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https://mathoverflow.net/questions/664 | 15 | Erdős's 1947 probabilistic trick provided a lower exponential bound for the Ramsey number $R(k)$. Is it possible to explicitly construct 2-colourings on exponentially sized graphs without large monochromatic subgraphs?
That is, can we explicitly construct (edge) 2-colourings on graphs of size $c^k$, for some $c>0$, w... | https://mathoverflow.net/users/416 | Can one make Erdős's Ramsey lower bound explicit? | I believe the answer is "no"; the best known constructions only give no clique or independent set of size about $2^\sqrt{n}$ in a graph with $2^n$ vertices. Bill Gasarch has a page on the subject [here](http://www.cs.umd.edu/~gasarch/const_ramsey/const_ramsey.html), although I don't know how frequently it updates.
| 5 | https://mathoverflow.net/users/382 | 667 | 378 |
https://mathoverflow.net/questions/652 | 22 | [This question](https://mathoverflow.net/questions/640/what-is-cohomology-and-how-does-a-beginner-gain-intuition-about-it) reminded me of a possibly stupid idea that I had a while back.
On page 2 of [this paper](http://arXiv.org/abs/math/0502016v1), while discussing Euclid's axioms of plane geometry and spatial geome... | https://mathoverflow.net/users/83 | Homological algebra and calculus (as in Newton) | I saw the notion of an "n-complex" once in this preprint by Peter Olver: <http://www.math.umn.edu/~olver/a_/hyper.pdf>
I only ever studied sections 5 and 6 of this paper, so I don't know what he's actually doing in the other sections (he introduces "hypercomplexes" which contain "n-complexes" as subcomplexes in sect... | 5 | https://mathoverflow.net/users/321 | 670 | 381 |
https://mathoverflow.net/questions/672 | 11 | I have just started reading Elementary Algebraic Geometry by Hulek. It is a nice book but I find that it doesn't give many problems (about 10 to 15 per chapter), and that the exercises present are a bit boring (mainly specific case work, seemingly arbitrary curves, etc.).
This is in stark contrast with say, Modern Gr... | https://mathoverflow.net/users/416 | Are good introductory/pedagogical problems in algebraic geometry rare? | I highly recommend Fultons book "Algebraic curves" It's available on his [webpage](http://www.math.lsa.umich.edu/~wfulton/)
It's a very good introduction, and in the first chapter there are 54 exercises.
| 11 | https://mathoverflow.net/users/135 | 675 | 384 |
https://mathoverflow.net/questions/674 | 37 | I have a few elementary questions about cup-products.
Can one develop them in an axiomatic approach as in group cohomology itself, and give an existence and uniqueness theorem that includes an explicitly computable map on cochains? Second, how do they relate to cup-products in algebraic topology? In general, are ther... | https://mathoverflow.net/users/344 | What is a cup-product in group cohomology, and how does it relate to other branches of mathematics? | You can identify group cohomology with $Ext\_{kG}(k,k)$ where $k$ is your base ring, and $kG $ is the group algebra. Here, the cup product given by Yoneda product (which there's some material on [here](https://en.wikipedia.org/wiki/Ext_functor#Ring_structure_and_module_structure_on_specific_Exts)). You should think Ext... | 21 | https://mathoverflow.net/users/66 | 679 | 387 |
https://mathoverflow.net/questions/592 | 9 | I've heard it stated that if you take the moduli of elliptic curves with some level structure imposed (as a moduli scheme over Spec(Z)), there is a logarithmic structure that you can impose at the cusps so that the natural projection maps obtained by forgetting the level structure are log-etale (at least away from prim... | https://mathoverflow.net/users/360 | Logarithmic structures on moduli of elliptic curves over Z | I thing Kato's log purity theorem gives you this. See, for instance, Theorem B in Mochizuki's "Extending Families of Curves over Log Regular Schemes." I think all you need is that the cusps form a normal crossings divisor on X(1) [if you're worried about X(1) being a stack rather than a scheme, you can start with a bit... | 4 | https://mathoverflow.net/users/431 | 693 | 398 |
https://mathoverflow.net/questions/696 | 22 | This is probably quite easy, but how do you show that the Euler characteristic of a manifold *M* (defined for example as the alternating sum of the dimensions of integral cohomology groups) is equal to the self intersection of *M* in the diagonal (of *M* × *M*)?
The few cases which are easy to visualise (ℝ in the pla... | https://mathoverflow.net/users/362 | Euler characteristic of a manifold and self-intersection | The normal bundle to $M$ in $M\times M$ is isomorphic to the tangent bundle of $M$, so a tubular neighborhood $N$ of $M$ in $M\times M$ is isomorphic to the tangent bundle of $M$. A section $s$ of the tangent bundle with isolated zeros thus gives a submanifold $M'$ of $N \subset M\times M$ with the following properties... | 22 | https://mathoverflow.net/users/317 | 700 | 403 |
https://mathoverflow.net/questions/691 | 41 | How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a [commutative] ring and a simplicial [commutative] ring?
| https://mathoverflow.net/users/83 | Simplicial objects | One could say many things about this, and I hope you get many replies! Here are some remarks, although much of this might already be familiar or obvious to you.
In some vague sense, the study of simplicial objects is "homotopical mathematics", while the study of objects is "ordinary mathematics". Here by "homotopical... | 35 | https://mathoverflow.net/users/349 | 705 | 407 |
https://mathoverflow.net/questions/707 | 15 | I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following:
>
> Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant $\Delta$, let $p$ be a prime, at which $E$ has potentially multiplicative reduction and let $\ell$ be a prime different th... | https://mathoverflow.net/users/2 | Reference for the `standard' Tate curve argument. | I think most people just mentally have in mind the argument you give.
In my thesis I actually wrote this down semi-carefully (including the case l = p, in which case what you want to say is that E[l] is finite over Zp, where E is now the Neron model of your elliptic curve over Q\_p.) Or rather I wrote down the direct... | 10 | https://mathoverflow.net/users/431 | 708 | 409 |
https://mathoverflow.net/questions/636 | 6 | Let Cat denote the 1-category of small categories. The functor Mor : Cat -> Set which assigns to a category its set of morphisms (aka Hom([• -> •], -)) does not commute with most colimits. Does it commute with quotients by free group actions? In other words, if C is a small category and G is a group acting on C such th... | https://mathoverflow.net/users/126667 | Quotient of a category by a free group action | OK, I found a "high-tech" argument which is maybe more convincing.
The external input to the argument is that if X -> Z <- Y is a diagram of G-sets with G acting freely on Z (and hence on X and Y) then (X xZ Y)/G = (X/G) x(Z/G) (Y/G). This is pretty clear because we can always write our original X -> Z <- Y in the fo... | 2 | https://mathoverflow.net/users/126667 | 717 | 413 |
https://mathoverflow.net/questions/400 | 45 | Around these parts, the aphorism "A gentleman never chooses a basis," has become popular.
>
> **Question.** Is there a gentlemanly way to prove that the natural map from $V$ to $V^{\*\*}$ is surjective if $V$ is finite-dimensional?
>
>
>
As in life, the exact standards for gentlemanliness are a bit vague. Some... | https://mathoverflow.net/users/27 | "A gentleman never chooses a basis." | Following up on Qiaochu's query, one way of distinguishing a finite-dimensional $V$ from an infinite one is that there exists a space $W$ together with maps $e: W \otimes V \to k$, $f: k \to V \otimes W$ making the usual [triangular equations](http://ncatlab.org/nlab/show/triangle+identities) hold. The data $(W, e, f)$... | 39 | https://mathoverflow.net/users/2926 | 718 | 414 |
https://mathoverflow.net/questions/686 | 30 | I subscribe to feeds from the [arXiv Front](https://arxiv.org/abs/math) for a number of subject areas, using [Google Reader](http://www.google.com/reader). This is great, but there is one problem: when a new preprint is listed in several subject categories, it gets listed in several feeds, which means I have to spend m... | https://mathoverflow.net/users/349 | Handling arXiv feeds to avoid duplicates | Unless the arXiv has changed recently, articles are published daily which means that the feeds and the email are completely in step.
The problem with the duplicates is that each feed is a separate request to the arXiv for information. The arXiv doesn't know that you are going to merge these results, and I've never he... | 24 | https://mathoverflow.net/users/45 | 726 | 418 |
https://mathoverflow.net/questions/711 | 8 | In analogy with the Hodge diagram for ordinary de Rham cohomology, we should have some kind of diagram for Alexander-Spanier cohomology. Doing all the relevant duality stuff and assuming that now our space is a *noncompact* Calabi-Yau manifold, we get a reduced Hodge diamond, to which mirror symmetry probably applies. ... | https://mathoverflow.net/users/436 | Mirror symmetry for noncompact Calabi-Yau manifolds | There is a version of mirror symmetry, called "local mirror symmetry", for certain non-compact Calabi-Yaus, for example the total space of the canonical bundle of P^2 (exercise: show this is CY). The mirror (or rather one possible mirror) of this non-compact Calabi-Yau is an affine elliptic curve in (C^\*)^2. I don't t... | 7 | https://mathoverflow.net/users/83 | 738 | 426 |
https://mathoverflow.net/questions/723 | 7 | Is the $n$-dimensional Fourier transform of $\exp(-\|x\|)$ always non-negative, where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$? What is its support?
| https://mathoverflow.net/users/447 | Is the Fourier transform of $\exp(-\|x\|)$ non-negative? | This Fourier transform is positive, supported everywhere, and has polynomial decay. It is the Poisson kernel evaluated at time 1, up to some rescaling.
<http://en.wikipedia.org/wiki/Poisson_kernel>
| 8 | https://mathoverflow.net/users/373 | 752 | 435 |
https://mathoverflow.net/questions/743 | 6 | Additionally, is there any intuitive way to visualize the cardinalities that result?
| https://mathoverflow.net/users/441 | What do models where the CH is false look like? | I think your reading is wrong. Set theorists have studied all sorts of additional axioms, some implying CH, some being strictly weaker than CH, and many contradicting CH. My understanding is that most set theorists today, if they have any opinion on the matter, prefer to think that CH is false. In particular, Woodin ha... | 14 | https://mathoverflow.net/users/75 | 753 | 436 |
https://mathoverflow.net/questions/731 | 38 | Why were algebraic geometers in the 19th Century thinking of
m-Spec as the set of points of an affine variety associated to the
ring whereas, sometime in the middle of the 20 Century, people started to think Spec was more appropriate as the "set of points".
What are advantages of the Spec approach? Specific theorems?... | https://mathoverflow.net/users/416 | "Points" in algebraic geometry: Why shift from m-Spec to Spec? | The basic reason in my mind for using Spec is because **it makes the category of affine schemes equivalent to the category of commutative rings**. This means that if you get confused about what's going on geometrically (which you will), you can fall back to working with the algebra. And if you have some awesome results... | 63 | https://mathoverflow.net/users/1 | 756 | 439 |
https://mathoverflow.net/questions/747 | 14 | Could anyone point to good readable references for learning about syntomic cohomology?
| https://mathoverflow.net/users/349 | References for syntomic cohomology | I like the notes [here](http://math.arizona.edu/~swc/notes/files/DLSTsuji2.pdf), though I must confess I haven't read them completely. It has a nice introduction and explains the context.
| 4 | https://mathoverflow.net/users/2 | 758 | 441 |
https://mathoverflow.net/questions/274 | 6 | Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is non-zero? What about $H^{j}\left(\mathbb{R},F\right)$ for integers $j\ge 2$ ?
(Here cohomology means derived functor cohomolog... | https://mathoverflow.net/users/450 | Non-zero sheaf cohomology | The sheaf cohomology ${H}^i(X,F)$ of a (topological) manifold $X$ of dimension $n$ vanishes for $i > n$. This is a topological version of Grothendieck's vanishing theorem above. You can find this result in Kashiwara-Schapira's "[Sheaves on manifolds](http://books.google.com/books?id=qfWcUSQRsX4C&lpg=PA475&ots=Dk-Z-Jgqh... | 11 | https://mathoverflow.net/users/322 | 770 | 447 |
https://mathoverflow.net/questions/760 | 6 | More precisely, does there exist a sequence $G\_1 < G\_2 < \cdots$ of finite groups such that the irreducible representations of $G\_n$ are parameterized by the plane partitions of total size $n$?
| https://mathoverflow.net/users/290 | Does there exist a sequence of groups whose representation theory is described by plane partitions? | Not if you want the direct analogue of the branching rule to hold: namely, if V is the representation of Gn corresponding to a plane partition A of n, then the restriction of V to Gn-1 is the direct sum of one copy of the representation corresponding to each plane partition of n-1 contained in A. That would allow you t... | 11 | https://mathoverflow.net/users/126667 | 779 | 454 |
https://mathoverflow.net/questions/406 | 33 | There is a folk — I can't call it a theorem — "fact" that the mathematical relationship between Complex and Tropical geometry is analogous to the physical relationship between Quantum and Classical mechanics. I think I first learned about this years ago on *This Week's Finds*. I'm wondering if anyone can give me a prec... | https://mathoverflow.net/users/78 | How is tropicalization like taking the classical limit? | The analogy I've worked out from pieces here and there goes like this:
using the logarithm and exponential, we define for two real numbers x and y the following binary operation $x §\_h y := h .ln( e^{x/h} + e^{y/h} )$ which depends on some positive real parameter $h$. Then we observe that as $h \rightarrow 0$ the nu... | 12 | https://mathoverflow.net/users/469 | 791 | 465 |
https://mathoverflow.net/questions/457 | 14 | Let A be a non-negative integer square matrix with eigenvalues x1, x2, ... xn. Any symmetric function of these eigenvalues with integer matrices is an integer. I'm aware of the following results regarding the combinatorial interpretation of these integers:
* If A is the adjacency matrix of a finite directed graph G, ... | https://mathoverflow.net/users/290 | What are the Schur functions of the eigenvalues of a non-negative integer matrix counting? | This is moving far enough afield from my own understanding that I'm not sure how useful it is, but have you looked at the Ph.D. thesis of Andrius Kulikauskas? He seems to be giving some sort of combinatorial interpretation in terms of Lyndon words for the Schur functions.
<http://www.selflearners.net/uploads/Andrius... | 6 | https://mathoverflow.net/users/405 | 793 | 467 |
https://mathoverflow.net/questions/775 | 7 | What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical? Wikipedia suggests that any simple ring with a nontrivial right ideal would work, but this is clearly false (take a matrix ring over a field, for instance).
Benson's *Representations and Cohomo... | https://mathoverflow.net/users/396 | What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical? | Certainly every maximal ideal is the annihilator of a simple R-module, but the converse isn't true. See Exercise 4.8 in Lam's "[Exercises in Classical Ring Theory](http://books.google.com/books?id=S3pZbAByfDgC&pg=PA56&lpg=PA56&dq=exercises+in+classical+ring+theory&printsec=frontcover&source=bl&ots=8Fgz07rq_Y&sig=ilAV-8... | 7 | https://mathoverflow.net/users/430 | 798 | 471 |
https://mathoverflow.net/questions/812 | 77 | What is the purpose of the "teaching statement" or "statement of teaching philosophy" when applying for jobs, specifically math postdocs? I *am* applying for jobs, and I need to write one of these shortly.
Let us assume for the sake of argument that I *have* a teaching philosophy; I am not asking you to tell me what ... | https://mathoverflow.net/users/143 | Teaching statements for math jobs? | Having been on both sides of the issue, I might say that having considered it for some time, I really don't know! But in reality if you are looking for a position at a research university, the Dean will want to have evidence (or the non-research faculty will want to have evidence) that you care about teaching. More pre... | 48 | https://mathoverflow.net/users/36108 | 813 | 480 |
https://mathoverflow.net/questions/815 | 32 | Someone recently [said](https://mathoverflow.net/questions/329/what-is-koszul-duality) "derived/triangulated categories are an abomination that should struck from the earth and replaced with dg/A-infinity versions".
I have a rough idea why this is true ("don't throw away those higher homotopies -- you might need them s... | https://mathoverflow.net/users/284 | triangulated vs. dg/A-infinity | I don't really think that triangulated categories are abominable, but they certainly have their problems which are a result of having forgotten the higher homotopies. For instance, non-functoriality of mapping cones can be fixed via dg-enhancement.
Another problem has to do with localization for triangulated categori... | 21 | https://mathoverflow.net/users/310 | 825 | 490 |
https://mathoverflow.net/questions/631 | 4 | This question is based on [a blog post of Qiaochu Yuan](http://qchu.wordpress.com/2009/08/12/krafts-inequality-for-prefix-codes/).
Let P be a locally finite\* graded poset with a minimal element, and w be a weight function on the elements of P. Suppose that the total weight of the elements of rank k is bounded by 1.... | https://mathoverflow.net/users/382 | Is there a "universal LYM inequality?" | An approach to the strong form of the property, based on a probabilistic proof of the LYM inequality (apparently due to Bollobas, can be found in Tao and Vu, Ch. 7):
Consider a graded poset P and let G be a compact topological group that acts on P. Construct a weight function as follows. Fix a distinguished (saturate... | 1 | https://mathoverflow.net/users/382 | 826 | 491 |
https://mathoverflow.net/questions/836 | 6 | Suppose we are given a diagram $X \to Z \gets Y$ of $G$-spaces ($G$ a discrete group). Let $(- \times^h -)$ denote homotopy pullback. Is $(X \times^h\_Z Y)\_{hG}$ weakly equivalent to $X\_{hG} \times^h\_{Z\_{hG}} Y\_{hG}$?
| https://mathoverflow.net/users/126667 | Do homotopy pullbacks commute with homotopy orbits (in spaces)? | Yes. A sketch:
Taking products with the free $G$-space $EG$ commutes with the pullback diagram (because product is also a limit) and so you can assume they're free, and one of the maps is a fibration.
Having done this, there is a natural long exact sequence of homotopy groups
$\to \pi\_\* (U) \to \pi\_\*(U\_{hG}... | 7 | https://mathoverflow.net/users/360 | 839 | 499 |
https://mathoverflow.net/questions/828 | 11 | What I had in mind was something like the following:
X is separated/proper iff for all curves C and all maps f : C \ c -> X, f extends to C in at most/exactly one way.
Is there a good reason why this cannot possibly be true?
Here X denotes a reduced scheme of finite type of a field k (I guess people usually call... | https://mathoverflow.net/users/438 | Is there a version of the valuative criteria for separateness/properness for varieties? | If you make the statement
Fix an algebraically closed base field k and let X be a scheme of finite type over k. Then
X/k is proper iff for all smooth quasi-projective curves C/k and all maps f: C\c -> X then f extends uniquely to f': C -> X.
it seems true to me. This should basically comes down to the fact that on... | 4 | https://mathoverflow.net/users/310 | 841 | 500 |
https://mathoverflow.net/questions/730 | 11 | While reading a [blog post](http://sbseminar.wordpress.com/2007/12/14/the-nullstellensatz-and-partitions-of-unity/) on partitions of unity at the Secret Blogging Seminar the following question came into my mind.
Let $n$ be a positive integer and let $B\_1$ and $B\_2$ be $n \times n$ matrices with integer entries. Is ... | https://mathoverflow.net/users/296 | An "existence contra partition of unity" statement for integer matrices? | I believe that what you say is true. I'll sketch an argument.
Let f:Zn ---> Z2n be the map of free Z-modules given by the matrices B1, B2 put in column (i.e. the direct sum of the morphisms given by B1 and B2). Now we rephrase conditions (1) and (2) in a slightly more abstract way:
* (1) fails to hold if, and only ... | 5 | https://mathoverflow.net/users/322 | 865 | 517 |
https://mathoverflow.net/questions/873 | 13 | Are there any good **short** expositions of planar algebras out there? I am interested primarily in seeing the main definition and some explicit examples.
| https://mathoverflow.net/users/498 | Short Introduction to Planar Algebras | My paper with Emily and Noah about the [`D_{2n}` planar algebra](http://arxiv.org/abs/0808.0764) includes our attempt at a friendly explanation. It's all about arguably the simplest non-trivial example of a *subfactor* planar algebra.
| 13 | https://mathoverflow.net/users/3 | 874 | 521 |
https://mathoverflow.net/questions/868 | 27 | In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation with the Artin-Schreier sequence I believe you can show that, at least, the abelianization of the absolute Galois group is... | https://mathoverflow.net/users/360 | Etale covers of the affine line | Indeed, you can get whatever genus you want even with a fixed Galois group G, so long as its order is divisible by p: this is a result of Pries: .pdf [here.](http://www.math.colostate.edu/~pries/Preprints/00DecPreprints/06genus1_05.pdf)
In fact, Pries has lots of papers about exactly what can happen; looking at her p... | 17 | https://mathoverflow.net/users/431 | 877 | 524 |
https://mathoverflow.net/questions/847 | 23 | Apologies in advance if this is obvious.
| https://mathoverflow.net/users/290 | Is any representation of a finite group defined over the algebraic integers? | Not a satisfying argument: We can, first of all, find a basis in which the entries lie in some algebraic number field $K$. Let $\mathcal{O}$ be the ring of integers of $K$.
Then there is a locally free $\mathcal{O}$-module $M$ of rank $n$ preserved by $G$: add up all the translates of $\mathcal{O}^n$ under $G$. Now, $M... | 22 | https://mathoverflow.net/users/513 | 882 | 528 |
https://mathoverflow.net/questions/827 | 6 | Is it true that the pro-objects of an abelian category form a category with enough projectives?
In general, given an abelian category A, is there a canonical way to embed it a bigger abelian category A' with enough projectives (or injectives) and such that A' is universal with respect to this property?
| https://mathoverflow.net/users/344 | Embedding abelian categories to have enough projectives | It seems that Pro(A) does not have enough projectives in general. In Kashiwara-Schapira's book "Categories and Sheaves" they prove (corollary 15.1.3) that Ind(k-Mod) does not have enough injectives. This means, taking opposite categories, that Pro(k-Mod^{op}) does not have enough projectives.
I don't know of any univ... | 3 | https://mathoverflow.net/users/322 | 892 | 536 |
https://mathoverflow.net/questions/840 | 3 | As I see it, the core question of topology is to figure out whether a homeomorphism exists between two topological spaces.
To answer this question, one defines various properties of a space such as connectedness, compactness, the fundamental group, betti numbers etc.
However, it seems that these properties can at ... | https://mathoverflow.net/users/490 | The core question of topology | As others have noted, it's hopeless to try to answer this question for general topological spaces. However, there are a few positive results if you assume, say, that X and Y are both simply connected closed manifolds of a given dimension. For example, Freedman showed that if X and Y are oriented and have dimension four... | 14 | https://mathoverflow.net/users/424 | 894 | 537 |
https://mathoverflow.net/questions/903 | 44 | I've been doing functional programming, primarily in OCaml, for a couple years now, and have recently ventured into the land of monads. I'm able to work them now, and understand how to use them, but I'm interested in understanding more about their mathematical foundations. These foundations are usually presented as com... | https://mathoverflow.net/users/550 | Resources for learning practical category theory | Online resources:
* [The Catsters channel](http://www.youtube.com/user/TheCatsters)
* [MATH198 course notes](http://haskell.org/haskellwiki/User:Michiexile/MATH198) - examples in Haskell
* [Rydehard, Burstall: Computional Category Theory](http://www.cs.man.ac.uk/%7Edavid/categories/book/) - examples in ML (free repri... | 46 | https://mathoverflow.net/users/158 | 906 | 545 |
https://mathoverflow.net/questions/908 | 15 | I understand why primes are useful numbers and also why the product of large primes are useful such as for application in public key cryptography, but I am wondering why it is useful to continue the search for larger and larger prime numbers such as in the [GIMPS project](http://www.mersenne.org/). It would seem to me ... | https://mathoverflow.net/users/538 | Why the search for ever larger primes? | Well the M in GIMPS stands for Mersenne, and it hasn't been proven that there are infinitely many Mersenne primes. But it's widely believed to be true--in fact there is a conjectural [estimate](http://en.wikipedia.org/wiki/Lenstra%E2%80%93Pomerance%E2%80%93Wagstaff_conjecture#Lenstra.E2.80.93Pomerance.E2.80.93Wagstaff_... | 11 | https://mathoverflow.net/users/126667 | 914 | 551 |
https://mathoverflow.net/questions/924 | 1 | In the chapter 6.4 on normal and self-adjoint operators, there is an example of an infinite dimensional inner product space H that has a normal operator but that has no eigenvectors.
The space is the set of functions f\_n(t) = e^(int) , t in [0,2pi]
with inner product = 1/2pi \* integral\_0\_2pi(e^(-int) e^(imt))dt
... | https://mathoverflow.net/users/570 | Friedberg, Insel, and Spence Linear Algebra example | The space being defined is the **span** of the functions fn, and in the definition of the span we only allow finite sums of the basis vectors.
Edit: I should also mention that the notion of infinite sum in an inner product space doesn't make sense unless the space is also complete with respect to the induced norm, i.... | 4 | https://mathoverflow.net/users/290 | 925 | 559 |
https://mathoverflow.net/questions/849 | 4 | For any two matrices $P,Q \in SU(2)$, with $tr(P)=tr(Q)=0$, does there always exist some $G\in SU(2)$ such that $G P G^{-1} = -P$, and $G Q G^{-1} = -Q\ ?$
| https://mathoverflow.net/users/492 | Conjugation in SU(2) | It's not hard to explicitly construct G using the quaternions, assuming P is not \pm Q, and I think this is worth working out in detail because I really like this picture of SU(2). Identify SU(2) with the unit quaternions by the isomorphism
```
[a b ]
[-\bar{b} \bar{a}] --> a+bj
```
and since the tra... | 4 | https://mathoverflow.net/users/428 | 942 | 575 |
https://mathoverflow.net/questions/929 | 5 | Once I found by accident an article by MacPherson: "Classical projective geometry and modular varieties", in "Algebraic analysis, geometry, and number theory" (Baltimore, MD, 1988), whose introduction thrilled me and I'm still curious about how that developed and if now a general theory of configurations as continuatio... | https://mathoverflow.net/users/451 | a general theory of configurations? | We model theorists have been studying such things for the past couple of decades, so I know something about this.
Suppose that (S, cl) is a matroid -- i.e. a set S endowed with a closure operator satisfying a couple of natural axioms; canonical examples are where S is a vector space and cl(X) = Span(X), or when S a "... | 4 | https://mathoverflow.net/users/93 | 949 | 578 |
https://mathoverflow.net/questions/947 | 22 | I'm looking for the algorithm that efficiently locates the "loneliest person on the planet", where "loneliest" is defined as:
Maximum minimum distance to another person — that is, the person for whom the closest other person is farthest away.
Assume a (admittedly miraculous) input of the list of the exact latitude/... | https://mathoverflow.net/users/587 | How does one find the "loneliest person on the planet"? | The paper *Vaidya, Pravin M.*, [**An $O(n \log n)$ algorithm for the all-nearest-neighbors problem**](https://doi.org/10.1007/BF02187718), Discrete Comput. Geom. 4, No. 2, 101-115 (1989), [ZBL0663.68058](https://zbmath.org/?q=an:0663.68058) gives an $O(n \log n)$ algorithm for the "all-nearest-neighbors" problem: given... | 29 | https://mathoverflow.net/users/126667 | 955 | 581 |
https://mathoverflow.net/questions/953 | 52 | The connection between the fundamental group and covering spaces is quite fundamental. Is there any analogue for higher homotopy groups? It doesn't make sense to me that one could make a branched cover over a set of codimension 3, since I guess, my intuition is all about 1-D loops, and not spheres.
| https://mathoverflow.net/users/353 | Analogue to covering space for higher homotopy groups? | There's certainly a homotopy-theoretic analogue. A universal cover of a connected space $X$ is (up to homotopy) a simply connected space $X'$ and a map $X' \to X$ which is an isomorphism on $\pi\_n$ for $n \geq 2$. We could next ask for a $2$-connected cover $X''$ of $X'$: a space $X''$ with $\pi\_kX = 0$ for $k \leq 2... | 62 | https://mathoverflow.net/users/126667 | 957 | 583 |
https://mathoverflow.net/questions/951 | 4 | I'm aware of the great body of work on [Legendrian knot](http://en.wikipedia.org/wiki/Legendrian_knot) theory in contact geometry, but suppose I'm curious just about homotopy and not isotopy. How does one understand the space of Legendrian loops based at a point in a contact manifold? Can that be made into a "Legendria... | https://mathoverflow.net/users/353 | Legendrian homotopy of curves in a contact structure? | In general, the (parametric) h-principle for Legendrian immersions implies that Legendrian immersions f:L->(M,\xi) are classified up to homotopy (through Legendrian immersions) by the following bundle-theoretic invariant: Choosing a compatible almost complex structure on \xi allows one to complexify the differential of... | 4 | https://mathoverflow.net/users/424 | 963 | 588 |
https://mathoverflow.net/questions/983 | 16 | So I'm not much of a math guy but I've really enjoyed programming in Lisp and have become interested in the ideas of lambda calculus which it is based.
I was wondering if anyone had a suggestion where I should go from here if I'm interested in learning about similar fields. If would be nice if I could relate it back ... | https://mathoverflow.net/users/621 | What is lambda calculus related to? | To get you started, you could try this:
<http://math.ucr.edu/home/baez/week240.html>
It has a whole lot of references, some of which you might want to follow up. In particular, some of them describe the close relations between functional programming, lambda calculus and cartesian closed categories. One reference th... | 18 | https://mathoverflow.net/users/586 | 987 | 604 |
https://mathoverflow.net/questions/968 | 14 | In the answers to [Qiaochu's post](https://mathoverflow.net/questions/847/is-any-representation-of-a-finite-group-defined-over-the-algebraic-integers) on defining representations of finite groups over the algebraic integers, it came out that which fields a representation of a finite group is defined over might depend o... | https://mathoverflow.net/users/66 | What is the Hilbert class field of a cyclotomic field? | Giving an "explicit" description of the Hilbert class field of a number field K (or, more generally, all abelian extensions of K) is Hilbert's 12th problem, and has only been solved for Q and for imaginary quadratic fields. The Hilbert class field H of Q(zeta) will only be contained in a cyclotomic field if H = Q(zeta)... | 14 | https://mathoverflow.net/users/nan | 999 | 611 |
https://mathoverflow.net/questions/915 | 76 | The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}\_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, but an extra statement for $2$. So in these examples characteristic $2$ is a messy special case.
On the other hand, ce... | https://mathoverflow.net/users/290 | Is there a high-concept explanation for why characteristic 2 is special? | I think there are two phenomena at work, and often one can separate behaviors based on whether they are "caused by''one or the other (or both). One phenomenon is the smallness of $2$, i.e., the expression $p-1$ shows up when describing many characteristic $p$ and $p$-adic structures, and the qualitative properties of t... | 63 | https://mathoverflow.net/users/121 | 1003 | 614 |
https://mathoverflow.net/questions/994 | 9 | The valuative criterion for separatedness (resp. properness) says that a morphism of schemes (resp. a quasi-compact morphism of schemes) f:X→Y is separated (resp. proper) if and only if it satisfies the following criterion:
>
> For any valuation ring R (with K=Frac(R)) and any morphisms Spec(R)→Y and Spec(K)→X maki... | https://mathoverflow.net/users/1 | Example where you *need* non-DVRs in the valuative criteria | You can probably just take Y to be the spectrum of a valuation ring A which is not a DVR,
for example the integral closure of C[[t]] in an algebraic closure of C((t)). In this case
any homomorphism from A to a DVR R has to factor through the quotient field or the residue field of A.
For an explcit example, Let X be t... | 14 | https://mathoverflow.net/users/519 | 1042 | 642 |
https://mathoverflow.net/questions/889 | 8 | Mumford's book Abelian Varieties asserts that for a line bundle L on a projective variety (if necessary, you can assume it is as nice as possible), the Euler characteristic $\chi(L^k)$ of tensor powers of $L$ is a polynomial in $k$. If L is very ample, this is just the Hilbert polynomial, and this can be proven by an i... | https://mathoverflow.net/users/75 | Why is the Euler characteristic of powers of a line bundle a polynomial in the power? | OK, here is another way to see it more in line with what you had in mind I think. Write your $L$ as $\mathcal O(D)$ for some divisor $D$ on $X$. Set $J\_1$ to be the ideal sheaf defined by $\mathcal O(-D) \cap \mathcal O\_X$ and $J\_2$ to be the ideal sheaf defined by $\mathcal O(D) \cap \mathcal O\_X$ (intersections t... | 5 | https://mathoverflow.net/users/397 | 1044 | 643 |
https://mathoverflow.net/questions/1039 | 6 | Let f:X→Y be a semismall resolution of singularities. Then the pushforward of the constant sheaf on X is a semisimple perverse sheaf on Y. Under these conditions, I know how to calculate the direct summands of the pushforward f\*ℚX[dim X].
My question is as follows: What more general statements are there that enable ... | https://mathoverflow.net/users/425 | Explicit Direct Summands in the Decomposition Theorem | I'm a little confused about your question. If you just want to know what the summands are there's nothing special about f\*ℚX[dim X]. The only way I know of understanding that sheaf is a general algorithm for understanding all semi-simple perverse sheaves. The only fact you use is that a simple perverse sheaf is concen... | 2 | https://mathoverflow.net/users/66 | 1045 | 644 |
https://mathoverflow.net/questions/1024 | 2 | Let G be the [gamma function](http://en.wikipedia.org/wiki/Gamma_function), and b be a constant in (-2,inf). Let
H(n, i) = G(i+1+b) \* G(n-i+1+b) / [G(i+1) \* G(n-i+1)]
for integers n > i > 0. Let
S(n) = \sum\_{i=1}^{i=n-1} H(n, i).
Let x\_ n = H(n,1) / S(n). Note x\_ 2 = 1, x\_ 3 = 1/2 for all b.
I am convi... | https://mathoverflow.net/users/650 | Limit of sequence involving gamma functions | Using Mathematica and using reflection formulae for Gamma one finds:
x[n,b] = (b+1) n/(n+b) G[n+b+1]/G[n+2b+2] / ( G[b+1]/G[2b+2] - 2 G[n+b+1]/G[n+2b+2] )
Now, observe that for b<-1 the quotients G[n+b+1]/G[n+2b+2] tend to infinity as n->oo (this follows from Stirling's approximation). Accordingly, for such b,
x[... | 3 | https://mathoverflow.net/users/359 | 1049 | 645 |
https://mathoverflow.net/questions/1048 | 46 | When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do
$$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$.
Why do we need to square and then square-root the numbers?
| https://mathoverflow.net/users/668 | Why is it so cool to square numbers (in terms of finding the standard deviation)? | One answer is mathematical convenience. The theory is much simpler using powers of two rather than other powers.
There are justifications. Squaring makes small numbers smaller and makes big numbers bigger. You could argue that a useful measure of dispersion should be forgiving of small errors but weigh larger errors ... | -4 | https://mathoverflow.net/users/136 | 1057 | 650 |
https://mathoverflow.net/questions/530 | 18 | More precisely, what is the smallest exponent e such that, for every n, there exists a group of size at most Cn^e for some absolute constant C and with an n-dimensional irreducible complex representation?
I know that e ≤ 3. For various special values of n we can attain the lower bound e ≥ 2. For example, if n+1 = q ... | https://mathoverflow.net/users/290 | How small can a group with an n-dimensional irreducible complex representation be? | I [posted](https://mathoverflow.net/questions/834/arithmetic-progressions-without-small-primes) a question about primes in arithmetic progressions and was told my understanding is wrong. Although proving this would be harder than the generalized Riemmann hypothesis, the expectation of experts is that, for any n, there ... | 6 | https://mathoverflow.net/users/297 | 1066 | 657 |
https://mathoverflow.net/questions/1075 | 49 | Let G be a finite group of order n. Must every automorphism of G have order less than n?
(David Speyer: I got this question from you long ago, but I don't know whether you knew the answer. I stil don't!)
| https://mathoverflow.net/users/126667 | Order of an automorphism of a finite group | Yes every automorphism has order bounded by |G|-1, provided G is not the trivial group. A reference is
M V Horoševskiĭ 1974 Math. USSR Sb. 22 584-594
which can be found at
<http://www.iop.org/EJ/abstract/0025-5734/22/4/A08/>
It is even shown that the upper bound is reached only for elementary abelian groups.... | 49 | https://mathoverflow.net/users/310 | 1077 | 664 |
https://mathoverflow.net/questions/1095 | 5 | Sorry if the terminology's wrong, I don't know differential topology. Also, this is more of a brain-teaser than a bona fide research question, but it's hopefully a "real mathematician"-level brain-teaser.
So the crossing number inequality gives a lower bound for how many intersections you have to have if you draw a g... | https://mathoverflow.net/users/382 | Smooth immersion(?) of graphs into the plane | If a graph can be embedded into the plane with smooth edges and one point of crossing, then it can be embedded smoothly into the projective plane without crossings, by blowing up the crossing point (and conversely). But not every graph can be embedded into the projective plane, so not every graph has a smooth embedding... | 6 | https://mathoverflow.net/users/440 | 1098 | 676 |
https://mathoverflow.net/questions/1058 | 92 | The Cantor-Bernstein theorem in the category of sets (A injects in B, B injects in A => A, B equivalent) holds in other categories such as vector spaces, compact metric spaces, Noetherian topological spaces of finite dimension, and well-ordered sets.
However, it fails in other categories: topological spaces, groups, ... | https://mathoverflow.net/users/416 | When does Cantor-Bernstein hold? | Whenever the objects in your category can be classified by a bounded collection of cardinal invariants, then you should expect to have the Schroeder-Bernstein property.
For example, vector spaces (over some fixed field $K$) or algebraically closed fields (of some fixed characteristic) can each be classified by a sing... | 79 | https://mathoverflow.net/users/93 | 1101 | 679 |
https://mathoverflow.net/questions/1076 | 9 | The pentagon and hexagon axioms in the definition of a symmetric monoidal category are one example that I was thinking of here; the axioms of a weak 2-category are another. I understand that it can be checked laboriously that these few coherence axioms are sufficient to show, e.g. in the first case, that all coherence ... | https://mathoverflow.net/users/344 | Motivation for coherence axioms | I agree with Viritrilbia: there is in general no "nice" or canonical choice of coherence axioms. By "in general" I mean for an arbitrary 2-dimensional theory, such as the theory of weak 2-categories, monoidal categories, braided monoidal categories with duals, etc. This is true for the same reason that there is in gene... | 10 | https://mathoverflow.net/users/586 | 1104 | 680 |
https://mathoverflow.net/questions/1124 | 13 | In Bonn, we've been have a discussion on the topic in the title:
>
> Suppose that $A$ and $B$ are classes and that there are injections from $A$ to $B$ and from $B$ to $A$. Does it follow that there is a bijection between $A$ and $B$?
>
>
>
Example: Let $A$ the class of sets of cardinality one and let $B$ be t... | https://mathoverflow.net/users/296 | Does Cantor-Bernstein hold for classes? | Ignoring set-theoretic technicalities of formulating the question properly, I see no reason that the usual proof of Schroder-Bernstein wouldn't work.
(Set-theoretic technicalities: In the standard language of set theory, you can't quantify over classes, so you can't quite state this. However, you can prove a metatheo... | 7 | https://mathoverflow.net/users/75 | 1127 | 691 |
https://mathoverflow.net/questions/1129 | 3 | Sorry if this is obvious. I'd like to understand why the map
WC(E/Q) -> H^1(Gal(**Q**/Q), E(**Q**))
is bijective. Thanks.
| https://mathoverflow.net/users/436 | Weil-Châtelet group | This should be in chapter 10 of Silverman's AEC (sorry if I'm wrong about this!).
| 4 | https://mathoverflow.net/users/2 | 1130 | 693 |
https://mathoverflow.net/questions/1108 | 9 | I've been wondering about this ever since I was a little kid and I used to ride in the back of the car and my mom would speed like hell towards a green light, only to slam on the brakes when she realized she wasn't going to make it.
Here is my question, loosely phrased: Given that we want to make it across the inters... | https://mathoverflow.net/users/303 | easy(?) probability/diff eq. question | I think there's potentially an interesting set of problems here - I thought about it myself some time ago, and I'm [not the only one](http://terrytao.wordpress.com/2008/12/09/an-airport-inspired-puzzle/#comment-33878) (if you're too lazy to follow the link, this is Tim Gowers posing exactly this question on Terry Tao's... | 6 | https://mathoverflow.net/users/25 | 1134 | 695 |
https://mathoverflow.net/questions/1081 | 9 | It would also be nice if someone can explain this comment appearing on the Wikipedia page on CW-complexes:
"The homotopy category of CW complexes is, in the opinion of some experts, the best if not the only candidate for the homotopy category."
| https://mathoverflow.net/users/689 | What is an example of a topological space that is not homotopy equivalent to a CW-complex? | My favorite example of a space which is not homotopy equivalent to a CW complex is the Long Line. All it's homotopy groups vanish (exercise 1) but the long line is not contractible (exercise 2). It's too long!
| 10 | https://mathoverflow.net/users/184 | 1137 | 698 |
https://mathoverflow.net/questions/1142 | 6 | **Edit:** Apparently the answer is "no", so what is an example of two curves of genus g, and a divisor of degree d on each, such that one is very ample and the other is not?
Question as originally stated:
Suppose X is a complete nonsingular curve (smooth proper integral scheme of dimension 1 over C) and D ∈ DivX.
... | https://mathoverflow.net/users/84526 | Is very ampleness of a divisor on a curve determined entirely by degree and genus? | Let $C\_1$ be a hyperelliptic curve of genus $g \geq 3$ (example: $y^7 = x^2 + 1$ for $g = 3$), and $C\_2$ be a non-hyperelliptic curve of the same genus $g$ (for example, the Klein quartic with $g = 3$ again: I'll use it in the form $y^7 = x^2(x-1)$).
Then let $K\_1$, $K\_2$ be the canonical divisors of $C\_1$, $C\_... | 11 | https://mathoverflow.net/users/422 | 1158 | 713 |
https://mathoverflow.net/questions/1159 | 32 | A lot of times I see theorems stated for local rings, but usually they are also true for "graded local rings", i.e., graded rings with a unique homogeneous maximal ideal (like the polynomial ring). For example, the Hilbert syzygy theorem, the Auslander-Buchsbaum formula, statements related to local cohomology, etc.
B... | https://mathoverflow.net/users/321 | Graded local rings versus local rings | One small thing I know of which changes is that if one has a Z-graded-commutative noetherian ring (where Z is the integers) Matlis' classification of indecomposable injective modules goes through but with one small hiccup.
Every indecomposable injective is isomorphic to E(R/p)[n] for some unique homogeneous prime ide... | 8 | https://mathoverflow.net/users/310 | 1163 | 715 |
https://mathoverflow.net/questions/1151 | 58 | In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come into play? It is very confusing and awkward to me that why taking injective stuff into consideration would allow you to "e... | https://mathoverflow.net/users/nan | Sheaf cohomology and injective resolutions | Since everybody else is throwing derived categories at you, let me take another approach and give a more lowbrow explanation of how you might have come up with the idea of using injectives. I'll take for granted that you want to associate to each object (sheaf) $F$ a bunch of abelian groups $H^i(F)$ with $H^0(F)=\Gamma... | 92 | https://mathoverflow.net/users/1 | 1165 | 717 |
https://mathoverflow.net/questions/1102 | 16 | Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular construction if G is finitely presented, etc.
What I'm wondering about, is there a notion of a smooth classifying space? Th... | https://mathoverflow.net/users/343 | Smooth classifying spaces? | The answer to this does depend highly on the category in which you are prepared to work. If by "smooth structure" you mean "when is BG a **finite dimensional** manifold" then the answer is, as Andy says, "not many".
However if you are prepared to admit that there are more things that deserve the name "smooth" than ju... | 13 | https://mathoverflow.net/users/45 | 1177 | 722 |
https://mathoverflow.net/questions/1166 | 2 | More precisely, is every concrete category C isomorphic to a category C' of small categories such that the morphisms between two elements of C are precisely the functors between their images in C'?
At some point I started adopting this point of view as a philosophy without ever bothering to actually verify it.
| https://mathoverflow.net/users/290 | Can the objects of every concrete category themselves be realized as small categories? | No. There is no full subcategory of Cat equivalent to • ⇉ •, because for any two objects C and D of Cat, either both Hom(C, D) and Hom(D, C) are nonempty, or one of C and D is the empty category and then one of Hom(C, D) and Hom(D, C) is empty and the other is the category •.
| 4 | https://mathoverflow.net/users/126667 | 1178 | 723 |
https://mathoverflow.net/questions/1184 | 23 | There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and then ask how far they can be generalized.
* A good place to start is [Hadwiger's theorem](http://en.wikipedia.org/wiki... | https://mathoverflow.net/users/78 | Is there a topological description of combinatorial Euler characteristic? | the best approach to the geometric euler characteristic comes from the theory of o-minimal structures.
the best reference in this area is the book "tame topology and o-minimal structures" by lou van den dries. requires very little background to understand.
in brief: an o-minimal structure is collection of boolean ... | 15 | https://mathoverflow.net/users/15245 | 1192 | 733 |
https://mathoverflow.net/questions/1162 | 56 | Every year or so I make an attempt to "really" learn the Atiyah-Singer index theorem. I always find that I give up because my analysis background is too weak -- most of the sources spend a lot of time discussing the topology and algebra, but very little time on the analysis. Question : is there a "fun" source for readi... | https://mathoverflow.net/users/317 | Atiyah-Singer index theorem | I found Booss, Bleecker: "Topology and analysis, the Atiyah-Singer index formula and gauge-theoretic physics" ([review](http://projecteuclid.org/euclid.bams/1183553481)) very beautifull and had read it just for fun. It is a very nice piece of exposition, motivates everything and demands from the reader only very little... | 29 | https://mathoverflow.net/users/451 | 1193 | 734 |
https://mathoverflow.net/questions/1186 | 3 | I'm trying to compute the singular cohomology of SO(4), just as practice for using spectral sequences. I got H0=**Z**, H1=0, H2=**Z**/2**Z**, H3=**Z**⊕**Z**, H4=0, H5=**Z**/2**Z**, and H6=**Z**. Are these correct? I'm not sure if I'm reading it right, but these calculations seem to disagree with [this](http://www.math.... | https://mathoverflow.net/users/303 | singular cohomology of SO(4) | Your calculation is correct. In Hatcher's description of integral cohomology mod torsion the last generator has to be interpreted as an extra generator in addition to the previous ones.
The integral cohomology ring of the limit is in general not isomorphic to the E\_{\infty} ring,
but the E\_{\infty} ring is the ass... | 4 | https://mathoverflow.net/users/86 | 1200 | 739 |
https://mathoverflow.net/questions/1123 | 13 | Let $S$ be a base scheme, let $A/S$ be an abelian scheme, and let $\mathbf{G}\_m/S$ be the multiplicative group; consider $A$ and $\mathbf{G}\_m$ as objects in the abelian category of sheaves of abelian groups on the fppf site of $S$, and take $\mathrm{Ext}$'s between them. We know that $\mathrm{Ext}^1(A,\mathbf{G}\_m)... | https://mathoverflow.net/users/307 | What are the higher $\mathrm{Ext}^i(A,\mathbf{G}_m)$'s, where $A$ is an abelian scheme? | It seems that here you are talking about the local exts, i.e. about the sheaves $\underline{Ext}^{i}(A,G\_m)$ on $S$. The question is rather tricky actually. The problem is that before we try to answer it, we should first specify what we mean by $\underline{Ext}^{i}(A,G\_m)$. We could mean exts in the category of sheav... | 11 | https://mathoverflow.net/users/439 | 1220 | 746 |
https://mathoverflow.net/questions/1238 | 4 | This is a pretty basic question but I have been stuck on it for a while.
Given an abstract simplicial complex X and a subcomplex A, why does \* suffice to show that the map |A|->|X| induced by inclusion is a homotopy equivalence:
* Let g: (|K|,|L|) -> (|X|,|A|) be a continuous map, where K is a **finite** simplicia... | https://mathoverflow.net/users/353 | proving that an inclusion map from a subcomplex is a homotopy equivalence | By taking K = a simplex and L = its boundary you can show that |A| -> |X| is an isomorphism on all homotopy groups (do surjectivity and injectivity separately). Then apply Whitehead's theorem.
| 3 | https://mathoverflow.net/users/126667 | 1244 | 760 |
https://mathoverflow.net/questions/1237 | 72 | The automorphism group of the symmetric group $S\_n$ is $S\_n$ when $n$ is not $2$ or $6$, in which cases it is respectively $1$ and the semidirect product of $S\_6$ with the (cyclic) group of order $2$. (For this famous outer automorphism, see for instance wikipedia or Baez's thoughts on the number $6$.)
On the othe... | https://mathoverflow.net/users/336 | Is ${\rm S}_6$ the automorphism group of a group? | ${\rm S}\_6$ is not the automorphism group of a finite group.
See H.K. Iyer, *On solving the equation Aut(X) = G*, Rocky Mountain J. Math. 9 (1979), no. 4, 653--670, available online
[here](http://dx.doi.org/10.1216/RMJ-1979-9-4-653).
This paper proves that for any finite group $G$, there are finitely many
finite gr... | 89 | https://mathoverflow.net/users/428 | 1245 | 761 |
https://mathoverflow.net/questions/605 | 13 | One can easily prove that factors have no nontrivial ultraweakly closed 2-sided ideals as these are equivalent to nontrivial central projections. One can also show type $I\_n$, type $II\_1$, and type $III$ factors are algebraically simple (any 2-sided ideal must contain a projection. All projections are comparable in a... | https://mathoverflow.net/users/351 | Ideals in Factors | Blackadar in his textbook on operator algebras gives a complete classification of norm-closed ideals in factors.
See Proposition III.1.7.11.
| 12 | https://mathoverflow.net/users/402 | 1258 | 768 |
https://mathoverflow.net/questions/1206 | 9 | Let $\Lambda(n)$ be the von Mangoldt function, i.e., $\Lambda(n) = \log p$ for $n$ a prime power $p^k$ and $\Lambda(n) = 0$ for all $n$ that not prime powers. Let
$$S(\alpha) = \sum\_{n \leq N} \Lambda(n) e(\alpha n).$$
Now, using, say, Lemma 7.15 in Iwaniec-Kowalski (or the same result in Montgomery), we get
$$... | https://mathoverflow.net/users/398 | The large sieve for primes | Heh, I think I know why you are interested in this question, Harald, as Ben and I thought about essentially the same problem for what I suspect to be the same reason :-)
Anyway, we were able to get rid of the e^gamma factor. One way to proceed is to work not with the exponential sums, but rather the inner product of ... | 10 | https://mathoverflow.net/users/766 | 1276 | 780 |
https://mathoverflow.net/questions/1051 | 20 | Can one partition the set of positive integers into finitely many Pythagorean-triple-free subsets? If so, what is the smallest number of such subsets? Taking a wild guess, I would
be least surprised if the answer were 3.
Notice that the 2 subsets of integers such that highest power of 5 that divides them is
a) even
b... | https://mathoverflow.net/users/2480 | Splitting Pythagorean triples | This problem appears in Croot and Lev's 2007 "Open Problems in Additive Combinatorics" (<http://people.math.gatech.edu/~ecroot/E2S-01-11.pdf> ), where it is attributed to Erdos and Graham (the latter of whom offers $250 for its solution).
Other references may include Cooper and Poirel's "Notes on the Pythagorean Tri... | 18 | https://mathoverflow.net/users/405 | 1279 | 783 |
https://mathoverflow.net/questions/854 | 2 | I am programming an algorithm where I have broken up the surface of a sphere into grid points (for simplicity I have the grid lines are parallel and perpendicular to the meridians). Given a point A on the sphere, I would like to be able to efficiently take any grid "square" and determine the point B in the square with ... | https://mathoverflow.net/users/494 | Closest grid square to a point in spherical coordinates | The tricky part is to find the nearest point on a meridian to a given point P. Let's fix the meridian M at φ=0, to keep the algebra simple.
Suppose P has spherical coordinates (θ, φ), with 0 ≤ θ ≤ π and -π < φ ≤ π. Let C be the great circle through P which is perpendicular to M; then we are looking for an intersecti... | 2 | https://mathoverflow.net/users/767 | 1281 | 785 |
https://mathoverflow.net/questions/1283 | 10 | I've seen the following lower bound for the complementary error function ([erfc](http://en.wikipedia.org/wiki/Erfc)) but I haven't been able to prove it. Does anyone know how to establish the following?
$$erfc(x) > \frac{ x \exp(-x^2) }{ \pi(1 + 2x^2) }$$
| https://mathoverflow.net/users/136 | erfc lower bound | Durrett, *Probability: Theory and Examples*, 3rd edition, p. 6 gives
$$(x^{-1} - x^{-3}) e^{-x^2/2} \le \int\_x^\infty e^{-y^2/2} \: dy $$
The proof Durrett gives is from the observation that
$$ \int\_x^\infty (1-3y^{-4}) e^{-y^2/2} \: dy = \left( x^{-1} + x^{-3} \right) e^{-x^2/2} $$
which I suspect can be fou... | 7 | https://mathoverflow.net/users/143 | 1290 | 791 |
https://mathoverflow.net/questions/1267 | 3 | How to classify **K3 surfaces** over an arbitrary field *k*?
| https://mathoverflow.net/users/65 | K3 over fields other than C? | The "standard" definition of a K3 surface is field independent (unless you are a physicist):
$p\_g=1, q=0$, and trivial canonical class.
Some results:
* Mumford and Bombieri showed that you get (just as in the complex case) a 19 dimensional family of K3 surfaces for any degree (the 19 dimensional thingy is a defo... | 8 | https://mathoverflow.net/users/404 | 1299 | 793 |
https://mathoverflow.net/questions/1291 | 127 | Unfortunately this question is relatively general, and also has a lot of sub-questions and branches associated with it; however, I suspect that other students wonder about it and thus hope it may be useful for other people too. I'm interested in learning modern Grothendieck-style algebraic geometry in depth. I have som... | https://mathoverflow.net/users/344 | A learning roadmap for algebraic geometry | FGA Explained. Articles by a bunch of people, most of them free online. You have Vistoli explaining what a Stack is, with Descent Theory, Nitsure constructing the Hilbert and Quot schemes, with interesting special cases examined by Fantechi and Goettsche, Illusie doing formal geometry and Kleiman talking about the Pica... | 41 | https://mathoverflow.net/users/622 | 1305 | 799 |
https://mathoverflow.net/questions/1294 | 36 | Let's say that I have a one-dimensional line of finite length 'L' that I populate with a set of 'N' random points. I was wondering if there was a simple/straightforward method (not involving long chains of conditional probabilities) of deriving the probability 'p' that the minimum distance between any pair of these poi... | https://mathoverflow.net/users/774 | Mean minimum distance for N random points on a one-dimensional line | This can answered without any complicated maths.
It can be related to the following: Imagine you have $N$ marked cards in a pack of $m$ cards and shuffle them randomly. What is the probability that they are all at least distance $d$ apart?
Consider dealing the cards out, one by one, from the top of the pack. Every ti... | 60 | https://mathoverflow.net/users/1004 | 1308 | 801 |
https://mathoverflow.net/questions/316 | 6 | What's the most natural way to establish the asymptotics of $\Delta$ on a compact Riemannian manifold $M$ of dimension $N$? The asymptotics should be
$$ \#\{v < A^2\} = \mathrm{const}\ast\mathrm{vol}(M)\ast A^n + o(\mathrm{something}) $$
(Perhaps one could consider first the case of a Kähler manifold? The Laplacian is... | https://mathoverflow.net/users/65 | Eigenvalues of Laplacian | The most natural way is to study the short-time asymptotics of the heat or wave kernel on M. For example, you can use the heat kernel $p\_t(x,y) = \sum\_i e^{-\lambda\_i t} f\_i(x) \overline{f\_i(y)}$ where $f\_i$ are the eigenfunctions with eigenvalues $\lambda\_i$. This is a fundamental solution to the heat equation.... | 7 | https://mathoverflow.net/users/327 | 1315 | 806 |
https://mathoverflow.net/questions/385 | 35 | There is supposed to be a philosophy that, at least over a field of characteristic zero, every "deformation problem" is somehow "governed" or "controlled" by a differential graded Lie algebra. See for example <http://arxiv.org/abs/math/0507284>
I've seen this idea attributed to big names like Quillen, Drinfeld, and D... | https://mathoverflow.net/users/83 | Deformation theory and differential graded Lie algebras | I hope to write more on this later, but for now let me make some general assertions: there are general theorems to this effect and give two references: arXiv:math/9812034, DG coalgebras as formal stacks, by Vladimir Hinich, and the survey article arXiv:math/0604504, Higher and derived stacks: a global overview, by Bert... | 38 | https://mathoverflow.net/users/582 | 1320 | 811 |
https://mathoverflow.net/questions/1269 | 8 | Are **supersingular primes** and **supersingular elliptic curves** related?
(this was essentially a subquestion in [my earlier question](https://mathoverflow.net/questions/1249/ways-to-characterize-supersingular-primes), but still looks sufficiently different to me to deserve a separate post)
| https://mathoverflow.net/users/65 | What does "supersingular" mean? | Let F be the finite field with p elements.
A supersingular elliptic curve is an elliptic curve E/F with the property that the endomorphism ring (ring of homomorphisms from E to E) of E over the algebraic closure of F\_p is has rank 4 as a Z-module.
It is a theorem that End(E/F) has rank 2 or rank 4 (and that in th... | 10 | https://mathoverflow.net/users/683 | 1324 | 815 |
https://mathoverflow.net/questions/1325 | 2 | The mathedu mailing list has a recent longish thread at
<http://www.nabble.com/Why-do-we-do-proofs--to25809591.html>
which discussed among other things whether we should teach triangles as labeled or unlabeled to high school students (this is a vast oversimplification of the thread). I have long been concerned with... | https://mathoverflow.net/users/342 | Characterizing triangles unembeddedly | I'm not sure what you have in mind for circles--the only invariant of a connected compact Riemannian 1-manifold is its length; they have only extrinsic curvature (defined for a manifold embedded in Euclidean space) not intrinsic curvature. You'd have to consider Riemannian manifolds with some extra structure--perhaps j... | 4 | https://mathoverflow.net/users/126667 | 1341 | 829 |
https://mathoverflow.net/questions/1329 | 14 | So, when doing LaTeX, it is absolutely necessary for ones sanity to using a preview program which updates automatically every time you compile. Of course, any previewer designed for DVIs will do this, but as far I can tell, Adobe Acrobat not only does not automatically update, but will not let you change the PDF with i... | https://mathoverflow.net/users/66 | Automatically updating PDF reader for Windows | Use [SumatraPDF](http://blog.kowalczyk.info/software/sumatrapdf/index.html). It is a lightweight pdf viewer which updates automatically. It also allows syncing with [TeXnicCenter](http://transact.dl.sourceforge.net/project/texniccenter/Tutorials/How_to_Sumatra_EN_%282009.09.23%29.pdf) and [WinEdt](http://robjhyndman.co... | 18 | https://mathoverflow.net/users/261 | 1347 | 834 |
https://mathoverflow.net/questions/1312 | 13 | Why, from a string theory perspective, is it natural to consider the Deligne-Mumford (resp. Kontsevich) compactification of the moduli of curves (resp. maps [from curves to a target space X]) rather than some other compactification?
In any case, what other compactifications of the moduli of curves have been studied?... | https://mathoverflow.net/users/83 | Gromov-Witten theory and compactifications of the moduli of curves | I can give individual answers to a lot of your questions, but I can't answer any of them completely, nor can I fit all these answers together into a coherent whole.
For string theory, there does seem to be something special about the Deligne-Mumford compactification. Morally, what's going on is this: string theorists... | 12 | https://mathoverflow.net/users/35508 | 1357 | 839 |
https://mathoverflow.net/questions/1365 | 21 | I want to say that a [group object](http://en.wikipedia.org/wiki/Group_object) in a category (e.g. a discrete group, topological group, algebraic group...) is the image under a product-preserving functor of the "group object diagram", $D$. One problem with this idea is that this diagram $D$ as a category on its own doe... | https://mathoverflow.net/users/84526 | Is there a "universal group object"? (answered: yes!) | Yes, the category U is the opposite of the full subcategory of Grp on the free groups on 0, 1, 2, ... generators. This is an instance of Lawvere's theory of "theories". See [this nLab entry](http://ncatlab.org/nlab/show/Lawvere+theory) for a discussion (of this example in fact).
| 32 | https://mathoverflow.net/users/126667 | 1370 | 846 |
https://mathoverflow.net/questions/1363 | 11 | Let's say that I want to prove that a language is not regular.
The only general technique I know for doing this is the so-called "pumping lemma", which says that if $L$ is a regular language, then there exists some $n>0$ with the following property. If $w$ is a word in $L$ of length at least $n$, then we can write $w... | https://mathoverflow.net/users/317 | Regular languages and the pumping lemma | For necessary and sufficient conditions for a language to be regular (sometimes useful in proving nonregularity when simpler tricks like the pumping lemma fail) see the [Myhill–Nerode theorem](http://en.wikipedia.org/wiki/Myhill%E2%80%93Nerode_theorem).
| 18 | https://mathoverflow.net/users/440 | 1375 | 849 |
https://mathoverflow.net/questions/1367 | 29 | Note: This comes up as a byproduct of Qiaochu's question ["What are examples of good toy models in mathematics?"](https://mathoverflow.net/questions/1354/what-are-examples-of-good-toy-models-in-mathematics)
There seems to be a general philosophy that problems over function fields are easier to deal with than those ov... | https://mathoverflow.net/users/nan | Global fields: What exactly is the analogy between number fields and function fields? | There's a really nice table in section 2.6 of [these](http://www-math.mit.edu/~poonen/papers/curves.pdf) notes from a seminar that Bjorn Poonen ran at Berkeley a few years ago.
| 44 | https://mathoverflow.net/users/2 | 1376 | 850 |
https://mathoverflow.net/questions/1373 | 6 | Is there a good software package for doing computations in the cohomology ring of Grassmannians? Things like, I can write down a polynomial in, in fact, special Schubert classes, but it's one where doing the multiplication out is too tedious for me to have any chance at accuracy in the final answer, and want an efficie... | https://mathoverflow.net/users/622 | Is there a software package that does Schubert Calculus computations? | There's a Littlewood-Richardson calculator here:
<http://math.rutgers.edu/~asbuch/lrcalc/>
I usually use the "SchurRings" package in Macaulay 2 ( <http://www.math.uiuc.edu/Macaulay2/> ) though. No particular reason why, just that Macaulay 2 is something I am used to using. It's very easy to use, here's an example (it... | 5 | https://mathoverflow.net/users/321 | 1379 | 853 |
https://mathoverflow.net/questions/1383 | 3 | I feel a little embarrassed to be asking this question here, since I think it should be much easier than I'm making it, but here goes:
Given a finite poset P, does there necessarily exist some chain that intersects every maximal antichain? (Note: By maximal antichain, I mean that there's no antichain strictly contain... | https://mathoverflow.net/users/382 | Chains intersecting antichains in finite posets | No.
Consider the poset of subsets of {x,y,z} under inclusion. The maximal chain Ø, {x}, {x,y}, {x, y, z} does not intersect every maximal antichain: it misses the maximal antichain {y}, {x,z}. By symmetry every other maximal chain also misses some maximal antichain.
| 5 | https://mathoverflow.net/users/440 | 1386 | 857 |
https://mathoverflow.net/questions/1346 | 9 | I know that the individual cohomology groups are representable in the homotopy category of spaces by the Eilenberg-MacLane spaces. Is it also true that the entire cohomology ring is representable? If so, is there a geometric interpretation of the cup product as an operation on the representing space?
| https://mathoverflow.net/users/788 | Representablity of Cohomology Ring | The total cohomology of spaces should be thought of as a **graded ring**, or even more precisely as a **graded E\* algebra** where E\* is the cohomology of a point. It is representable in the sense that there is a *graded E\* algebra object* in hTop representing it.
Let's unpack that a little.
First, you have to un... | 9 | https://mathoverflow.net/users/45 | 1392 | 861 |
https://mathoverflow.net/questions/1387 | 9 | Does anybody know good references to learn about Lie superalgebras? I started with Howe's "Remarks on classical invariant theory", which contains a study of osp(m,2n), and now I am reading Kac's '77 Advances paper. I wonder if there are other helpful sources. I am especially interested in getting a feel for the represe... | https://mathoverflow.net/users/316 | References for Lie superalgebras | Have you seen the survey by Frappat-Sciarrino-Sorba, "Dictionary on Lie Superalgebras" listed [here](http://ncatlab.org/nlab/show/super+Lie+algebra)?
When you have collected more references, please feel encouraged to add them to that list there...
| 3 | https://mathoverflow.net/users/381 | 1400 | 868 |
https://mathoverflow.net/questions/1359 | 20 | Given a morphism of schemes f: U → X, can one determine when f is an isomorphism of U onto an open subscheme of X in terms of some induced functors between the categories of quasicoherent modules Qcoh(U) and Qcoh(X)?
---
To begin, it might be helpul to simply assume that U ⊆ X is an open subscheme, and consider s... | https://mathoverflow.net/users/778 | Functorial characterization of open subschemes? | The abelian category of quasicoherent sheaves on a schemes determine the scheme. This is an old result of Gabriel ("[des categories abeliennes](http://www.numdam.org/item?id=BSMF_1962__90__323_0)" 1962), proved in full generality by [Rosenberg](http://www.mpim-bonn.mpg.de/preprints/send?bid=3948). This means that, $\op... | 22 | https://mathoverflow.net/users/322 | 1417 | 883 |
https://mathoverflow.net/questions/1438 | 28 | This is in the same vein as my previous question on the representability of the cohomology ring. Why are the homology groups not corepresentable in the homotopy category of spaces?
| https://mathoverflow.net/users/788 | Why is homology not (co)representable? | Corepresentable functors preserve products; homology does not.
One replacement is the following. Let X be a CW-complex with basepoint. Then the spaces {K(Z,n)} represent reduced integral homology in the sense that for sufficiently large n, the reduced homology Hk(X) coincides with the homotopy groups of the smash pro... | 24 | https://mathoverflow.net/users/360 | 1444 | 902 |
https://mathoverflow.net/questions/1443 | 3 | I am given a graph defined by vertexes and edges. I have to obtain all the cycle bases in a network. **No coordinates will be given for the nodes.**
Here's a [sketch](http://deluxecourse.com/network.png) that illustrates my point.
**Note that inside a cycle it must not contain any edge**
| https://mathoverflow.net/users/807 | Algorithm to find all the cycle bases in a graph | Maybe what you want is a cycle basis? That is, a set of cycles such that any other cycle can be found by adding and subtracting combinations of cycles in the basis. One can find a cycle basis easily for any graph by finding a spanning tree and then, for each edge that's not in the tree, reporting the cycle formed by th... | 11 | https://mathoverflow.net/users/440 | 1466 | 919 |
https://mathoverflow.net/questions/1420 | 111 | For my purposes, you may want to interpret "best" as "clearest and easiest to understand for undergrads in a first number theory course," but don't feel too constrained.
| https://mathoverflow.net/users/66 | What's the "best" proof of quadratic reciprocity? | I think by far the simplest easiest to remember elementary proof of QR is due to Rousseau ([On the quadratic reciprocity law](https://stacky.net/files/115/RousseauQR.pdf "G. Rousseau: On the quadratic reciprocity law")). All it uses is the Chinese remainder theorem and Euler's formula $a^{(p-1)/2}\equiv (\frac{a}{p}) \... | 135 | https://mathoverflow.net/users/22 | 1472 | 923 |
https://mathoverflow.net/questions/1440 | 16 | Is there a nice analog of the Freyd-Mitchell theorem for triangulated categories (potentially with some requirements)? Freyd-Mitchell is the theorem which says that any small abelian category is a fully faithful, exact embedding into the module category of some ring.
Therefore, I'd like a theorem like this:
Any small... | https://mathoverflow.net/users/750 | Freyd-Mitchell for triangulated categories? | There are some things like what you ask for but as Tyler points out one needs restrictions on the categories one can consider.
Any algebraic triangulated category which is well generated is equivalent to a localization of the derived category of a small DG-category - this is a theorem of Porta (ref is M. Porta, The P... | 9 | https://mathoverflow.net/users/310 | 1478 | 928 |
https://mathoverflow.net/questions/1465 | 19 | Polynomials in $\mathbb Z[t]$ are categorified by considering Euler characteristics of complexes of finite-dimensional graded vector spaces. Now, given a rational function that has a power series expansion with integer coefficients, it seems natural to consider complexes of (locally finite-dimensional) graded vector sp... | https://mathoverflow.net/users/813 | Can we categorify the equation (1 - t)(1 + t + t^2 + ...) = 1? | Yes, the particular equation you wrote is categorified by the free resolution of k as module over k[x] by the complex $k[x] \overset{x}\longrightarrow k[x]$ given by multiplication by x. It also appears in the numerical criterion for Koszulity of k[x] (see the paper of [Beilinson, Ginzburg and Soergel](http://mathaware... | 22 | https://mathoverflow.net/users/66 | 1481 | 929 |
https://mathoverflow.net/questions/1479 | 5 | I was planning on figuring this problem out for myself, but I also wanted to try out mathoverflow. Here goes:
I wanted to know the asymptotics of the sum of the absolute values of the Fourier-Walsh coefficients of the "Majority" function on 2n+1 binary inputs. Long story short, that boiled down to finding the asympto... | https://mathoverflow.net/users/658 | Asymptotics of a hypergeometric series/Taylor series coefficient. | Okay, you want the asymptotics of
[z^n] 1/((1-2z) sqrt(1-4z)).
In a neighborhood of z = 1/4, you have
1/((1-2z) sqrt(1-4z)) = 2/sqrt(1-4z) \* (1+o(1))
Then a theorem of Flajolet and Odlyzko (see Flajolet and Sedgewick, *Analytic Combinatorics*, Cor. VI.1; the book is available for free download from [Flajolet's... | 4 | https://mathoverflow.net/users/143 | 1486 | 934 |
https://mathoverflow.net/questions/1489 | 71 | Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup generated?
| https://mathoverflow.net/users/828 | Cohomology and fundamental classes | Rene Thom answered this in section II of "Quelques propriétés globales des variétés différentiables." Every class $x$ in $H\_r(X; \mathbb Z)$ has some integral multiple $nx$ which is the fundamental class of a submanifold, so the homology is at least rationally generated by these fundamental classes.
Section II.11 wo... | 71 | https://mathoverflow.net/users/428 | 1495 | 940 |
https://mathoverflow.net/questions/834 | 16 | The following question came up in the discussion at [How small can a group with an n-dimensional irreducible complex representation be?](https://mathoverflow.net/questions/530/how-small-can-a-group-with-an-n-dimensional-irreducible-complex-representation-be) :
Is it known that there are infinitely many primes p for w... | https://mathoverflow.net/users/297 | Arithmetic progressions without small primes | It is firmly expected that for every \epsilon > 0 each aritmetic progression with difference q and terms coprime with q will contain a prime <<{\epsilon} q^{1 + \epsilon}.
This is a direct consequence of a conjecture of H. L. Montgomery (not the one on pair correlations, another one), which implies both the GRH and th... | 13 | https://mathoverflow.net/users/3304 | 1498 | 942 |
https://mathoverflow.net/questions/1492 | 10 | Using a minimum of technical vocabulary, give a summary of why it is that the moduli space of genus g complex curves with n marked points has a natural compactification that is isomorphic (as a complex orbifold) to a projective algebraic variety.
| https://mathoverflow.net/users/683 | Moduli spaces of complex curves as algebraic varieties | The classical (pre Deligne-Mumford) approach is to map $\mathcal{M}\_g$ into $\mathcal{A}\_g$ using the Torreli map. Whereas the classical way to see that $\mathcal{A}\_g$ is quasi-projective is to define it as the Siegel upper half space $\mathcal{H}\_g$ ($g$ by $g$ complex matrices with positive define imaginary part... | 12 | https://mathoverflow.net/users/404 | 1500 | 943 |
https://mathoverflow.net/questions/1504 | 34 | More precisely, how does one characterize integrally closed finitely generated domains (say, over C) based on geometric properties of their varieties? Given a finitely generated domain A and its integral closure A' (in its field of fractions), what's the geometric relationship between V(A) and V(A')?
If you can, phr... | https://mathoverflow.net/users/290 | What is the geometric meaning of integral closure? | The property you are interested in is known as being *normal*. For affine varieties, the definition of normal is just that the coordinate ring is integrally closed, and the operation on varieties that corresponds to taking the integral closure of the coordinate ring is known as *normalization* (a general variety is sai... | 29 | https://mathoverflow.net/users/422 | 1512 | 951 |
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