Change rates of the 2nd Chebyshev function | There is the asymptotic estimate $\psi(x+h) - \psi(x) \sim h$ for $x^{7/12 + \epsilon} \leq h \leq x$, valid for any $\epsilon > 0$. This is due to M. N. Huxley, and dates to 1972. I am not aware of any better range for $h$ if you want asymptotic equality. But if you are satisfied with an order of magnitude result, you... | 7 | https://mathoverflow.net/users/3304 | 5836 | 3,970 |
https://mathoverflow.net/questions/5833 | 7 | Let V and W be irreducible representations of $S\_n$ and $S\_m$ over a field of characteristic 0. Then the Littlewood-Richardson coefficients allow us to compute the isomorphism type of the induced $S\_{n+m}$-module V⊗W↑. This induction comes from the inclusion
$S\_n\times S\_m \rightarrow S\_{n+m}$.
Now suppose V=... | https://mathoverflow.net/users/109 | Symmetric Powers, Tableau and Wreath Products | You want to read [Splitting the square of a Schur function into its symmetric and antisymmetric parts](http://www.ams.org/mathscinet-getitem?mr=1331743), where this question is answered in terms of domino tableaux. I am also a big fan of [Domino tableaux, Schützenberger involution, and the symmetric group action](http:... | 7 | https://mathoverflow.net/users/297 | 5837 | 3,971 |
https://mathoverflow.net/questions/5845 | 2 | Can anyone give an explicit basis of the universal (noncommutative) differential calculus over an algebra $A$ with basis ${e\_i}$. (The universal calculus over $A$ is the kernel of the multiplication map $m:A \otimes A \to A$.)
| https://mathoverflow.net/users/1867 | Basis for Universal Calculus | You can describe $\Omega\_A=\ker(m:A\otimes A\to A)$ as the quotient of the free $A$-bimodule generated by symbols $d(a)$, one for each element $a\in A$, by the sub-bimodule generated by the elements of the form $$d(ab)-d(a)\\,b-a\\,d(b), \qquad a,b\in A,$$ together with the elements of the form $$d(\lambda 1), \qquad ... | 3 | https://mathoverflow.net/users/1409 | 5849 | 3,980 |
https://mathoverflow.net/questions/5841 | 2 | This question is motivated from my [last question](https://mathoverflow.net/questions/4317/integrally-closed-factor-rings-and-projective-modules) here. I wonder if one has a ring A and an over-ring of this ring say B, and if we know that B is a projective A-module can we have a particular idea of how Spec B would look ... | https://mathoverflow.net/users/1245 | Spectra of rings that are projective module over a subring | So, for example, A could be a field k, and B could be any k-algebra whatsoever. Basically you would be trying to recover all of classical algebraic geometry from Spec k. It does not seem likely.
| 6 | https://mathoverflow.net/users/1698 | 5858 | 3,987 |
https://mathoverflow.net/questions/5865 | 2 | As is well known, every differential calculus $(\Omega,d)$ over an algebra $A$ is a quotient of the universal calculus $(\Omega\_A,d)$, by some ideal $I$. In the classical case, when $A$ is the coordinate ring of a variety $V(J)$ (for some ideal of polynomials $J$), and $(\Omega,d)$ is its ordinary calculus, how is $I$... | https://mathoverflow.net/users/1867 | Classical Calculi as Universal Quotients | In the classical case, if $\Omega(A)$ is the kernel of the multiplication map $m:A\otimes A\to A$, then—since $A$ is commutative, so that $m$ is not only a map of $A$-bimodules but also a morphism of $k$-algebras,—it turns out that $\Omega(A)$ is an *ideal* of $A\otimes A$, not only a sub-$A$-bimodule. In particular, y... | 4 | https://mathoverflow.net/users/1409 | 5870 | 3,995 |
https://mathoverflow.net/questions/5857 | 5 | In a [previous question](https://mathoverflow.net/questions/3887/quantum-frobenius), I asked how Lusztig's quantum Frobenius generalizes the classical Frobenius map on a variety over a finite field. I was directed to a very interesting [paper](http://arxiv.org/abs/math/0005246) by Kumar and Littlemann in which a quanti... | https://mathoverflow.net/users/1095 | Quantum Frobenius II | In general, the idea of the Kumar-Littelmann paper is the following: For a semisimple group G, set $V := \displaystyle \bigoplus\_{n \geq 0} H^0(\lambda)$, where $\lambda$ is a fixed regular dominant weight for G. Then $V$ is the projective coordinate ring of $G/B$ under the embedding $G/B \hookrightarrow \mathbb P( V(... | 9 | https://mathoverflow.net/users/1528 | 5883 | 4,008 |
https://mathoverflow.net/questions/5867 | 3 | I started writing [nLab:Theta space](http://ncatlab.org/nlab/show/Theta+space). Not done yet, but while I am working on it:
is there a good proposal for what the "$(n+1,r+1)$-$\Theta$-space of all $(n,r)$-$\Theta$-spaces" would be?
| https://mathoverflow.net/users/381 | (n+1,r+1)-Theta space of (n,r)-Theta spaces? | Let me assume $n=\infty$, to make things easier to write, so "$(\infty,r)$-$\Theta$-space" equals "$r$-$\Theta$-space".
The totality of $r$-$\Theta$-spaces forms a (large) category enriched over $r$-$\Theta$-spaces, which I'll call $C$. Given this, you can form a presheaf of spaces $X$ on the category $\Theta\_{r+1}$... | 5 | https://mathoverflow.net/users/437 | 5886 | 4,010 |
https://mathoverflow.net/questions/5885 | 4 | In classical geometry the calculation of the Chern classes of a vector bundle using a connection is independent of the choice of connection. Does any such result hold for projective modules in non-commutative geometry?
| https://mathoverflow.net/users/1095 | Does the non-commutative Chern class depend on the choice of connection? | Yes.
You can see the construction in detail, for example, in Max Karoubi's ‘Homologie cyclique et $K$-theorie’ (Asterisque 149, SMF; you can get this from his web page), where he constructs the Chern classes $K\_0(A)\to H(A)$ using connections much à Chern-Weyl (Here $H(A)$ is the non-commutative de Rham theory, or o... | 7 | https://mathoverflow.net/users/1409 | 5887 | 4,011 |
https://mathoverflow.net/questions/5847 | 10 | Faa di Bruno's formula ([MathWorld](http://mathworld.wolfram.com/FaadiBrunosFormula.html), [Wikipedia](http://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula)) gives the kth derivative of f(g(t)) as a sum over set partitions. (I'm not sure how well-known the combinatorial interpretation is; for example see [a 2006 ... | https://mathoverflow.net/users/143 | Is there a Faa di Bruno-like formula for composition of three functions? | The Faa di Bruno formula can be well reinterpreted in terms of trees, and then the generalization is obvious. Here's how:
First of all, the $k$th derivative of $f\circ g \circ h$ depends only on the first $k$ derivatives of $f,g,h$, evaluated at the appropriate spots. So we can without loss of generality identify smo... | 22 | https://mathoverflow.net/users/78 | 5888 | 4,012 |
https://mathoverflow.net/questions/5889 | 2 | What's the correct mathematical name for the partial ordering on vectors based on what is sometimes called "Pareto Dominance"?
Does Pareto Dominance have an alternative name in fields other than economics?
For two vectors of the same dimension, one Pareto Dominates the other if all its elements are greater than th... | https://mathoverflow.net/users/1875 | Is there an agreed name for partial ordering based on Pareto Dominance relation? | Depending on if you allow "greater" to mean "greater or equal" or to mean "strictly greater" we have two answers coming from Order Theory, respectively:
* The product order: <http://en.wikipedia.org/wiki/Product_order>
* The reflexive closure of the direct product of two strict total orders
Those two, along with t... | 1 | https://mathoverflow.net/users/1234 | 5898 | 4,017 |
https://mathoverflow.net/questions/5897 | 6 | In Hartshorne p. 109 he defines a sheaf $\mathcal{F}$ of $O\_X$-modules to be locally free if there is an open cover of $X$, s.t. on each $U$, $\mathcal{F}|\_U$ is a free $O\_X|\_U$ module of rank $I$. Then if $X$ is connected, rank $I$ is globally well-defined. Here $(X,O\_X)$ is any ringed topological space (e.g. not... | https://mathoverflow.net/users/1877 | Why is the rank of a locally free sheaf well-defined? | Actually, there are two different restriction maps:
1. The first one (the one you correctly say is neither surjective nor injective in general) is that on **sections**: for $\mathcal{F}$ a sheaf on a scheme $X$ and two open subsets $V \subseteq U$, there is a map $\mathcal{F}(U) \to \mathcal{F}(V)$.
2. On the other h... | 19 | https://mathoverflow.net/users/1797 | 5905 | 4,020 |
https://mathoverflow.net/questions/5895 | 12 | * What are tame and wild hereditary algebras?
* Are they related to hereditary rings? (Those are rings for which every left (resp. right) ideal is projective, equivalently, for which every left (resp. right) submodule of a projective module is again projective).
* Googling them I can see they seem related to the "tame ... | https://mathoverflow.net/users/1234 | What are tame and wild hereditary algebras? | An $k$-algebra $A$ is *tame* (or, equivalently, it has tame representation type) if, for every dimension $d\geq0$, you can parametrize all isoclasses of indecomposable $A$-modules of dimension $d$, apart from a finite number of them, by a finite number of $1$-parameter families. On the other hand, a finite dimensional ... | 32 | https://mathoverflow.net/users/1409 | 5906 | 4,021 |
https://mathoverflow.net/questions/5907 | 4 | Let $X$ be a set and let $f: X\longrightarrow X$ be a function on $X$. Introduce a topology on $X$ by the following basis of open sets: for any subset $S$ of $X$, let $B\_S$ be the set of forward images of $S$ under $f$; i.e. $$B\_S = \{f^n(s): s\in S, n\in \textbf{Z}^+\}.$$
My question is, is this topology well-known ... | https://mathoverflow.net/users/960 | Is there a name for this topology? | This topology is rather combinatorial in nature. You certainly could call it well-understood, but I would not expect it to have a name in the context of topology. It is essentially a special case of a concept in combinatorics known as an [order ideal](http://en.wikipedia.org/wiki/Ideal_%28order_theory%29), or if you li... | 13 | https://mathoverflow.net/users/1450 | 5910 | 4,024 |
https://mathoverflow.net/questions/5892 | 190 | If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f\*g)(x)$, the convolution of $f$ and $g$, is the probability distribution of $X+Y$. This is the only intuition I have for what convolution means.
Are there any other intuitive models f... | https://mathoverflow.net/users/812 | What is convolution intuitively? | I remember as a graduate student that Ingrid Daubechies frequently referred to convolution by a bump function as "blurring" - its effect on images is similar to what a short-sighted person experiences when taking off his or her glasses (and, indeed, if one works through the geometric optics, convolution is not a bad fi... | 228 | https://mathoverflow.net/users/766 | 5916 | 4,029 |
https://mathoverflow.net/questions/5249 | 16 | Everybody knows that there are $D\_n=n! \left( 1-\frac1{2!}+\frac1{3!}-\cdots+(-1)^{n}\frac1{n!} \right)$ [derangements](http://en.wikipedia.org/wiki/Derangement) of $\{1,2,\dots,n\}$ and that there are $D\_n(q)=(n)\_q! \left( 1-\frac{1}{(1)\_q!}+\frac1{(2)\_q!}-\frac1{(3)\_q!}+\cdots+(-1)^{n}\frac1{(n)\_q!} \right)$ e... | https://mathoverflow.net/users/1409 | Derangements and q-variants | Both of Reid's suggestions sort-of work and lead to the same formula. However, it's easier to first change the question to a $q$-analogue of the difference
$$D^{\pm}\_n = D^+\_n - D^-\_n = (-1)^n(n-1).$$
(Also, the formula for $D\_n(q)$ is missing a factor of $q^{n(n-1)/2}$.)
You can find $D^{\pm}\_n$ by the inclusion-... | 5 | https://mathoverflow.net/users/1450 | 5929 | 4,038 |
https://mathoverflow.net/questions/5893 | 10 | I have just created a presentation using beamer, and I want the
"one" command at the top of the file that creates a printable version. It is true that I can recompile having searched for all the \pause commands and percent them out, but I remember there is an elegant way of doing this.
For the record, I am using the ... | https://mathoverflow.net/users/36108 | Beamer printout | The first answer is the basic correct answer, but there are variants.
1. Don't change the background colour (waste of ink), rather use pgfpages to put a border around each frame (this isn't one of the standard page-type declarations, but it isn't hard and I can make mine available if anyone wants it).
2. It's possibl... | 10 | https://mathoverflow.net/users/45 | 5933 | 4,042 |
https://mathoverflow.net/questions/5954 | 49 | According to the [Norwegian meterological institute](http://www.yr.no/nyheter/1.6870525), the answer is that it is best to run. According to Mythbusters (quoted in the comments to that article), the answer is that it is best to walk.
My guess would be that this is something that can be properly modelled mathematicall... | https://mathoverflow.net/users/45 | Is it best to run or walk in the rain? | try this, the latest in a long line of recreational mathematics on the topic
"Keeping Dry: The Mathematics of Running in the Rain"
Dan Kalman and Bruce Torrence,
Mathematics Magazine, Volume 82, Number 4 (2009) 266-277
| 40 | https://mathoverflow.net/users/454 | 5962 | 4,062 |
https://mathoverflow.net/questions/5955 | 14 | May be it's not the right place for this, but I don't know the right definition of a strange attractor. Wikipedia states that "An attractor is informally described as strange if it has non-integer dimension or if the dynamics on it are chaotic." In the Tucker's paper about Lorenz system is written that "an attractor is... | https://mathoverflow.net/users/1888 | Definition of a strange attractor. | This is a good question. For some reason, terminology in dynamical systems is not standardized at all--and it's interesting to disentangle various definitions. A good book to look at is [Differential equations, dynamical systems, and an introduction to chaos](http://books.google.com/books?id=GNOmchErrMgC). The authors ... | 15 | https://mathoverflow.net/users/1227 | 5972 | 4,070 |
https://mathoverflow.net/questions/5975 | 1 | Given two positive integers `a,b` what is the minimal integer `n`, so that there exist two positive integers `u,v` for which `n=au=av`?
It is easy to verify that `n=ab/gcd(a,b)`.
But what happens if instead of requiring `au=bv`, or `|au-bv|`≤`0`, we require that `|au-bv|`≤`k` for some number k?
That is, given two... | https://mathoverflow.net/users/55 | For which integers u,v does au=bv *approximately*? | A good heuristic is to compute the [continued fraction](http://en.wikipedia.org/wiki/Continued_fraction) of $a/b$ and drop the last few terms. The continued fraction will equal $u/v$ with $a/b-u/v$ very small. Since $a/b-u/v=(av-bu)/bv$, we will have $au-bv$ small.
| 2 | https://mathoverflow.net/users/297 | 5978 | 4,074 |
https://mathoverflow.net/questions/5915 | 8 | Here is a topological question which seems quite elementary. The answer to this question may be useful e.g. in estimating the orders of the automorphism groups of some algebraic varieties and in computing the cohomology of some moduli spaces.
Let $K$ be a compact Lie group, $T$ a maximal torus of $K$ and $V$ a comple... | https://mathoverflow.net/users/2349 | Cohomology map induced by the group actions on homogeneous vector bundles | I'll answer in the case of K = U(n) because I'm less familiar with the cohomology of the other compact Lie groups. Let BK and BT be the classifying spaces.
The representation of T on V gives rise to a vector bundle on BT with total space F, and complement of the zero section F0. There is an Euler class c in H`*`(BT).... | 5 | https://mathoverflow.net/users/360 | 5983 | 4,078 |
https://mathoverflow.net/questions/5790 | 9 | I seem to recall that there is a straightforward subfactor construction that yields fusion categories given by G-graded vector spaces and representations of G, for finite groups G. Is there an analogous construction for 2-groups?
Some background: A 2-group is a monoidal groupoid, for which the isomorphism classes of ... | https://mathoverflow.net/users/121 | Is there a subfactor construction involving 2-groups? | This is a standard construction in Subfactor theory see the intro of <http://arxiv.org/abs/0811.1084v2> for details. The construction goes back a long long way (if I remember correctly both Vaughan Jones and Adrian Ocneanu's theses were related to this question, but I could be wrong there).
From a category theory per... | 6 | https://mathoverflow.net/users/22 | 5984 | 4,079 |
https://mathoverflow.net/questions/5198 | 2 | Is there a specific name for matrices with nonsingular principal submatrices?
| https://mathoverflow.net/users/1172 | Is there a specific name for matrices with nonsingular principal submatrices? | I've heard them called "strongly nonsingular matrices" in numerical linear algebra. Google that and you'll find some literature.
| 4 | https://mathoverflow.net/users/1898 | 5988 | 4,081 |
https://mathoverflow.net/questions/5986 | 14 | I want a good introduction to localization in equivariant $K$-theory. This introduction can be simple in several ways:
1. I only care about torus actions.
2. I only care about $K^0$.
3. I only care about very nice spaces. I would be fine if the only spaces considered were $G/P$'s and smooth projective toric varieties... | https://mathoverflow.net/users/297 | References for equivariant K-theory | I like the book by Chriss and Ginzburg (Representation Theory and Complex Geometry, <https://doi.org/10.1007/978-0-8176-4938-8>) very much, and I think it fits many of your requirements.
| 8 | https://mathoverflow.net/users/582 | 5991 | 4,083 |
https://mathoverflow.net/questions/5993 | 26 | What is known about the classification of n-transitive group actions for n large without using the classification of finite simple groups? With the classification of finite simple groups a complete list of all 2-transitive group actions is known, in particular there are no 6-transitive groups other than the symmetric g... | https://mathoverflow.net/users/22 | Highly transitive groups (without assuming the classification of finite simple groups) | Marshall Hall's *The Theory of finite groups* only cites an asymptotic bound: a permutation group of degree n that isn't Sn or An can be at most t-transitive for t less than 3 log n. I suppose that was the state of the art at the time (late 1960s). There is an earlier paper by G.A. Miller on JSTOR that you can find by ... | 18 | https://mathoverflow.net/users/121 | 5996 | 4,085 |
https://mathoverflow.net/questions/5997 | 21 | One time I heard a talk about "the" random tree. This tree has one vertex for each natural number, and the edges are constructed probabilistically. Connect vertex $2$ to vertex $1$. Connect vertex $3$ to vertex $1$ or $2$ with probability $\frac{1}{2}$. Connect vertex $n+1$ to exactly one of vertices $1,\dots, n$ with ... | https://mathoverflow.net/users/1345 | "The" random tree | I learned about [The random graph](http://en.wikipedia.org/wiki/Rado_graph) a week ago from [the blog post on n-cafe](http://golem.ph.utexas.edu/category/2009/11/fraisse%5Flimits.html) (thanks to Andrew and sdcvvc for the links!).
Your construction, while slightly different, can be examined in the same way as the Rad... | 11 | https://mathoverflow.net/users/65 | 5999 | 4,087 |
https://mathoverflow.net/questions/6009 | 8 | What is the best algorithm for computing covariance that would be accurate for a large number of values like 100,000 or more?
| https://mathoverflow.net/users/812 | What's an efficient way to calculate covariance for a large data set? | Check out [How to calculate correlation accurately](http://www.johndcook.com/blog/2008/11/05/how-to-calculate-pearson-correlation-accurately/). There are two common formulas that are algebraically equivalent but one has much better numerical properties than the other.
| 7 | https://mathoverflow.net/users/136 | 6013 | 4,097 |
https://mathoverflow.net/questions/6019 | 18 | How would you calculate the order of a list of reviews sorted by "Most Helpful" to "Least Helpful"?
Here's an example inspired by product reviews on Amazon:
Say a product has 8 total reviews and they are sorted by "Most Helpful" to "Least Helpful" based on the part that says "x of y people found this review helpful... | https://mathoverflow.net/users/1901 | Calculating the "Most Helpful" review | See [How not to sort by average ranking](http://www.evanmiller.org/how-not-to-sort-by-average-rating.html).
| 27 | https://mathoverflow.net/users/297 | 6021 | 4,101 |
https://mathoverflow.net/questions/5813 | 8 | It seems that there is no g.r.r for stack yet according to dejong. Does anyone know anything about it? But as you might know, there are some complex manifold which is not scheme having atiyah singer index theorem. So I was wondering if there exists some analogue of g.r.r for special stack.
| https://mathoverflow.net/users/1851 | Is there any Grothendieck Riemman Roch theorem for general stack ? | If you work with the naive Chow-groups and allow non-representable morphisms the GRR-Theorem does not hold! In the paper by Toen quoted above and in some of the papers by Joshua there are explicit counterexamples. They always involve non-representable morphisms.
There are two ways to get around this.
The first is... | 12 | https://mathoverflow.net/users/473 | 6027 | 4,106 |
https://mathoverflow.net/questions/6026 | 7 | According to the Wikipedia page on [generalized continued fractions](http://en.wikipedia.org/wiki/Generalized_continued_fraction), $\pi$ can be given several GCF representations which have very regular structures; for example, one has the partial denominators as (1, 2, 2, 2, ...) and the partial numerators as (4, 12, 3... | https://mathoverflow.net/users/1455 | Patterns in Generalized Continued Fractions | I think under any reasonable definition there will be only countably many explicit formulas or patterns. So in fact most reals can't be expressed this way. (See also the concept of ["computable number."](http://en.wikipedia.org/wiki/Computable_number))
| 15 | https://mathoverflow.net/users/1227 | 6031 | 4,109 |
https://mathoverflow.net/questions/6010 | -3 | Here is something that isn't yet very clear to me. Say, I've got a commutative ring A. I consider the affine scheme from A, so it's a sheaf of rings over Spec A.
**EDIT**: And additionally let's say Spec A is Hausdorff.
Now additionally let's say I know an A-module M and from that I can make a sheave of modules o... | https://mathoverflow.net/users/1245 | Dense section of sheaves of modules | If U is an open set in X, but U isn't X, then there are non-zero sheaves on X whose support lies outside U. Now add O\_X to one of these to get a counterexample.
| 2 | https://mathoverflow.net/users/1384 | 6034 | 4,111 |
https://mathoverflow.net/questions/6033 | 6 | Let $G$ be a locally compact totally disconnected group, and to make life easy let's suppose its Haar measure is bi-invariant. Let $C\_c(G)$ be the space of locally constant complex functions on $G$ with compact support, which forms an algebra under convolution. Suppose $e \in C\_c(G)$ is an idempotent, so that $H = eC... | https://mathoverflow.net/users/379 | Inverses in convolution algebras | I don't have a solution, but here are some thoughts which might be of use or interest.
You may have seen this already, but if your group is *discrete* then its group von Neumann algebra $VN(G)$ is "directly finite" - that is, every left invertible element is invertible. I think this property is inherited by the algeb... | 4 | https://mathoverflow.net/users/763 | 6044 | 4,115 |
https://mathoverflow.net/questions/6006 | 9 | Any map of finite graphs (1-dimensional CW-complexes) factors as a composition of
1. a finite sequence of folds;
2. an inclusion; and
3. a finite-to-one covering map.
There should be a corresponding result for handlebodies, which presumably should say that, after a homotopy, a continuous map of handlebodies factors... | https://mathoverflow.net/users/1463 | Factoring maps of handlebodies | Suppose that we are given a PL map from a handlebody $W$ to a handlebody
$V$. Choose a spine for $W$. Homotope the map until the image is a
regular neighborhood of the image of the spine. By general position,
our map is now an embedding. Fix a pants decomposition of disks $D =
(D\_i)$, for $V$. Suppose that $P$ is a co... | 4 | https://mathoverflow.net/users/1650 | 6047 | 4,116 |
https://mathoverflow.net/questions/6043 | -1 | I need a way to split output pdf-file (a book) into chapters on such a way that cross-references will survive.
A simple example with a **solution** (based on answers below) can be found [here](http://www.math.psu.edu/petrunin/papers/splitting-pdf-latex/)
| https://mathoverflow.net/users/1441 | Splitting book into chapters | If you are using a book documentclass, then you can create a template file, with \include{chapter3} or \includeonly{chapter2}. I don't know if this works particularly. A hacky way to do it is to save your .aux file as say a .auk file, process, and then write over the .aux file with the saved version.
If you are on a... | 6 | https://mathoverflow.net/users/36108 | 6049 | 4,118 |
https://mathoverflow.net/questions/6052 | 2 | I read the following fact: if $U$ is an open subset of $P\_k^1$ and $f: U \to U$ is an automorphism of schemes, then $f$ extends to an automorphism of $P\_k^1$. Thus I was curious: is there a general criteria for when a continuous map defined on an open subset $U \subset X$ extends to $X$ (especially in non-Hausdorff s... | https://mathoverflow.net/users/344 | How do we know that a map $f: U \to Y$ extends to $\bar{U}$? | Your example is in the category of schemes, but you then ask about maps of topological spaces.
Assuming you care about schemes, the condition is that the source is one dimensional and regular, and the target is proper. That you can extend in this case is essentially the valuative criterion of properness; that you ca... | 5 | https://mathoverflow.net/users/297 | 6056 | 4,123 |
https://mathoverflow.net/questions/6020 | 2 | I'm not the person to understand everything in *[Geometric Endoscopy and Mirror Symmetry](http://arxiv.org/abs/0710.5939)*, but some parts of it are reasonably clear to me.
In particular, one of the main objects, mathematically speaking, is the category of coherent sheaves on an orbifold point $\mathrm{pt}/\Gamma$, w... | https://mathoverflow.net/users/65 | Understanding formula in Frenkel-Witten | It's a typo. R always denotes representations of Gamma, and representations of LG are named with a V.
| 2 | https://mathoverflow.net/users/121 | 6059 | 4,125 |
https://mathoverflow.net/questions/6057 | 25 | [*Edit (June 20, 2010):* I posted [an answer](https://mathoverflow.net/a/28891/14094) to this which summarizes one that I received verbally a few weeks after posting this question. I hope it is useful to someone.]
I am presently seeking references which introduce "formal geometry". So far as I can tell, this idea was... | https://mathoverflow.net/users/1040 | Formal geometry | This was briefly discussed [previously on this site](https://mathoverflow.net/questions/518/what-is-the-connection-between-d-modules-and-coordinate-bundles/523). You can find discussions of it in:
1. Chapters 6 and 17 of Frenkel and Ben-Zvi: *Vertex algebras and algebraic curves.*
2. Section 2.6.5 of Beilinson and Dr... | 8 | https://mathoverflow.net/users/121 | 6060 | 4,126 |
https://mathoverflow.net/questions/6061 | 9 | I am currently trying to apply some results from Choquet theory - i.e., the generalisation of results by Minkowski and Krein-Milman for representing points in a compact, convex set C by probability measures over its extreme points, ext C = { x ∈ C : C - { x } is convex }.
My main problem is with explicitly describing... | https://mathoverflow.net/users/1915 | Explicitly describing extreme points of infinite dimensional convex sets | There are more candidates for the extreme points. Take any compact convex subset of the simplex, then take the infimum of all affine functions that are ≥ the characteristic function of the subset. You could start with more arbitrary subsets, but the result is the same.
As far as general techniques go, I don't have an... | 7 | https://mathoverflow.net/users/802 | 6063 | 4,128 |
https://mathoverflow.net/questions/5739 | 8 | Hi,
I have recently got interested in multi-index (multi-dimensional) Dirichlet series, i.e. series of the form $F(s\_1,...,s\_k)=\sum\_{(n\_1,...,n\_k)\in\mathbb{N}^k}\frac{a\_{n\_1,...,n\_k}}{n\_1^{s\_1}...n\_k^{s\_k}}$. I found some papers suggesting that multi-index Dirichlet series are in fact a distinct subfield ... | https://mathoverflow.net/users/1849 | multi-index Dirichlet series | De la Breteche proved recently a Tauberian theorem for multiple Dirichlet series (MR1858338 (2002j:11106)). This is useful stuff in applications. It fails shortly of proving the main result in Balazard, et. al recent paper: <http://iml.univ-mrs.fr/~balazard/pdfdjvu/19.pdf> (but does so assuming the Riemann Hypothesis).... | 3 | https://mathoverflow.net/users/1919 | 6065 | 4,130 |
https://mathoverflow.net/questions/6070 | 84 | I know (at least I think I know) that some of the main motivating problems in the development of etale cohomology were the Weil conjectures. I'd like to know what other problems one can solve using the machinery of etale cohomology. I know a little bit about how etale cohomology groups appear in algebraic number theory... | https://mathoverflow.net/users/493 | Etale cohomology -- Why study it? | $\DeclareMathOperator{\gal}{Gal}$
Here's a comment which one can make to differential geometers which at least explains what etale cohomology "does". Given an algebraic variety over the reals, say a smooth one, its complex points are a complex manifold but with a little extra structure: the complex points admit an auto... | 120 | https://mathoverflow.net/users/1384 | 6076 | 4,135 |
https://mathoverflow.net/questions/6082 | 2 | We have a probability game, where we have $N$ number of events, each of which outcome can be $A,B$ or $C$. We do/will NOT know real probabilities afterwards: only the discrete outcome ($A, B$ or $C$) of each event.
Player 1 forecasts these events with certain probabilities (not only guess what is the outcome, but giv... | https://mathoverflow.net/users/1928 | Which fortuneteller is better | I could be missing something here but I'd compute
P[These events occur|Fortune teller 1 is telling the truth] and P[These events occur|Fortune teller 2 is telling the truth].
More explicitly
Let N\_a, N\_b and N\_c be the number of times outcomes A,B and C occured. Then
P(This happened given fortune teller 1 told... | 1 | https://mathoverflow.net/users/1000 | 6087 | 4,141 |
https://mathoverflow.net/questions/6079 | 45 | I would like to study/understand the (complete) classification of compact lie groups. I know there are a lot of books on this subject, but I'd like to hear what's the best route I can follow (in your opinion, obviously), since there are a lot of different ideas involved.
| https://mathoverflow.net/users/1049 | Classification of (compact) Lie groups | First, here's a rough outline of how the classification works:
1. Prove that if G and H are simply connected and have the same Lie algebra, then G and H are isomorphic as Lie groups.
2. Prove that if G is any Lie group, its universal cover $\tilde{G}$ inherits a natural Lie group structure for which G = $\tilde{G}/Z$... | 43 | https://mathoverflow.net/users/1708 | 6095 | 4,145 |
https://mathoverflow.net/questions/6089 | 5 | Propositional dynamic logic (PDL) is an example of a (multi)modal logic with a structure on the set of modalities. In particular, the set of its modalities is indexed by "programs" and one can use program constructors such as composition, choice and iteration to make new programs out of old (of course, there is a set o... | https://mathoverflow.net/users/1716 | Applications of propositional dynamic logic | The practical applications might be more obvious once you observe that these propositional "programs" are *regular expressions* -- which is to say, state machines. So you can expect it to have applications in the study of things [like program analyses](http://www.cs.cornell.edu/~kozen/papers/opti.pdf) and verifying con... | 5 | https://mathoverflow.net/users/1610 | 6098 | 4,148 |
https://mathoverflow.net/questions/6093 | 5 | Does it help to learn statistical mechanics or thermodynamics (as in physics or mathematical physics) in order to learn thermodynamic formalism: the study of equilibrium states, Gibbs measure, maximal measures mostly on shift spaces or ℤn?
I've not taken a course on thermodynamics, but so far my learning of the conce... | https://mathoverflow.net/users/1354 | Does it help to learn statistical mechanics in order to learn thermodynamic formalism? | If you're learning thermodynamic formalism in order to apply it to physics then it's fairly clear that you ought to learn the physical context, so I infer from this that you are more interested in using it either to understand dynamical systems, or for the sake of its applications to other areas of mathematics entirely... | 5 | https://mathoverflow.net/users/1840 | 6109 | 4,155 |
https://mathoverflow.net/questions/6088 | 6 | Hey
A friend and I are thinking of having an algebraic statistics seminar next semester. Does anyone know of a good book to try learn it out of?
| https://mathoverflow.net/users/1000 | Algebraic Statistics textbook | I've heard that [the book of Sturmfels and Pachter](http://bio.math.berkeley.edu/ascb/) is supposed to be good. But it's a bit slanted towards biological applications, which may or may not be what you're into.
| 3 | https://mathoverflow.net/users/143 | 6119 | 4,161 |
https://mathoverflow.net/questions/6111 | 10 | Does anyone know of a global proof (involving no local argument) of Serre duality at the level of varieties or manifolds (as opposed to schemes).
| https://mathoverflow.net/users/1095 | Global proof of Serre duality | You might like the proof in section 5.3 of Voisin's book *Hodge theory and complex algebraic geometry*.
| 9 | https://mathoverflow.net/users/297 | 6120 | 4,162 |
https://mathoverflow.net/questions/6108 | 9 | Let $S$ the blow up of $P^2$ in nine points. Why is the anticanonical divisor $-K\_S$ not semiample?
| https://mathoverflow.net/users/1937 | Anticanonical divisor of the blow up of P^2 in 9 points | Your nine points must be [**EDIT: very**] general [see MP's answer], otherwise it *can* be semiample.
The only effective anticanonical divisor is then (the strict transform of) the cubic C through the nine points. Since there is no other cubic curve cutting C in your nine points, the restriction of -K\_S to C is a no... | 10 | https://mathoverflow.net/users/1939 | 6121 | 4,163 |
https://mathoverflow.net/questions/6074 | 71 | What's the relationship between Kahler differentials and ordinary differential forms?
| https://mathoverflow.net/users/1867 | Kahler differentials and Ordinary Differentials | Let $M$ be a differentiable manifold, $A=C^\infty (M)$ its ring of global differentiable functions and $\Omega^1 (M)$ the A-module of global differential forms of class $C^\infty$.
The A-module of Kähler differentials $\Omega\_k(A)$ is the free A-module over the symbols $db$ ($b \in A$) divided out by the relations
... | 64 | https://mathoverflow.net/users/450 | 6138 | 4,178 |
https://mathoverflow.net/questions/6132 | 24 | Is it true that *any finitely presented group can be realized as fundamental group of compact 3-manifold **with boundary***?
| https://mathoverflow.net/users/1441 | Fundamental group of 3-manifold with boundary | A couple of extra points.
Any compact 3-manifold with boundary $M$ can be doubled to give a closed 3-manifold $D$. As $M$ is a retract of $D$, it follows that $\pi\_1(M)$ injects into $\pi\_1(D)$. Therefore, any "poison subgroup" (such as the Baumslag--Solitar groups that Autumn mentions above) applies just as well t... | 24 | https://mathoverflow.net/users/1463 | 6143 | 4,181 |
https://mathoverflow.net/questions/6142 | 12 | Does anybody know if orientable, closed $3$-manifolds that are circle bundles over $RP^2$ have been classified?
One can determine the isomorphism classes of bundles using obstruction theory, but I am interested in what total spaces can appear.
I am not assuming the bundle is principal.
Thank you.
| https://mathoverflow.net/users/1944 | Circle bundles over $RP^2$ | Such manifolds are examples of Seifert fibered spaces, which have, indeed, been classified. A good reference is Montesinos "Classical Tessellations and Three-Manifolds". Basically, such manifolds (over any nonorientable surface base) are classified by their Euler class, which measures the obstruction to the existence o... | 17 | https://mathoverflow.net/users/1672 | 6145 | 4,182 |
https://mathoverflow.net/questions/1959 | 22 | There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne.
Here's a refresher: by $IC$ one means the intersection complex, which is just $\mathbb Q$ for a smooth scheme but more complicated for others, and by $IC\_i$ one denotes the complex constructed from a pair ($Y\_i$, $\math... | https://mathoverflow.net/users/65 | Examples for Decomposition Theorem | I can think of several additions to your list which don't seem to be represented yet.
1. Semismall resolutions
------------------------
This first example is rather general, but afterward I will discuss how it is used in Springer theory.
First, suppose that $f:X \to Y$ is a proper map of stratified irreducible c... | 15 | https://mathoverflow.net/users/916 | 6153 | 4,189 |
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