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https://mathoverflow.net/questions/6165 | 1 | I have a general question in Riemannian geometry:
Let M be a compact manifold and $\partial M \neq \emptyset$. Then shoot a geodesic from any boundary point perpendicularly into the interior of M. How can one prove it will end at boundary? If so, it induces a transformation of $\partial M$, does anyone know any result ... | https://mathoverflow.net/users/1947 | Transformations induced by geodesics of boundary | It doesn't seem true to me that the geodesic will always return to the boundary. It may end up accumulating around a closed geodesic in the interior.
For example, consider the hyperboloid of one sheet
$$x^2+y^2-z^2=1$$
in $\mathbb{R}^3$. This is a surface of revolution and so one can talk of the angular momentum of... | 9 | https://mathoverflow.net/users/380 | 6167 | 4,199 |
https://mathoverflow.net/questions/6180 | 39 | In the same vein as [Kate](https://mathoverflow.net/questions/544/why-are-subfactors-interesting) and [Scott](https://mathoverflow.net/questions/2046/how-do-i-describe-a-fusion-category-given-a-subfactor)'s questions, why are fusion categories interesting? I know that given a "suitably nice" fusion category (which prob... | https://mathoverflow.net/users/351 | Why are fusion categories interesting? | Fusion categories (over $\mathbb{C}$) are a natural generalization of finite groups and their behavior over $\mathbb{C}$. The complex representation theory of a finite group is a fusion category, but there are many others. In fact, you can think of a fusion category as a non-commutative, non-cocommutative generalizatio... | 48 | https://mathoverflow.net/users/1450 | 6183 | 4,208 |
https://mathoverflow.net/questions/6186 | 4 | I need to compute the number of solutions to the equation $x^{p+1} = y^4$ in the field with $p^2$ elements (for p sufficiently large). The form of the equation suggests to me that the solution would depend on the congruence class of p mod 4, but I have reason to believe that the answer is a single polynomial in p.
I ... | https://mathoverflow.net/users/10273 | Counting solutions to x^{p+1}=y^4 in a finite field | Let g be a generator of the multiplicative group of the field; assuming x and y are nonzero, we can write x=ga and y=gb with 0 <= a,b < p2-1, and then xp+1=y4 becomes ga(p+1)=g4b, or equivalently a(p+1) = 4b (mod p2-1).
From this we see that p+1 | 4b is necessary, and if 4b=k(p+1) then (a,b) gives a solution iff a=k ... | 17 | https://mathoverflow.net/users/428 | 6196 | 4,217 |
https://mathoverflow.net/questions/6194 | 15 | Okay, that's a misleading title. This is a somewhat subtler problem than undergraduate linear algebra, although I suspect there's still an easy answer. But I couldn't resist :D.
Here's the actual problem: We're given a *black-box* linear transformation from $V \rightarrow W$, where $V, W$ are vector spaces of dimensi... | https://mathoverflow.net/users/382 | How to compute the rank of a matrix? | I think there would be a problem if the transformation was almost independent. If one vector were a combination of the others but otherwise there was independence. I think you would have to compute the image of basis to test for this.
If you want to have a high probability for any every black box function it will hav... | 8 | https://mathoverflow.net/users/1098 | 6197 | 4,218 |
https://mathoverflow.net/questions/6168 | 3 | Is there a closed form for $E(Y^n)$, where $Y$ is a random variable with a gamma distribution with parameters $\alpha$, $\beta$?
| https://mathoverflow.net/users/1646 | Expected value of a gamma-distributed random variable to the n-th power? | If the shape parameter is $\alpha$ and the scale parameter $\beta$, then $E(Y^r) = \beta^r \Gamma(\alpha + r)/\Gamma(\alpha)$ for real $r > 0$.
| 1 | https://mathoverflow.net/users/136 | 6199 | 4,220 |
https://mathoverflow.net/questions/6139 | 12 | There is a [wikipedia article](https://en.wikipedia.org/wiki/Trace_diagram). There is [a paper by Elisha Peterson](https://arxiv.org/abs/0910.1362). I tried reading these but they don't seem to click for me.
Are there books or other resources for learning how to do linear algebra with trace diagrams?
| https://mathoverflow.net/users/812 | How can I learn about doing linear algebra with trace diagrams? | The best resource I can point a beginner to is the first few chapters of Stedman's book "Group Theory". He focuses on the specific example of 3-vector diagrams, and does a good job of including lots of sample calculations. Unfortunately, it's not available online. I have found Cvitanovic' book fascinating but tough to ... | 13 | https://mathoverflow.net/users/498 | 6203 | 4,223 |
https://mathoverflow.net/questions/6175 | 16 | Let $A$ be an associative algebra over a commutative ring $k$. I've read statements saying that Hochschild (co)homology is the "right" notion of (co)homology for associative algebras. When $A$ is projective over $k$, the Hochschild cohomology, say, can be written as $Ext^\*\_{A \otimes A^{op}}(A,A)$, where $A^{op}$ is ... | https://mathoverflow.net/users/1800 | Cohomology of associative algebras | For good algebras, Hochschild cohomology computes ‘all’ other interesting cohomologies. For example, it is already in Cartan-Eilenberg that if $M$ and $N$ are left $A$-modules, then $\mathrm{Ext}\_A^\bullet(M,N)=H^\bullet(A,\hom(M,N))$, where on the right $H^\bullet(A,\mathord-)$ is Hochschild cohomology with coefficie... | 14 | https://mathoverflow.net/users/1409 | 6212 | 4,231 |
https://mathoverflow.net/questions/6200 | 83 | I would like to know what quantization is, I mean I would like to have some elementary examples, some soft nontechnical definition, some explanation about what do mathematicians quantize?, can we quantize a function?, a set?, a theorem?, a definition?, a theory?
| https://mathoverflow.net/users/1651 | What is Quantization ? | As I'm sure you'll see from the many answers you'll get, there are lots of notions of "quantization". Here's another perspective.
Recall the primary motivation of, say, algebraic geometry: a geometric space is determined by its algebra of functions. Well, actually, this isn't quite true --- a complex manifold, for ex... | 133 | https://mathoverflow.net/users/78 | 6216 | 4,235 |
https://mathoverflow.net/questions/6217 | 14 | To motivate my question, I will describe a related problem and then give a solution to it. My question will then be a variant of this problem.
N individuals sit around a table and want to compute the average of their salaries. They wish to do this in a manner such that no private information is leaked. This is to say... | https://mathoverflow.net/users/630 | Computing the maximum salary | These questions (and many others) are studied in the literature under the heading of secret-sharing or common-knowledge protocols. A nice but short review appears in chapter 4 of David Gale's "Tracking the Automatic Ant".
The "sum protocol" you presented can be modified to determine how many people have salary x (wit... | 12 | https://mathoverflow.net/users/25 | 6219 | 4,236 |
https://mathoverflow.net/questions/6222 | 15 | What can one say about the set of continued fractions $[0;a\_1,a\_2,\ldots]$, where $a\_1,a\_2,\ldots$ are a *permutation* of the set of natural numbers?
| https://mathoverflow.net/users/824 | Continued fractions using all natural integers | Yeah, it has measure 0 by what Qiaochu said, although I strongly suspect there's probably a more elementary way to prove this (or at least give a strong heuristic argument.)
It's also uncountable, which means that in particular: 1. There exist numbers of this type that are transcendental, 2. There exist numbers of th... | 6 | https://mathoverflow.net/users/382 | 6228 | 4,242 |
https://mathoverflow.net/questions/6248 | 3 | I've been attending a series of lectures on Cryptography from an engineering perspective, which means that most of the assertions made are supplied without proof... here's one that the lecturer couldn't recall the reason for, nor original source of.
Given an unfactored $n=pq$, computing $\phi(n)$ is as hard as findin... | https://mathoverflow.net/users/987 | Recovering $\Phi(n)$ from a multiple? | While I cannot immediately see an easy way to find Φ(n) from (t-u)Φ(n), assuming t-u is also of "cryptographic size" of course, an attacker will probably not have to.
Depending on how RSA-like the cryptosystem is, knowing a multiple of Φ(n) might well be enough to decrypt. After all, given the public exponent e, the ... | 13 | https://mathoverflow.net/users/1972 | 6254 | 4,260 |
https://mathoverflow.net/questions/6227 | 15 | I hope I'm using the terminology correctly. What I mean is this: fix $K = \mathbb{R}$ or $\mathbb{C}$ (I'm interested in both cases). Which topological spaces $X$ have the property that for every open set $U$, every continuous function $f : U \to K$ is a quotient of continuous functions $\frac{g}{h}$ where $g, h : X \t... | https://mathoverflow.net/users/290 | Which topological spaces have the property that their sheaves of continuous functions are determined by their global sections? | This isn't a complete answer, but I think that whatever the family is, it contains compact metric (metrisable) spaces. With a paracompactness argument, I suspect that it would extend to locally compact, and I would not be surprised if one could replace "metrisable" by something weaker (though I think that it would need... | 6 | https://mathoverflow.net/users/45 | 6257 | 4,263 |
https://mathoverflow.net/questions/6250 | 26 | Are there any good examples of theorems in reasonably expressive theories (like Peano arithmetic) for which it is substantially easier to prove (in a metatheory) that a proof exists than it is actually to find the proof? When I say "substantially easier," the tediousness of formalizing an informal proof should not be c... | https://mathoverflow.net/users/302 | Are there any good nonconstructive "existential metatheorems"? | Set theory provides a good example. It is often convenient in set theory to work with the concept of "classes" and treat them as mathematical objects of their own kind. The standard axiomatization of set theory with classes is called Goedel-Bernays set theory, denoted GBC, whereas the usual ZFC axioms have only set obj... | 26 | https://mathoverflow.net/users/1946 | 6268 | 4,272 |
https://mathoverflow.net/questions/6278 | 12 | Suppose $\cal{X}$ is a DM-stack, and *X* its coarse moduli space. Let *F* be a sheaf on $\cal{X}$, and $\pi : \mathcal{X} \to X$ the projection. In all examples I have seen, it has been true that
$H^i(\mathcal{X},F) = H^i(X,\pi\_\ast F)$.
Is there a simple example where this fails? Are there easy conditions where t... | https://mathoverflow.net/users/1310 | When can cohomology be calculated on the coarse moduli space? | Let k be a field and your DM-smack be [Spec(k)//G] for a trivial action of a group G. A sheaf on this stack is roughly a sheaf with a group action, and cohomology is group cohomology. If you consider a sheaf where multiplication by |G| is invertible, then group cohomology vanishes in high degrees, but otherwise there i... | 11 | https://mathoverflow.net/users/360 | 6286 | 4,284 |
https://mathoverflow.net/questions/6287 | 1 | Let $A$ be a ring and $E$ a module. If $\mathrm{spec} A$ is connected, then so is $\mathrm{spec} S^\bullet E$. If this is not true in general, then what are some minimal conditions that make it true?
| https://mathoverflow.net/users/21 | Is the total space of a module connected? | If I'm not mistaken the geometric fibers are nonempty affine spaces (hence connected, even if E is the zero module), and the augmentation gives you a zero section everywhere. I think that means the answer is yes.
| 2 | https://mathoverflow.net/users/121 | 6291 | 4,288 |
https://mathoverflow.net/questions/6279 | 10 | The graduate students here at MIT have been thinking about questions like the following: Over $\mathbb{F}\\_q$, how many symmetric matrices are there with nonzero determinant and $0$'s on the diagonal? They are doing brute force computer searches; checking every point in $\mathbb{A}^N(\mathbb{F}\\_q)$ and seeing whethe... | https://mathoverflow.net/users/297 | Counting points on varieties of low codimension | For a general variety and for a fixed small value of $q$, there isn't going to be a very good algorithm. That is because you can encode a Boolean formula in a single polynomial equation with minimal overhead. You are therefore counting solutions to a general logical expression, which is not only #P-hard, but also moral... | 9 | https://mathoverflow.net/users/1450 | 6295 | 4,291 |
https://mathoverflow.net/questions/6292 | 20 | Seriously. As an undergrad my thesis was on elliptic curves and modular forms, and I've done applied industrial research that invoked toric varieties, so it's not like I'm a partisan here. But this can't be a representative cross-section of mathematical questions. How can this be fixed? (I mean the word "fixed".)
| https://mathoverflow.net/users/1847 | Why is algebraic geometry so over-represented on this site? | I agree that the founder effect is a significant factor, but I also have another (more crackpot-ish) theory. I think some disciplines, like algebraic geometry, are harder to pick up in a traditional classroom setting, or out of a book. In practice, they are more often learned like a language, through repeated exposure ... | 17 | https://mathoverflow.net/users/750 | 6297 | 4,293 |
https://mathoverflow.net/questions/6281 | 50 | When I think of a "simplicial complex", I think of the geometric realization of a simplicial set (a simplicial object in the category of sets). I'll refer to this as "the first definition".
However, there is another definition of "simplicial complex", e.g. [the one on wikipedia](http://en.wikipedia.org/wiki/Simplicia... | https://mathoverflow.net/users/83 | Definition of "simplicial complex" | Simplicial sets and simplicial complexes lie at two ends of a spectrum, with Delta complexes, which were invented by Eilenberg and Zilber under the name "semi-simplicial complexes", lying somewhere in between. Simplicial sets are much more general than simplicial complexes and have the great advantage of allowing quoti... | 75 | https://mathoverflow.net/users/23571 | 6302 | 4,298 |
https://mathoverflow.net/questions/6303 | 8 | I am currently following a course on differential equations and difference equations (recurrence relations).
The teacher tries to make parallels between the two concepts, because the methods for solving both of these kinds of equations are essentially the same(sub-question : is there a deeper reason of why this is so?)... | https://mathoverflow.net/users/1619 | What is the term analogous to "Wronskian" for difference equations? | You mean "Casoratian"?
| 14 | https://mathoverflow.net/users/441 | 6304 | 4,299 |
https://mathoverflow.net/questions/4661 | 16 | [The Sylvester-Gallai theorem](http://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theorem) asserts that for every collection of points in the plane, not all on a line, there is a line containing **exactly** two of the points.
One high dimensional extension asserts that for every collection of points not all on a... | https://mathoverflow.net/users/1532 | The Sylvester Gallai Theorem and Sections of Varieties with "Simple Topology". | There are many cases of the question as stated that follow quickly from the standard Sylvester-Gallai theorem. If $V$ is an $r$-dimensional variety, then its intersection with a generic $(n-r)$-plane is a finite set of points. You can then apply the standard Sylvester-Gallai theorem, or the high-dimensional generalizat... | 7 | https://mathoverflow.net/users/1450 | 6310 | 4,304 |
https://mathoverflow.net/questions/6332 | 6 | As is very well known, the algebraic variety $S^2$ is isomorphic to projective variety $\mathbb{CP}^1$ as a complex manifold. As is also well known, the coordinate ring of $S^2$ is given by $< x,y,z > / < x^2 + y^2 +z^2 - 1 >$ and the function field of $\mathbb{CP}^1$ is $\mathbb{C}(\mathbb{CP}^1)$ (its coordinate ring... | https://mathoverflow.net/users/1977 | The 2-sphere and $\mathbb{CP}^1$ | Any projective variety is also a real affine variety, by using the real and imaginary parts of the coordinates $x\_{jk} = z\_j\overline{z\_k}$. You should first normalize the projective coordinates to have Hermitian-Euclidean length 1. Ordinarily the projective coordinates are sections of a line bundle that is only def... | 10 | https://mathoverflow.net/users/1450 | 6335 | 4,319 |
https://mathoverflow.net/questions/6316 | 24 | Some background on GIT
----------------------
Suppose G is a reductive group acting on a scheme X. We often want to understand the quotient X/G. For example, X might be some parameter space (like the space of possible coefficients of some polynomials which cut out things you're interested in), and the action of G on ... | https://mathoverflow.net/users/1 | When are GIT quotients projective? | I'm not sure if this is the sort of thing you are after but one can say the following.
Suppose we work over a base field $k$. If $X$ is proper over $k$ and the $G$-linearized invertible sheaf $L$ is ample on $X$ then the uniform categorical quotient of $X^{ss}(L)$ by $G$ is projective and so it gives a natural compac... | 6 | https://mathoverflow.net/users/310 | 6336 | 4,320 |
https://mathoverflow.net/questions/6343 | 16 | Let $\{A\_i\}$ be a collection of $m$ hyperplanes in $\mathbb{C}^n$ which all pass through the origin (a **central hyperplane arrangement**). Such an arrangement is called **Coxeter** if reflecting across any hyperplane in $\{A\_i\}$ sends the arrangement to itself (and so the reflections automatically will generate a ... | https://mathoverflow.net/users/750 | Coxeter Arrangements and an Identity | **Proof that Coxeter arrangements obey this identity:** Group together summands according to the two-plane spanned by $n\_i$ and $n\_j$. For any two-plane $H$, every summand coming from that two-plane is divisible by $\prod\_{n\_k \not \in H} \langle n\_k, v \rangle$. Factoring out this common summand, the contribution... | 15 | https://mathoverflow.net/users/297 | 6344 | 4,326 |
https://mathoverflow.net/questions/6179 | 15 | More precisely, does there exist an unbounded sequence $a\_0, a\_1, ... \in \mathbb{N}$ of primes such that the function
$\displaystyle O(z) = \sum\_{n \ge 0} a\_n z^n$
is meromorphic on $\mathbb{C}$?
[A previous version of the question also asked about the exponential generating function of $(a\_n)$. However, s... | https://mathoverflow.net/users/290 | Does there exist a meromorphic function all of whose Taylor coefficients are prime? | Borel proved the following much stronger result: if a power series with integer coefficients
represents a function f(z) that is meromorphic in a disk of radius >1, then f(z) extends to a rational function on all of C. I found this result without a reference on page 3 of www.mathematik.uni-bielefeld.de/~anugadre/Adeles.... | 19 | https://mathoverflow.net/users/2807 | 6347 | 4,327 |
https://mathoverflow.net/questions/6346 | 1 | Notation question:
What does $(\Delta^1)^{ \{1, \ldots,n-1 \}}$ denote? **UPDATE**: I (David Speyer) tried to fix the LaTeX. Please see if I got it right.
Vocabulary question:
Suppose $z:\Delta^{n+1} \rightarrow S$ is a morphism of simplicial sets. What does the following translate to in algebraic terms: $z|\Delt... | https://mathoverflow.net/users/1353 | Simplicial set notation and vocabulary question. | Okay, I've found the relevant notation in Higher Topos Theory. The first is at the end of 1.1.5.10, and is the simplicial set of maps from an n-1 element set to the 1-simplex (i.e., an n-1-cube). The second is at the end of warning 1.2.2.2, and describes a constant map of simplicial sets whose image is x. The curly bra... | 5 | https://mathoverflow.net/users/121 | 6349 | 4,329 |
https://mathoverflow.net/questions/4961 | 3 | I am looking for some simple and concrete -- but still non-trivial and illustrative -- applications of Atiyah-Bott localization in the context of equivariant cohomology.
Do you know any good ones?
| https://mathoverflow.net/users/83 | Simple applications of Atiyah-Bott localization | You want to read [A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications](http://www.jstor.org/pss/1970721), Atiyah and Bott, Ann. of Math., Vol. 88, No. 3 (Nov., 1968), pp. 451-491
| 2 | https://mathoverflow.net/users/297 | 6372 | 4,346 |
https://mathoverflow.net/questions/6373 | 12 | First a little background for those unaware. The Stasheff polytopes (or associahedra) are certain convex polytopes that arise in the theory of $A\_\infty$-algebras. There is one polytope for each $n\geq 2$ and is denoted by $K\_n$. The $K\_n$'s essentially encode the homotopies, higher homotopies and so on of the assoc... | https://mathoverflow.net/users/1993 | Combinatorics of the Stasheff polytopes | After a change of variables, the answer is [sequence A033282](http://www.research.att.com/~njas/sequences/A033282) in the Encyclopedia of Integer Sequences: $T(n,k)$ is the number of diagonal dissections of a convex $n$-gon into $k+1$ regions. The page gives the wonderful formula,
$$T(n,k) = \frac{1}{k+1}\binom{n-3}{k}... | 18 | https://mathoverflow.net/users/1450 | 6374 | 4,347 |
https://mathoverflow.net/questions/6376 | 50 | for forgetful functors, we can usually find their left adjoint as some "free objects", e.g. the forgetful functor: AbGp -> Set, its left adjoint sends a set to the "free ab. gp gen. by it". This happens even in some non-trivial cases. So my question is, why these happen? i.e. why that a functor forgets some structure (... | https://mathoverflow.net/users/1657 | Why forgetful functors usually have LEFT adjoint? | Forgetful functors usually have a left adjoint because they usually preserve limits. For example, the underlying set of the direct product of two groups is the direct product of the underlying sets, and similarly for equalizers (that gives you all finite limits).
However, functors that preserve limits don't have to ... | 44 | https://mathoverflow.net/users/342 | 6378 | 4,349 |
https://mathoverflow.net/questions/6379 | 154 | What is an **integrable system**, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" and "chaotic"? (There is an interesting [Wikipedia article](https://en.wikipedia.org/wiki/Integrable_system), but I do... | https://mathoverflow.net/users/1532 | What is an integrable system? | This is, of course, a very good question. I should preface with the disclaimer that despite having worked on some aspects of integrability, I do not consider myself an expert. However I have thought about this question on and (mostly) off.
I will restrict myself to integrability in classical (i.e., hamiltonian) mecha... | 70 | https://mathoverflow.net/users/394 | 6382 | 4,352 |
https://mathoverflow.net/questions/6377 | 42 | I'm studying Steenrod operations from Hatcher's book. Like homology, one can use them only knowing the axioms, without caring for the actual construction. But while there are plenty of intuitive reasons to introduce homology, I cannot find any for the Steenrod operations. I can follow the steps in the proofs given by H... | https://mathoverflow.net/users/828 | Why does one think to Steenrod squares and powers? | Steenrod operations are an example of what's known as a *power operation*. Power operations result from the fact that cup product is "commutative, but not too commutative". The operations come from a "refinement" of the operation of taking $p$th powers (squares if $p=2$), whose construction rests on this funny version ... | 58 | https://mathoverflow.net/users/437 | 6384 | 4,354 |
https://mathoverflow.net/questions/5108 | 3 | Is this possible to do constructively? The only sources that I have for the possibility of this construction is an exercise in Lang's Algebra (on p. 598, I believe) which states that one can be constructed, and then a construction given in Milnor and Hussemoller's book which only applies in the case that the elementary... | https://mathoverflow.net/users/1703 | How do you construct a symplectic basis on a lattice? | Here is an algorithm, it may not be a good one. I will only explain how to find a basis $e\_i$, $f\_i$ such that $\langle e\_i, e\_j \rangle = \langle f\_i, f\_j \rangle =0$ and $\langle e\_i, f\_j \rangle = c\_i \delta\_{ij}$ for some constants $c\_i$. I will punt on explaining how to make sure that $c\_1$ divides $c\... | 9 | https://mathoverflow.net/users/297 | 6391 | 4,359 |
https://mathoverflow.net/questions/6388 | 8 | If we ask which natural numbers n are not expressible as $n = ab + bc + ca$ ($0 < a < b < c$) then this is a well known open problem. Numbers not expressible in such form are called Euler's "numerus idoneus" and it is conjectured that they are finite.
If we omit the condition $a < b < c$ and assume $0 < a \leq b \leq... | https://mathoverflow.net/users/1737 | Prime numbers $p$ not of the form $ab + bc + ac$ $(0 < a < b < c )$ (and related questions) | Looking at the encyclopedia of integer sequnces, I've found that the set of numbers not expressible as ab + bc + ac 0 < a < b < c is finite.
<http://oeis.org/A000926>
Chowla showed that the list is finite and Weinberger showed that there is at most one further term.
| 10 | https://mathoverflow.net/users/1737 | 6408 | 4,373 |
https://mathoverflow.net/questions/6394 | 31 | Next term I am supposed to teach a course on representation of finite groups. This is a third year course for undegrads. I was thinking to use the book of Grodon James and Martin Liebeck "Representations and characters of groups", but also looking for other references.
The question is: could you advise some other bo... | https://mathoverflow.net/users/943 | Lecture notes on representations of finite groups | Some material from the undergrad rep theory course in Cambridge: [Example sheets](https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/), A [recent set of notes](http://tartarus.org/gareth/maths/notes/ii/Repn_Theory.pdf) (by Martin), and a [less recent (but very nice) set of notes](http://math.berkeley.edu/~telema... | 15 | https://mathoverflow.net/users/349 | 6409 | 4,374 |
https://mathoverflow.net/questions/6071 | 13 | In Connes work on the Riemann Hypothesis he talks about constructing $\overline{\text{Spec}\mathbb{Z}}$ as a curve over the field with one element. I just want to know what Spec means. Is the same as spectrum from algebraic geometry?
| https://mathoverflow.net/users/1867 | What is $\overline{\text{Spec}\mathbb{Z}}$? | If you remove the overline, you have the affine scheme Spec Z. It is the spectrum of a noetherian domain of Krull dimension one. This description also holds for any affine algebraic curve over a field, so we have the basis for an analogy. Z has many structural features in common with the ring of polynomials with coeffi... | 18 | https://mathoverflow.net/users/121 | 6417 | 4,380 |
https://mathoverflow.net/questions/6418 | 12 | What is the fastest algorithm that exists to solve a particular NP-Complete problem? For example, a naive implementation of [travelling salesman](http://en.wikipedia.org/wiki/Travelling_salesman_problem) is $O(n!)$, but with dynamic programming it can be done in $O(n^2 2^n)$. Is there any "easier" NP-Complete problem t... | https://mathoverflow.net/users/1646 | Best-case Running-time to solve an NP-Complete problem | If P is an NP-complete problem, then define Pk = instances of P in which the instances have been blown up from size n to size nk by padding them with blanks. Then Pk is also NP-complete, but if P takes time exp(p(n)) to solve where p is some polynomial then Pk can be solved in time essentially exp(p(n1/k)) (there's a l... | 25 | https://mathoverflow.net/users/440 | 6420 | 4,381 |
https://mathoverflow.net/questions/6438 | 7 | Prove or counterexample: If A is a commutative ring and $A\_p$ is a finitely generated algebra over A for all prime ideal p of A, then A is a product of local rings.
| https://mathoverflow.net/users/2008 | When is a localization of a commutative ring finitely generated as an algebra? | Here is a counterexample. Let p and q be distinct prime numbers, let S denote the complement of {p,q} in the set of all primes, and let A denote Z[1/S]. Then the prime ideals of A are pA, qA, and {0}, the first two of which are maximal. Also, since A is domain (being a subring of Q) with two maximal ideals, it's not a ... | 11 | https://mathoverflow.net/users/1114 | 6451 | 4,393 |
https://mathoverflow.net/questions/6450 | 1 | If $\sum\_{n=1}^\infty \frac{a\_n}{n^s} $ converges, does
$\sum\_{n=1}^\infty \frac{a\_n}{(n+1)^s} $ also converge?
| https://mathoverflow.net/users/2003 | Shifted Dirichlet series | Yes, because $(n+1)^{-s} = n^{-s} + sn^{-s-1} + O(|s|^2n^{-\sigma - 2})$. The first series necessarily converges in the open half-plane strictly to the right of s, and converges absolutely in the half-plane strictly to the right of s + 1. I hope I am not doing homework from a course in analytic number theory here.
| 4 | https://mathoverflow.net/users/3304 | 6454 | 4,395 |
https://mathoverflow.net/questions/6457 | 15 | I just wonder about Kapranov's ["Analogies between Langlands Correspondence and topological QFT"](http://books.google.de/books?id=TOPa9irmsGsC&pg=PA119&lpg=PA119&dq=Kapranov+%22%22Analogies+between+Langlands+Correspondence+and+topological+quantum+field+theory%22%22&source=bl&ots=kGZ-Wnk_Fu&sig=G-E6CK1Rza5dfp2niaBOmIdoN... | https://mathoverflow.net/users/451 | Kapranov's analogies | There are notes from more recent (2000) lectures of Kapranov on the subject on [my webpage](http://www.math.utexas.edu/users/benzvi/GRASP/lectures/Langlands.html).
Despite this, as far as I know, there isn't any convincing argument at this point that there should be a general higher dimensional Langlands theory. What... | 26 | https://mathoverflow.net/users/582 | 6466 | 4,404 |
https://mathoverflow.net/questions/6427 | 4 | $$F\\,(s) = 1/1^s + 0/2^s + 1/3^s + 1/4^s + 0/5^s + 1/6^s + 0/7^s + 1/8^s + ... $$
The coefficients of this Dirichlet series are the base-2 digits of $\sqrt2$.
Does it have an analytic continuation into the critical strip?
Assuming so, can anything be said about the locations of its poles in this region? Are ther... | https://mathoverflow.net/users/2003 | Dirichlet series whose coefficients are the bits of sqrt(2) | David Speyer is right. A Dirichlet series with coefficients that are random choices from
$0,1$ does not count as a random Dirichlet series. Random choices from $-1,1$ would count as a random Dirichlet series. Taking his insight into account, I correct my prediction: With probability $1$ the Dirichlet series $\sum b\_n... | 5 | https://mathoverflow.net/users/3304 | 6467 | 4,405 |
https://mathoverflow.net/questions/6325 | 12 | So, let's say we have a symplectic variety over $\mathbb{C}$, $M$, of dimension $2n$, and $f\_1,\ldots,f\_n$ Poisson commuting functions with $df\_1\wedge\ldots\wedge df\_n$ generically nonzero. Further assume that the fibers of the map $f:M\to\mathbb{C}^n$ determined by the $f\_i$ is an open subset of an abelian varie... | https://mathoverflow.net/users/622 | Equations for Integrable Systems | I'm no expert on this, but since nobody answers:
one book which should definitely help is [this big introductory one by Babelon, Bernard and Talon](http://books.google.fr/books?id=c6JN6Gp4RBQC&pg=PP1&dq=INTRODUCTION+TO+CLASSICAL+INTEGRABLE+SYSTEMS#v=onepage&q=&f=false). There's also a very technical older [paper by ... | 5 | https://mathoverflow.net/users/469 | 6469 | 4,407 |
https://mathoverflow.net/questions/6455 | 17 | Is there a game-theoretic interpretation of nimber multiplication? There is such for addition (a single move in a+b is either a move in a or a move in b).
| https://mathoverflow.net/users/2014 | Nimber multiplication | As Alex says, there's no good one. One way to see the problem is to compare nimbers with (surreal) numbers. Addition in both is just game addition, so you can consider nimbers and numbers together and add them consistently. But there is no consistent multiplication. (What would the unit be?) A good game theoretic multi... | 12 | https://mathoverflow.net/users/78 | 6470 | 4,408 |
https://mathoverflow.net/questions/6447 | 10 | If we have the complete countably infinite bipartite graph $K\_{\omega,\omega}$ and we colour the edges with just two colours. Should we expect to get a monochromatic copy of $K\_{\omega,\omega}$.
Infinite Ramsey theorem gives us infinitely many edges of one colour but this is no enough.
| https://mathoverflow.net/users/2011 | Ramsey Theory, monochromatic subgraphs | The example given by Konstantin Slutsky is, however, essentially unique, in the following sense. Let $G$ be the complete bipartite graph with both vertex sets equal to (copies of) $\mathbb{N}$ and colour its edges red or blue. For each $m< n$, colour the set $\{m,n\}$ according to whether the edge $(m,n)$ is red or blu... | 15 | https://mathoverflow.net/users/1459 | 6472 | 4,410 |
https://mathoverflow.net/questions/6276 | 23 | I'd be grateful for a good reference on this, it feels like a classic subject yet I couldn't find much about it.
Polynomials in one variable of the form $x^n+a\_{n-1}x^{n-1}+\dots +a\_1 x+a\_0$ can be transformed into simpler expressions. For instance [it is apparently well-known](http://homepage.mac.com/ehgoins/ma5... | https://mathoverflow.net/users/469 | Least number of non-zero coefficients to describe a degree n polynomial | You might have a look at [Polynomial Transformations of Tschirnhaus, Bring and Jerrard](http://www.apmaths.uwo.ca/~djeffrey/Offprints/Adamchik.pdf) ([Internet Archive](http://web.archive.org/web/20180816061604/http://www.apmaths.uwo.ca/~djeffrey/Offprints/Adamchik.pdf)). It gives more explicit detail on why you can rem... | 16 | https://mathoverflow.net/users/3 | 6474 | 4,412 |
https://mathoverflow.net/questions/6144 | 53 | The Chern character is often seen as just being a convenient way to get a ring homomorphism from K-theory to (ordinary) cohomology.
The most usual definition in that case seems to just be to define the Chern character on a line bundle as $\mathrm{ch}(L) = \exp(c\_1(L))$ and then extend this; then for example $\mathrm... | https://mathoverflow.net/users/362 | Explanation for the Chern character | There is a nice discussion about multiplicative sequences, &c., in Lawson and Michelsohn's book "Spin Geometry". It discusses things like the Todd genus, the A-hat genus, and so on, but also the Chern character and the ring homomorphism from K-theory to ordinary cohomology. It is a readable exposition and perhaps "conn... | 12 | https://mathoverflow.net/users/1938 | 6479 | 4,414 |
https://mathoverflow.net/questions/6481 | 13 | This is a follow-up to the following answer:
[Solvable class field theory](https://mathoverflow.net/questions/4379/solvable-class-field-theory/4386#4386)
in which it is stated as a "folklore" conjecture that the maximal solvable extension of Q is pseudo algebraically closed (this means, in particular, any geometric... | https://mathoverflow.net/users/81 | Evidence for $Q^{\operatorname{solv}}$ being pseudo-algebraically-closed | So far as I know, there is no compelling evidence to support this conjecture. (And some leading arithmetic geometers think it is false.) Rather, there are some very interesting consequences of this conjecture, e.g. a solution of the Inverse Galois Problem over Q^{solv}: in other words, for any finite group G, there exi... | 8 | https://mathoverflow.net/users/1149 | 6486 | 4,417 |
https://mathoverflow.net/questions/6476 | 4 | What is the relationship between the Hausdorff dimension and cardinality of a set?
Specifically, assuming the Continuum Hypothesis, if a set has Hausdorff dimension greater than zero does, that imply that its cardinality is equal too or greater than that of $2^{\aleph\_0}$?
Or, does the negation of CH, imply the ex... | https://mathoverflow.net/users/1320 | Hausdorff dimension vs. cardinality | As stated, countable sets have Hausdorff dimension 0.
So any set $S$ with $\mathrm{HD}(S)>0$ has power $\ge \aleph\_1$.
No need for continuum hypothesis.
Without CH, though, we cannot say whether power $ \ge c = 2^{\aleph\_0}$ is required. But this is not about Hausdorff dimension, it is the same question for positiv... | 6 | https://mathoverflow.net/users/454 | 6501 | 4,428 |
https://mathoverflow.net/questions/6493 | 4 | Hi,
this is kind of continuation of [this thread](https://mathoverflow.net/questions/6429/asymptotically-multiplicative-functions-and-matrices) to concentrate on a specific problem from linear algebra and analysis that, I think, is rather interesting for itself.
Here we go:
1) Main problem: Let $(c\_n)\_{n\in\math... | https://mathoverflow.net/users/1849 | factorization of the product of a matrix element and its cofactor | For the main problem, we can use expansion by minors along the ith row to compute
$\det(A\_n) = (-1)^{i+1}(a(n)\_{i1}\widetilde{a(n)\_{i1}} - a(n)\_{i2}\widetilde{a(n)\_{i2}} + a(n)\_{i3}\widetilde{a(n)\_{i3}} - \dots)$
or det(An) = (-1)i+1ci(c1 - c2 + c3 - ...).
This is true for any i <= n, so whenever c1 - c2 +... | 3 | https://mathoverflow.net/users/428 | 6502 | 4,429 |
https://mathoverflow.net/questions/6440 | 7 | Suppose we fix a universe $U$ and a $U$-small category $C$. The regular Yoneda lemma gives us some locally small (not necessarily locally U-small?) functor category $C'=[C^{op},Sets]$ with a fully faithful embedding $C\rightarrow C'$ and the canonical bijection between $Nat(F,Hom(-,x))$ and $F(X)$. Suppose we consider ... | https://mathoverflow.net/users/1353 | Sets, Universes, and the small Yoneda Lemma | Well, for any universe U, the U-small sets satisfy the axioms of ZFC (or whatever your preferred set theory is). Therefore, anything that we can prove in ZFC about the Yoneda embedding into "all" presheaves will also be true about the Yoneda embedding into U-small presheaves. So on that score, the answer would seem to ... | 4 | https://mathoverflow.net/users/49 | 6505 | 4,432 |
https://mathoverflow.net/questions/5678 | 7 | Let $\ell$ be a positive integer greater than 1. The problem is to find a set of $n$ real positive numbers $x\_i$ and $n+1$ numbers $y\_i$ such that
$$\sum\_{i=1}^n x\_i^k= \sum\_{i=1}^{n+1} y\_i^k$$
for $k=\ell,\cdots,2\ell-1$. These $2n+1$ numbers need to be upper/lower bounded by a constant independent of $\ell$ [th... | https://mathoverflow.net/users/1837 | Simultaneous Equations Involving Power Sums | Actually Darsh gave an almost full solution. Let me fill in the minor technical details.
1) We need the following quantitative form of the inverse function theorem. Suppose that $F:\mathbb R^n\to \mathbb R^n$. Assume also that $\|DF(X)^{-1}\|\le C\_1$, that $\max\_{Y\in B(X, \delta)}\|D^2F(Y)\|\le C\_2$, and that $C\... | 11 | https://mathoverflow.net/users/1131 | 6509 | 4,435 |
https://mathoverflow.net/questions/6498 | 3 | One way to define the free abelian group on a set $S$ is as $F(S) := \mbox{Hom}\_{\text{Set}}(S, \mathbb{Z})$ with the group operation coming by composition with the map $\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$. The problem with this definition is that it's contravariant, whereas it seems to me that $F$ should rea... | https://mathoverflow.net/users/290 | What is the "right" definition of the free abelian group on a set? | As I understand things, the desire to find a more categorical *construction* of the free abelian group on a set S is not itself fully in the categorical spirit. Rather, for an object which is defined by a universal mapping property, you only need to convince yourself that it exists; you don't need to be bothered by loo... | 11 | https://mathoverflow.net/users/1149 | 6510 | 4,436 |
https://mathoverflow.net/questions/6480 | 8 | Are there any good books out there that can serve as an introduction to thermodynamical formalism in dynamical systems?
I know only Zinsmeister's short "Thermodynamical formalism and holomorphic dynamical systems", which is concerned mostly with holomorphic dynamics, and Ruelle's "Thermodynamical formalism", which i... | https://mathoverflow.net/users/1121 | Suggested reading for thermodynamic formalism | If you are interested in learning mathematical statistical physics (e.g. you would like to read Ruelle) then Minlos "Introduction to mathematical statistical physics" is a good choice for introduction to the area.
If you would like to learn the part of ergodic theory which is called "thermodanamic formalism" and you do... | 5 | https://mathoverflow.net/users/2029 | 6528 | 4,449 |
https://mathoverflow.net/questions/6535 | 2 | I recently was wondering if there was a name for sheaves which were locally constant on the open simplexes in a simplicial complex. After some googling I stumbled across simplicial sheaves. I am alright with the definition as the presheaves to simplicial sets, but now I wonder, is this the answer to my question?
Are ... | https://mathoverflow.net/users/348 | Simplicial Sheaves? | If I understand correctly, these are constructible sheaves with respect to the stratification of your simplicial complex by its skeleta. I think by a theorem of MacPherson the category of such sheaves is equivalent to the category of functors from the poset of faces and face inclusions to whatever category your sheaves... | 6 | https://mathoverflow.net/users/126667 | 6537 | 4,455 |
https://mathoverflow.net/questions/6475 | 25 | The following things are all called holonomic or holonomy:
1. A holonomic constraint on a physical system is one where the constraint gives a relationship between coordinates that doesn't involve the velocities at all. (ie, $r=\ell$ for the simple pendulum of length $\ell$)
2. A D-module is holonomic (I might not hav... | https://mathoverflow.net/users/622 | What is the relationship between various things called holonomic? | Not a complete answer, but you might wish to look at Historical Remarks at the beginning of Bryant's [Recent advances in the theory of holonomy](http://arxiv.org/pdf/math/9910059v2), which seems to offer a relationship between (1) and (4).
| 5 | https://mathoverflow.net/users/394 | 6550 | 4,462 |
https://mathoverflow.net/questions/6554 | 28 | A lot of the terminology of category theory has obvious antecedents in analysis: limits, completeness, adjunctions, continuous (functors), to name but a few. However, analysis and category theory *seem* to be at opposite poles of the spectrum.
Is there anything deep here, or is it a case of "it has wings, so let's ca... | https://mathoverflow.net/users/45 | Terminology in category theory | Names in category theory are often born when someone realizes that a concept in one particular topic can be generalized in a categorical way. The generally-defined concept is then named after the original narrowly-defined one.
The case of metric spaces provides a slightly notorious example. As discussed in that [othe... | 23 | https://mathoverflow.net/users/586 | 6564 | 4,470 |
https://mathoverflow.net/questions/6552 | 40 | After realising that I don't have an intuitive understanding of adjoint **functors**, I then realised that I don't have an intuitive understanding of adjoint **linear transformations**!
Again, I can use 'em, compute 'em, and convolute 'em, but I have no real intuition as to what is going on. My best attempts (when te... | https://mathoverflow.net/users/45 | What is an intuitive view of adjoints? (version 2: functional analysis) | Just to add to Yemon's answer, in the case of inner product spaces I think it may be helpful to understand the case of operators of finite rank. So let's start with the shockingly simple case where $\eta$ is a single vector in the inner product space $H$. I like to identify this with the linear map $t\mapsto t\eta$ fro... | 22 | https://mathoverflow.net/users/802 | 6573 | 4,478 |
https://mathoverflow.net/questions/6517 | 60 | This is something of a followup to the question ["Kapranov's analogies"](https://mathoverflow.net/questions/6457/kapranovs-analogies), where a connection between Cherednik's double affine Hecke algebras (DAHA's) and Geometric Langlands program was mentioned.
I am interested to hear about known connections of the doub... | https://mathoverflow.net/users/1040 | Double affine Hecke algebras and mainstream mathematics | Well the first thing to say is to look at the very enthusiastic and world-encompassing papers of Cherednik himself on DAHA as the center of the mathematical world (say his 1998 ICM).
I'll mention a couple of more geometric aspects, but this is a huuuge area..
There are at least three distinct geometric appearances o... | 53 | https://mathoverflow.net/users/582 | 6575 | 4,480 |
https://mathoverflow.net/questions/6543 | 11 | Algebraic geometry allows one to think of an $A$-module $M$ geometrically as a module of functions on the $A$-scheme $\mathrm{Spec}(\mathrm{Sym}(M))$.
I'm wondering if anything is lost in just replacing $M$ by this geometric object. Since nothing is lost in taking $\mathrm{Spec}$, this amounts to asking:
>
> (1) ... | https://mathoverflow.net/users/84526 | Can different modules have the same symmetric algebra? (answered: no) | I now believe a-fortiori's argument: translations are a problem, but, as a-fortiori observed, they are the only problem. Let me spell it out.
Say $f:Sym(M)\to Sym(N)$ is an isomorphism. For $m\in M$ write $f(m)=f\_0(m)+f\_1(m)+f\_{\geq2}(m)$ with obvious notation: $f\_0(m)$ is in $A$, $f\_1(m)$ is in $N$ and $f\_{\ge... | 6 | https://mathoverflow.net/users/1384 | 6582 | 4,483 |
https://mathoverflow.net/questions/6576 | 15 | Given a univariate polynomial with real coefficients, p(x), with degree n, suppose we know all the zeros xj, and they are all real. Now suppose I perturb each of the coefficients pj (for j ≤ n) by a small real perturbation εj. What are the conditions on the perturbations (**edit:** for example, how large can they be, b... | https://mathoverflow.net/users/1171 | Finding the new zeros of a "perturbed" polynomial | Assume that the roots of $p(x)$ are real and distinct. Then we may let $\mu > 0$ denote the minimum distance between any two roots. Let $q(x)$ be any polynomial of degree $< n$ such that $|q(\alpha)| < (\mu/2)^n$ for any root $\alpha$ of $p(x)$. Then I claim that $g(x) = p(x) + q(x)$ has real roots.
Note that for a ... | 15 | https://mathoverflow.net/users/nan | 6585 | 4,485 |
https://mathoverflow.net/questions/6560 | 10 | The Brauer-Nesbitt theorem (well, one of them) says that if $k$ is a field and I have two semisimple representations (on finite-dimensional $k$-vector spaces) $r\_1$ and $r\_2$ of a $k$-algebra $A$ with the property that the char polys of $r\_1(a)$ and $r\_2(a)$ coincide for all $a\in A$, then the representations are i... | https://mathoverflow.net/users/1384 | Version of Brauer-Nesbitt for summands | The result is true, regardless of characteristic.
Lemma: Let A be a k algebra and M a semi-simple A-module which is finite dimensional as a k-algebra. Then the image of A in $\mathrm{End}\_k(M)$ is a semi-simple ring.
So, by the Artin-Wedderburn theorem, this image is a direct sum of matrix algebras over division... | 6 | https://mathoverflow.net/users/297 | 6587 | 4,487 |
https://mathoverflow.net/questions/6593 | 14 | A fairly simple question: I've read in multiple sources that Godel proved that if we accept the axiom of constructibility in ZFC, then we can create an explicit formula that well-orders the real numbers. I tried searching for a paper or some other source that explains what this formula is, but I came up empty-handed. C... | https://mathoverflow.net/users/1455 | V=L and a Well-Ordering of the Reals | The order is very easy. Under $V=L$, the set-theoretic universe is built according the hierarchy $(L\_\alpha \mid \alpha \in \mathrm{Ord})$, where $L\_0$ is empty, $L\_{\alpha+1}$ consists of all definable subsets of $L\_\alpha$, and $L\_\lambda$ is the union of all earlier $L\_\alpha$ when $\lambda$ is a limit ordinal... | 29 | https://mathoverflow.net/users/1946 | 6600 | 4,494 |
https://mathoverflow.net/questions/6592 | 8 | I came along the following question while trying to understand and apply some ideas of Dugger's article *Universal Homotopy Theories*.
Suppose, we are given a nice model category $\mathcal{C}$, say left proper and cellular or combinatorial, so we have a good theory of localization. I am primarly thinking here of the... | https://mathoverflow.net/users/2039 | Localizing Model Structures | (Your question is basically about presentable (∞,1)-categories, so I will take the liberty of writing my answer in that language. Hopefully the translations to model category language will be straightforward.)
Inside $\mathcal{C}$ we have the full subcategories of $S$-local objects, $T$-local objects, and $(S \cup T)... | 9 | https://mathoverflow.net/users/126667 | 6610 | 4,503 |
https://mathoverflow.net/questions/6608 | 10 | Let $B$ be the graded ring $\bigoplus\_i B^i$ (with $B^k B^l \subset B^{k+l}$), and $B\_f$ the multiplicative group of all formal sums $1 + b\_1 + b\_2 + \cdots$ where $b\_i \in B^i$ for all $i$.
The idea when talking about genera (such as the Todd genus or the $L$ genus) is that we can take $B$ to be $H^{2 \bullet}(... | https://mathoverflow.net/users/362 | How are multiplicative sequences related to formal power series and genera of manifolds? | I think the idea is that using the splitting principle everything reduces to the first Chern class of line bundles: Chern classes of a general bundle $E$ are symetric functions of $c\_1(L\_i)$ where $\bigoplus L\_i = E$.
If $Q(z)$ is a power series with constant term 1, you can define $K\_n$ by the formula:
$$
\sum... | 6 | https://mathoverflow.net/users/1985 | 6614 | 4,507 |
https://mathoverflow.net/questions/6618 | 56 | I was wondering whether there is some notion of "vector bundle de Rham cohomology".
To be more precise: the k-th de Rham cohomology group of a manifold $H\_{dR}^{k}(M)$ is defined as the set of closed forms in $\Omega^k(M)$ modulo the set of exact forms. The coboundary operator is given by the exterior derivative.
Le... | https://mathoverflow.net/users/675 | de Rham cohomology and flat vector bundles | Warning: The first paragraph of the following is outside my expertise.
I am told this construction is very useful in PDE's. If you have a PDE on some manifold $M$, you can often formulate the vector space of solutions as the kernel of some flat connection on a vector bundle. In particular, I believe that the analytic... | 21 | https://mathoverflow.net/users/297 | 6625 | 4,514 |
https://mathoverflow.net/questions/6541 | 2 | Does anyone know of good references for nonstandard set theories and their applications to various branches of mathematics like category theory, algebra, geometry, etc.?
Edit: What I mean by "nonstandard set theory" is a formalization of the naive notion of sets that allows direct arguments about certain intuitive no... | https://mathoverflow.net/users/nan | nonstandard set theories | It seems that Lars Brünjes and Christian Serpé have a whole program for introducing non standard mathematics in algebraic geometry; they wish to play with non standard contructions, seen internally and externally (the interest of this game consists precisely to look at the non-classical logic (or internal) point of vie... | 7 | https://mathoverflow.net/users/1017 | 6628 | 4,516 |
https://mathoverflow.net/questions/6636 | 0 | Let $S\_n$ denote the symmetric group on $n$ letters, and let $S\_n(p)$ denote a Sylow $p$-subgroup. Why is the image of $H\_i(S\_n(p))$ in $H\_i(S\_n)$ the $p$-primary part of $H\_i(S\_n)$?
| https://mathoverflow.net/users/2046 | Homology of symmetric groups | That is the content of the first part of theorem 10.1 in Cartan-Eilenberg (up to the fact that their $\hat H^i$ is the same as $H\_{-i-1}$ for $i<-1$). In particular, this is true for all finite groups, not just the symmetric ones.
| 3 | https://mathoverflow.net/users/1409 | 6638 | 4,522 |
https://mathoverflow.net/questions/6637 | 11 | I'm reading Madsen and Tornehave's "From Calculus to Cohomology" and tried to solve this interesting problem regarding knots.
Let $\Sigma\subset \mathbb{R}^n$ be homeomorphic to $\mathbb{S}^k$, show that $H^p(\mathbb{R}^n - \Sigma)$ equals $\mathbb{R}$ for $p=0,n-k-1, n-1$ and $0$ for all other $p$. Here $1\leq k \l... | https://mathoverflow.net/users/2044 | The De Rham Cohomology of $\mathbb{R}^n - \mathbb{S}^k$ | To apply the Mayer-Vietoris sequence, you need subspaces whose *interiors* cover your space (see e.g. [Wikipedia](http://en.wikipedia.org/wiki/Mayer%E2%80%93Vietoris_sequence), or [Hatcher](http://www.math.cornell.edu/~hatcher/AT/ATpage.html), p. 149). This is not true in your example, because a *k*-disk in Rn has empt... | 15 | https://mathoverflow.net/users/250 | 6639 | 4,523 |
https://mathoverflow.net/questions/6647 | 13 | Let $H\leq G$ be an inclusion of finite groups. Define a map $E\colon \mathbb{C}[G]\to \mathbb{C}[H]$ to be the $\mathbb{C}$-linear extension of
$$
E(g)=\begin{cases}
g &\text{if } g\in H\\\
0 &\text{else,}
\end{cases}
$$
i.e., $E$ is the projection onto $\mathbb{C}[H]$. A finite subset $B\subset \mathbb{C}[G]$ will be... | https://mathoverflow.net/users/351 | Do subgroups have "two sided bases"? | If $H$ is a subgroup of finite index in a group $G$, there is a subset $\mathcal B$ of $G$ which serves both as a set of representatives for the left cosets of $H$ in $G$ and as a set of representatives for the right cosets of $H$ in $G$. (See, for example, Theorem 3, §4, Chap. I, in the book *The Theory of groups* by ... | 12 | https://mathoverflow.net/users/1409 | 6652 | 4,529 |
https://mathoverflow.net/questions/314 | 6 | Is there a good monoidal structure on a category of integrable representations of a quantum affine algebra? In the ordinary affine Kac-Moody case, there is the usual tensor product (symmetric, adds charges) and a fusion structure (braided, comes from G-bundles on curves, preserves central charge). In the quantum case, ... | https://mathoverflow.net/users/121 | Are there interesting monoidal structures on representations of quantum affine algebras? | The fusion product for affine Lie algebras is closely related to the existence of "evaluation homomorphisms" from the loop algebra to the finite-dimensional semisimple Lie algebra g, which split the natural inclusion of g as the subalgebra of constant loops. In the quantum case there is no evaluation map from the quant... | 5 | https://mathoverflow.net/users/1878 | 6655 | 4,532 |
https://mathoverflow.net/questions/6651 | 13 | Gauss's Lemma on irred. polynomial says,
Let R be a UFD and F its field of fractions. If a polynomial f(x) in R[x] is reducible in F[x], then it is reducible in R[x].
In particular, an integral coefficient polynomial is irreducible in Z iff it is irreducible in Q. For me this tells me something on how the horizont... | https://mathoverflow.net/users/1657 | "Counter"-example for Gauss's Lemma on irreducible polynomials | Gauss' Lemma over a domain R is usually taken to be a stronger statement, as follows:
If R is a domain with fraction field F, a polynomial f in R[T] is said to be *primitive* if the ideal generated by its coefficients is not contained in any proper principal ideal. One says that Gauss' Lemma holds in R if the product... | 22 | https://mathoverflow.net/users/1149 | 6659 | 4,536 |
https://mathoverflow.net/questions/6635 | 9 | I am looking for a certain masa in a $II\_1$ factor which is singular and has nontrivial Takesaki invariant.
For this I am looking for an example of an inclusion of groups $H\subset G$ such that:
* $G$ is a countable icc (infinite conjugacy class) group
* $H$ is abelian
* $\forall g\in G-H,\{ hgh^{-1} |h\in H \}$ is ... | https://mathoverflow.net/users/2045 | Does such a subgroup exist? | [generalization of Agol's answer]
Take $H$ a group and let $K$ act on $H$ by isomorphisms (write the action as $\sigma$) and consider $G=H\rtimes\_\sigma K$. Then
* condition 2 is satisfied if $H$ is abelian
* condition 4 is satisfied if $K$ contains at least 3 elements
* condition 5 is satisfied if $K$ acts non-tr... | 7 | https://mathoverflow.net/users/2055 | 6682 | 4,552 |
https://mathoverflow.net/questions/6685 | 5 | To establish the Weil conjectures for $n$-dimensional projective space over a finite field is elementary. Does there exist a simple direct proof of the conjectures for finite field Grassmannians?
| https://mathoverflow.net/users/1095 | Weil Conjectures for Grassmannians | The first result on the google search "zeta function of grassmannian" seems to contain quite a direct and not too long derivation of the zeta function for a grassmannian over a finite field:
<http://www.math.mcgill.ca/goren/SeminarOnCohomology/GrassmannVarieties%20.pdf>
From the zeta you see that it is rational, of... | 10 | https://mathoverflow.net/users/2024 | 6688 | 4,557 |
https://mathoverflow.net/questions/6674 | 9 | Given a number field K of dimension d over Q, and galois closure of dimension d! over Q (i.e galois group Sd), can we relate the discriminant of the galois closure to that of the discriminant of K? Assume no special ramification happens in the closure or the other subfields, for example if the discriminant of K is a pr... | https://mathoverflow.net/users/2024 | How do I calculate the discriminant of a galois closure and its other subfields? | It's true for $d = 2$. Even for $d = 3$, it fails miserably. If $K = \mathbf{Q}(p^{1/3})$ and $p \ne 3$ then $p^2$ exactly divides $\Delta\_{K}$, whereas $p^4$ exactly divides the discriminant of the Galois closure. (As David points out, things are even worse for $p = 3$.) Not to mention the fact that $p$ doesn't divid... | 11 | https://mathoverflow.net/users/nan | 6714 | 4,574 |
https://mathoverflow.net/questions/6711 | 50 | Is there a (preferably simple) example of a function $f:(a,b)\to \mathbb{R}$ which is everywhere differentiable, such that $f'$ is not Riemann integrable?
I ask for pedagogical reasons. Results in basic real analysis relating a function and its derivative can generally be proved via the mean value theorem or the fund... | https://mathoverflow.net/users/1044 | Integrability of derivatives | I believe this answers the question:
---
MR0425042 (54 #13000)
Goffman, Casper
A bounded derivative which is not Riemann integrable.
Amer. Math. Monthly 84 (1977), no. 3, 205--206.
In 1881 Volterra constructed a bounded derivative on $[0,1]$ which is not Riemann integrable. Since that time, a number of authors ... | 48 | https://mathoverflow.net/users/1149 | 6716 | 4,576 |
https://mathoverflow.net/questions/6701 | 31 | Is there a canonical definition of the concept of inner products for vector spaces over arbitrary fields, i.e. other fields than $\mathbb R$ or $\mathbb C$?
| https://mathoverflow.net/users/2058 | Definition of inner product for vector spaces over arbitrary fields | As Qiaochu wrote, the answer to the question is "not really, but..." Let me amplify on the "but" part.
Positive-definiteness inherently requires an ordering on your field. Conversely, if you have an ordered field, then the theory of inner products goes through verbatim. (The ordering does not have to be Archimedean, ... | 28 | https://mathoverflow.net/users/1149 | 6721 | 4,579 |
https://mathoverflow.net/questions/6704 | 45 | There are a few questions about CM rings and depth.
1. Why would one consider the concept of depth? Is there any geometric meaning associated to that? The consideration of regular sequence is okay to me. (currently I'm regarding it as a generalization of not-a-zero-divisor that's needed to carry out induction argumen... | https://mathoverflow.net/users/nan | How to think about CM rings? | "Life is really worth living in a Noetherian ring $R$ when all the local rings have the property that every s.o.p. is an R-sequence. Such a ring is called Cohen–Macaulay (C–M for short).": [Hochster, "Some applications of the Frobenius in characteristic 0", 1978](https://doi.org/10.1090/S0002-9904-1978-14531-5).
Sect... | 32 | https://mathoverflow.net/users/460 | 6724 | 4,580 |
https://mathoverflow.net/questions/6719 | 21 | I know that for a finitely presented $A$-module $M$ ($A$ a commutative ring), TFAE:
1. $M$ is *projective*;
2. $M$ is *max-locally free*, meaning that $M\_{\mathfrak m}$ is free for every maximal ideal $\mathfrak m$;
3. $M$ is *prime-locally free*, meaning that $M\_{\mathfrak{p}}$ is free for every prime ideal $\math... | https://mathoverflow.net/users/84526 | Is projectiveness a Zariski-local property of modules? (Answered: Yes!) | Being projective is indeed a local property for the Zariski topology. In fact, it is even local for the fpqc topology --- this is a famous theorem of Raynaud and Gruson (see MR0308104).
| 13 | https://mathoverflow.net/users/986 | 6738 | 4,592 |
https://mathoverflow.net/questions/6508 | 28 | If doing geometry over $\mathbb F\_p$ means also using its algebraic closure, it must be interesting to talk about the algebraic closure of $\mathbb F\_1$ - the field with one element.
I saw that the finite extensions of $\mathbb F\_1$ are considered as $\mu\_n$, but an article by Connes et al says that it is unjusti... | https://mathoverflow.net/users/2024 | What is the algebraic closure of the field with one element? | There have been several questions on mathoverflow about the field with one element. Of course, such a field doesn't really exist and the discussion must fray sooner or later. So here is a different kind of answer.
Besides finite fields, which are 0-manifolds, there are only two fields which are manifolds, $\mathbb{C}... | 12 | https://mathoverflow.net/users/1450 | 6745 | 4,597 |
https://mathoverflow.net/questions/6668 | 2 | Preface: I think this is interesting (and hopefully at least one other mathematician will agree!), but it's entirely possible that y'all will consider this too low-brow for MO. There isn't a completely definite answer, but I think there can probably be a near-consensus. If you're able, feel free to re-title if you thin... | https://mathoverflow.net/users/303 | determining a fair betting scheme | I think it is easier if each person chooses an order for all 5 people (12345) but you still call it after only three people get married. To determine the winner, you play out the best possibility for 4th and 5th, and then you are rewarded 1 point for every relationship you get correctly.
In other words, you choose on... | 1 | https://mathoverflow.net/users/2065 | 6752 | 4,601 |
https://mathoverflow.net/questions/6751 | 2 | ### Background
Consider an electron with mass $1$ moving in $\mathbb R^n$ in under the influence of a static electromagnetic field. Up to identifying vector fields with differential forms, Maxwell's equations state that the electric force is given by a closed one-form and the magnetic field is given by a closed two-f... | https://mathoverflow.net/users/78 | Is the Hessian of Hamilton's function positive-definite? | The answer to your bonus bonus question is negative, I'm afraid. The Euler-Lagrange equations only extremise the action in general, not minimise it. Nondegeneracy of the lagrangian and positive-definiteness of the Hessian are two separate notions: the former indicating only the absence of constraints. Just take a massi... | 2 | https://mathoverflow.net/users/394 | 6760 | 4,608 |
https://mathoverflow.net/questions/6759 | 10 | I'm inspired by Yuhao's question. The functor that takes a scheme S to the set of k-dimensional vector subbundles of C^n x S (understanding "subbundle" to mean that the quotient by it is another vector bundle) is represented by the Grassmannian G(k,n). What functor is represented by the Schubert subvarieties of G(k,n)?... | https://mathoverflow.net/users/1048 | What functor does a Schubert variety represent? | Everything you said should be fine. As for the case of not necessarily reduced schemes, we have to be careful, but I think the following will work.
Say E is our given vector bundle of rank n which is a subbundle of V which is some trivial bundle, and fix a trivial subbundle W of V. The condition $\dim(E(x) \cap W(x)... | 5 | https://mathoverflow.net/users/321 | 6767 | 4,611 |
https://mathoverflow.net/questions/6765 | 24 | I'm trying to understand Milnor's proof of the existence of exotic 7-spheres.
Milnor finds his examples among $S^{3}$ bundles over $S^{4}$ (with structure group $SO(4)$ ). Such a bundle can be described as follows:
Given $M$, an $S^{3}$ bundle over $S^{4}$, if we restrict $M$ to the northern (or southern) hemispher... | https://mathoverflow.net/users/1708 | Characteristic classes of sphere bundles over spheres in terms of clutching functions | There is a way to explain it that's similar to what you said about multiplication by $n$.
Let $G$ be a Lie group, and let $f\_1$ and $f\_2$ be any two clutching functions that describe $G$-bundles $E\_1$ and $E\_2$ on $S^n$. Suppose that $f\_1$ and $f\_2$ agree at a base point of $S^{n-1}$. Let $c$ be some characteri... | 24 | https://mathoverflow.net/users/1450 | 6769 | 4,613 |
https://mathoverflow.net/questions/6695 | 12 | If we replace projective variety with algebraic variety in the statement of the Weil conjectures what happens? To me it seems the statement still makes sense. But is it still true?
| https://mathoverflow.net/users/1095 | Weil Conjectures for nonprojective algebraic varieties | Correctly restated, the conjectures hold for any variety $V$ (not necessarily complete or nonsingular) over a finite field $k$.
Dwork proved that the zeta function $Z(V,t)$ of $V$ is a rational function of $t$.
Grothendieck (et al.) expressed $Z(V,t)$ as the alternating product of the characteristic
polynomials of ... | 22 | https://mathoverflow.net/users/930 | 6777 | 4,617 |
https://mathoverflow.net/questions/6762 | 10 | X is a Noetherian scheme, F is an injective object in the category of quasi-coherent sheaves on X. U is an open subset of X. Why F's restriction on U is still an injective object in the category of quasi-coherent sheaves on U?
| https://mathoverflow.net/users/2008 | Why is an injective quasi-coherent sheaf's restriction to an open subset still an injective object? | The restriction-by-zero type arguments can actually be made to work, with some effort and an extra hypothesis. Suppose $X$ is *locally Noetherian*, $j: U \to X$ the inclusion of an open subscheme.
Let $Mod(X)$ and $QCoh(X)$ be the categories of $O\_X$-modules, and quasi-coherent $O\_X$-modules, respectively.
The "s... | 25 | https://mathoverflow.net/users/1921 | 6778 | 4,618 |
https://mathoverflow.net/questions/6780 | 9 | Theorem (Triangle-free Lemma). For all $\eta>0$ there exists $c > 0$ and $n\_0$ so that every graph $G$ on $n>n\_0$ vertices, which contains at most $cn^3$ triangles can be made
triangle free by removing at most $\eta\binom{n}{2}$ edges.
I am trying to find some information related to this topic, I am unable to acces... | https://mathoverflow.net/users/2011 | Triangle-free Lemma | Possibly an even better place to look is in surveys on graph property testing; this is by far the most common use nowadays of the triangle removal lemma, and any sufficiently good introduction to the subject should have some information on it. (I haven't actually read it, but I believe the most recent edition of Alon a... | 7 | https://mathoverflow.net/users/382 | 6788 | 4,625 |
https://mathoverflow.net/questions/6810 | 8 | Does anyone know a nice description of a Seifert surface of a torus knot? I can construct such surfaces in band projection, but what I get is ugly and unwieldy. Is there some elegant description for Seifert surfaces for such knots which I'm missing? (I'm not sure precisely what I mean by elegant...)
| https://mathoverflow.net/users/2051 | Seifert surfaces of torus knots | There's the usual description of the Seifert surface for a general cable obtained by taking copies of a Seifert surface for the knot and a fiber for the cable in the solid torus. See [Ken Baker's discussion](http://sketchesoftopology.wordpress.com/2009/11/18/cabling-a-knots-surface/).
| 10 | https://mathoverflow.net/users/1335 | 6815 | 4,647 |
https://mathoverflow.net/questions/5982 | 3 | This is theorem 14.C on p.84 of Matsumura's commutative algebra.
>
> Let $A$ be a noetherian domain, and let $B$ be a finitely generated overdomain of $A$. Let $P \in Spec(B)$ and $p = P \cap A$. Then we have
> $ht(P) \leq ht(p) + tr.d.\_{A} B - tr.d.\_{K(p)} K(P)$ with equality holds when $A$ is universally cate... | https://mathoverflow.net/users/nan | Intuition for Nagata's altitude formula? | Put dim B=n for the dimension of the variety with coordinates ring B. Then
n-ht P ≥ ((n-tr deg A B)- ht p)+ tr deg *k*(p) *k*(P)
The first member of the inequality indicates the dimension of the subvariety definited by P.
The term (n-tr deg A B) in the second member is the dimension of the variety w... | 2 | https://mathoverflow.net/users/2040 | 6816 | 4,648 |
https://mathoverflow.net/questions/6802 | 17 | Usually when people talk on the absolute Galois group *G*ℚ of ℚ they have in mind two elements they can describe explicitly, namely the identity and complex conjugation (clearly, everything is up to conjugation), although the cardinality of the group is uncountable.
Can you describe other elements of *G*ℚ?
| https://mathoverflow.net/users/2042 | Element in the absolute Galois group of the rationals | See the last of this extended answer. I'm going to part company with everyone else and say that you **can** describe other elements of $\text{Gal}(\mathbb{Q})$. In other words, I claim that you can identity a specific element of $\text{Gal}(\mathbb{Q})$ in a wide range of ways, together with an algorithm to compute the... | 12 | https://mathoverflow.net/users/1450 | 6830 | 4,658 |
https://mathoverflow.net/questions/6827 | 3 | As is well known, the set
$\{a^ib^jc^k | i,j,k \in \mathbb{Z}\\_{\geq 0},k>0\} \cup \{b^lc^md^n | l,m,n \in \mathbb{Z}\\_{\geq 0}\}$
forms a basis for quantum $SU(2)$. Does anyone know of a basis for quantum $SU(n)$?
My guess would be that a similar result holds. Namely that the set made up of all products of p... | https://mathoverflow.net/users/1867 | Basis of quantum SU(n) | [edit: Following John's helpful comments below, I made this answer much more complete.]
Yes, this is the statement that $O\_q(G)$ is a flat deformation of $O(G)$ for any semi-simple group G. See the book by Klimyk and Schmuedgen, "Quantum Groups and Their Representations" for a proof of this: on page 311 they state t... | 8 | https://mathoverflow.net/users/1040 | 6831 | 4,659 |
https://mathoverflow.net/questions/6820 | 94 | Is there an **infinite** field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only finitely many points $y \in k$ do not lie in the image of $f$)?
For finite fields $k$, there are such polynomials $f$. I... | https://mathoverflow.net/users/296 | Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points? | Since such a polynomial would have to have degree at least 2, its existence implies that
the set of k-rational points of the affine line over k is *thin* in the sense of Serre's *Topics In Galois Theory*. It follows from the results presented in that book that this cannot be the case over any *Hilbertian field*. This ... | 33 | https://mathoverflow.net/users/1149 | 6832 | 4,660 |
https://mathoverflow.net/questions/6776 | 2 | I wonder whether following statements holds
If A is an abelian category(or quasi abelian category) having enough projectives, then category of pointed diagram(which means diagram has final object,or for simplicity, one assume the diagram is finite)(A\_D=(D--->A))has enough projectives.
I want to construct a pair o... | https://mathoverflow.net/users/1851 | How to construct pair of adjoint functors from category A to category A_D(category of diagrams) | If D is small and A has enough projectives and has infinite sums then $A^D$ has enough projectives. For the proof, see Weibel, "An introduction to homological algebra", 2.3.13 on p.43. It contains the adjoint you are apparently looking for.
The proof is a version of Godement's argument that the category of sheaves of... | 2 | https://mathoverflow.net/users/1784 | 6836 | 4,662 |
https://mathoverflow.net/questions/6833 | 11 | What is the difference between
connected
strongly-connected and
complete?
My understanding is:
**connected**: you can get to every vertex from every other vertex.
**strongly connected**: every vertex has an edge connecting it to every other vertex.
**complete**: same as strongly connected.
Is this corre... | https://mathoverflow.net/users/1983 | Difference between connected vs strongly connected vs complete graphs | * *Connected* is usually associated with undirected graphs (two way edges): there is a **path** between every two nodes.
* *Strongly connected* is usually associated with directed graphs (one way edges): there is a **route** between every two nodes.
* *Complete graphs* are undirected graphs where there is an **edge** b... | 22 | https://mathoverflow.net/users/507 | 6837 | 4,663 |
https://mathoverflow.net/questions/6789 | 147 | Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason!
Is there some geometric property corresponding to "flatness" (of morphisms, modules, whatever) that makes the choice of term... | https://mathoverflow.net/users/382 | Why are flat morphisms "flat?" | A lot of people will tell you that flatness means "continuously varying fibres" in some sense, and that flatness was invented to have correspondingly nice *consequences*, which is true. But there is a way to expect this (vague) interpretation *a priori* from an alternative, equivalent definition:
An $A$-module $M$ is... | 189 | https://mathoverflow.net/users/84526 | 6844 | 4,668 |
https://mathoverflow.net/questions/6845 | 5 | A random walk matrix has largest eigenvalue 1 with multiplicty 1 - why?
Let $G$ be a non-directed, regular connected graph with degree $d$. Let $A$ be its random walk matrix, i.e. it's adjacency matrix, with each entry divided by $d$.
i) It is easy to observe that $A$ is symmetric, hence normal, that it has real ei... | https://mathoverflow.net/users/2082 | A random walk matrix has eigenvalue 1 with multiplicty 1 - why? | Here is a simple proof.
Suppose $Ax = x$. Consider the entry of $x$ with the largest absolute value; lets use $x\_k$ to denote this entry (e.g. if $x=[1,2,-4,3]^T$, then $k=3, x\_k=-4$). Consider the $k$'th row of the equation $Ax=x$; it's telling you
that $x\_k$ is a convex combination of the $x\_i$'s of its neighb... | 8 | https://mathoverflow.net/users/1407 | 6856 | 4,676 |
https://mathoverflow.net/questions/6271 | 2 | Following on from my last two questions [link text](https://mathoverflow.net/questions/5865/classical-calculi-as-universal-quotients) and [link text](https://mathoverflow.net/questions/6074/kahler-differentials-and-ordinary-differentials): Is it correct (and useful) to say that the relationship between Connes' cyclic c... | https://mathoverflow.net/users/1867 | Connes v Woronowicz - Cyclic Cohomology v Diff Calculi | I'll answer here instead of in a comment, because of the character limit...
If $A$ is the coordinate algebra of an affine variety which is *smooth* and the base field $k$ contains $\mathbb{Q}$, then $$HC\\_n(A) \cong \Omega^n\\_{A/k} / d\Omega^{n-1}\\_{A/k} \oplus H\\_{\mathrm{dR}}^{n-2}(A) \oplus H\\_{\mathrm{dR}}^{... | 3 | https://mathoverflow.net/users/1409 | 6859 | 4,678 |
https://mathoverflow.net/questions/6442 | 9 | This is more of a request for pointers to *relevant* literature than a question per se. I am, erm, looking at a paper which uses a kind of iterated pushout construction to obtain a commutative monoid with certain desired properties. The particular "gluing" construction the authors want to do is handled by quite direct ... | https://mathoverflow.net/users/763 | References/literature for pushouts in category of commutative monoids? [ed. - amalgams] | Arthur Ogus wrote a book on logarithmic geometry, apparently soon to be published, and there is a [preprint version](http://math.berkeley.edu/~ogus/preprints/log_book/) on his webpage. The first chapter is about commutative monoids, and in particular, it has a bit about pushouts (starting on page 12, and you can tell y... | 5 | https://mathoverflow.net/users/121 | 6863 | 4,681 |
https://mathoverflow.net/questions/6851 | -3 | Now there are n independent projects, each project is composed by several steps, each step is labeled, there are k workers are going to work on these projects.
Assume that each person can do only one step, and each step can only be done by a specific worker(it will be declared in input file). Each project has a order f... | https://mathoverflow.net/users/1956 | A problem seeking for algorithm | I think this is a member of a class of problems called scheduling problems. In particular I think it is the job shop scheduling problem. There are machines instead of people but the basic structure is the same. If I am right about this the prospects of a solution of this type of problem is bleak as problems with three ... | 3 | https://mathoverflow.net/users/1098 | 6869 | 4,687 |
https://mathoverflow.net/questions/6870 | 66 | Consider an [elliptic curve](http://en.wikipedia.org/wiki/Elliptic_curve) $y^2=x^3+ax+b$. It is well known that we can (in the generic case) create an addition on this curve turning it into an abelian group: The group law is characterized by the neutral element being the point at infinity and the fact that $w\_1+w\_2+w... | https://mathoverflow.net/users/802 | Why is an elliptic curve a group? | Everything I am writing below is carried out explicitly in Chapter III of Silverman's book on elliptic curves. In the earlier chapters, he defines the Picard group.
For any curve over any field, algebraic geometers are interested in an associated group called the Picard group. It is a certain quotient of the free abe... | 71 | https://mathoverflow.net/users/1018 | 6872 | 4,689 |
https://mathoverflow.net/questions/6874 | 33 | In the early 1990's, Gil Kalai introduced me to a very interesting generalization of homology theory called intersection homology, which existed for like 10 years back then I believe. Defined initially by Goresky and MacPherson, this is a version of homology which agrees with ordinary homology on manifolds, but also re... | https://mathoverflow.net/users/25 | What (if anything) happened to Intersection Homology? | Intersection homology and cohomology are still around, but as a topic they have just substantially been renamed. They are part of the theory of perverse sheaves, which are widely used in the Langlands program, in algebraic geometry approaches to categorification, and elsewhere in algebraic geometry.
To the extent tha... | 37 | https://mathoverflow.net/users/1450 | 6875 | 4,691 |
https://mathoverflow.net/questions/6895 | 2 | (Edit: The first formulation is wrong. See the second answer) Does every totally ordered set contain an unbounded countable subset. In other words: If S is a totally ordered set, can we find a (edit: at most) countable subset A, such that for every $s \in S$, there is a $a \in A, a\geq s$?
| https://mathoverflow.net/users/2097 | Unbounded countable subset | There is a counterexample in the long line L. It is totally ordered and every sequence has a limit in L. see the following:
[http://en.wikipedia.org/wiki/Long\_line\_(topology)](http://en.wikipedia.org/wiki/Long_line_%28topology%29)
| 4 | https://mathoverflow.net/users/1098 | 6900 | 4,707 |
https://mathoverflow.net/questions/6890 | 33 | The famous Birkhoff-von Neumann theorem asserts that every doubly stochastic matrix can be written as a convex combination of permutation matrices.
The question is to point out different generalizations of this theorem, different "non-generalizations" namely cases where an expected generalization is false, and to br... | https://mathoverflow.net/users/1532 | Generalizations of the Birkhoff-von Neumann Theorem | I am cheating a little to give this answer, because I am fairly sure that it is part of Gil's motivation in asking the question. The most natural generalization of the Birkhoff hypothesis to quantum probability is only true for qubits. (It might also be true for a qubit tensor a classical system; I did not check that c... | 14 | https://mathoverflow.net/users/1450 | 6904 | 4,709 |
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