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https://mathoverflow.net/questions/2214 | 12 | Hello, the Green-Tao theorem says infinitely many k-term Arithmetic Progressions exist for any integer k.
My question is: can we actually partition the primes into 3-term APs only (or is there a simple reason why it cannot be expected) ? And if it were possible, then what would it mean for the set of primes ?
For ... | https://mathoverflow.net/users/469 | Covering the primes by 3-term APs ? | Using the greedy algorithm, this would follow if for any fixed prime q, there exist infinitely prime "pairs" of the form p and 2p-q. This follows from standard (difficult) conjectures if q is odd (for example, the case q = -1 corresponds to "Sophie Germaine Primes"). On the other hand, it would be an implication of suc... | 8 | https://mathoverflow.net/users/nan | 2219 | 1,419 |
https://mathoverflow.net/questions/2215 | 12 | I've read numerous *introductions* to finite fields, but I feel like my intuition about them is fairly lacking. Considering that finite fields are the the most "inert" objects in algebraic geometry, I think I could use a serious surge of perspective.
What I would like to read now is a comprehensive overview that tell... | https://mathoverflow.net/users/84526 | A comprehensive overview of finite fields | [Finite Fields](http://books.google.com/books?id=xqMqxQTFUkMC) by R. Lidl and H Niederreiter (CUP). Probably as comprehensive as it gets.
The ams review calls it the ``the Bible of finite fields''. You can find it (the review)[here](http://www.ams.org/bull/1999-36-01/S0273-0979-99-00768-5/S0273-0979-99-00768-5.pdf).... | 10 | https://mathoverflow.net/users/262 | 2223 | 1,423 |
https://mathoverflow.net/questions/2212 | 6 | I ran into this obstacle in a harmonic analysis problem; I know epsilon about coloring problems.
Is it possible to finitely color Z^2 so that the points (x,a) and (a,y) are differently colored for every x, y and a in the integers (excepting, of course, the trivial cases x=y=a)?
| https://mathoverflow.net/users/1071 | Can I finitely color Z^2 such that (x,a) and (a,y) are different for every x,y,a? | No, this isn't possible. First note that the question is the same if we replace Z with any infinite subset of Z. Now, suppose it were possible to color Z2 with k colors, where k is minimum. Pick any row r. We can find an infinite set S of integers not containing r such that for x in S, all the (x, r) have the same colo... | 11 | https://mathoverflow.net/users/1013 | 2226 | 1,426 |
https://mathoverflow.net/questions/2218 | 11 | What can you say about the complexity class $\text{P}^{\text{NP}}$, i.e. decision problems solvable by a polytime TM with an oracle for SAT? This class is also known as $\Delta\_2^p$.
Obviously $\text{P}^{\text{NP}}$ is in $\text{PH}$ somewhere between $\text{NP} \cup \text{coNP}$, and $\Sigma\_2^p \cap \Pi\_2^p$. Wh... | https://mathoverflow.net/users/1027 | Characterize P^NP (a.k.a. Delta_2^p) | The standard complete problem for the "function version" of P^NP is to find the lexicographically last satisfying assignment of a given boolean formula. To be more finicky, a complete language for P^NP is
{ n-variable boolean formulas phi : phi is satisfiable and phi's lexicographically last satisfying assignment ha... | 23 | https://mathoverflow.net/users/658 | 2240 | 1,437 |
https://mathoverflow.net/questions/2200 | 26 | What is a simplest example of a manifold $M^{2n}$ that admits two symplectic structures with isotopic almost complex structures, and such that Gromov-Witten invariants of these symplectic structures are different?
(unfortunately I don't know any example...) If we don't impose the condition that almost complex structur... | https://mathoverflow.net/users/943 | Manifolds distinguished by Gromov-Witten invariants? | The examples in Ruan's paper "Symplectic topology on algebraic 3-folds" (JDG 1994) seem to qualify: take any two algebraic surfaces V and W which are homeomorphic but such that V is minimal and W isn't. These are nondiffeomorphic, but VxS2 and WxS2 are diffeomorphic, and Ruan gives lots of examples (starting with V equ... | 21 | https://mathoverflow.net/users/424 | 2242 | 1,438 |
https://mathoverflow.net/questions/2245 | 3 | We call $S(u)$ the space complexity of the vEB tree holding elements in the range $0$ to $u-1$, and suppose without loss of generality that $u$ is of the form $2^{2^k}$.
It's easy to get the recurrence $S(u^2) = (1+u) S(u) + \Theta(u)$. (In Wikipedia's [article](https://en.wikipedia.org/w/index.php?title=Van_Emde_Boa... | https://mathoverflow.net/users/961 | Determining the space complexity of van Emde Boas trees | The recurrence S(u2) = (1+u) S(u) + Ξ(u) can be shown to be O(u) by the following method: First assume that the constant in the Ξ(u) is at most 1 and that S(4) is at most 1, by dividing through as necessary.
Then we can prove S(u) < u - 2 by induction. The base case S(4) holds by the above assumption. The inductive c... | 4 | https://mathoverflow.net/users/1079 | 2251 | 1,444 |
https://mathoverflow.net/questions/2173 | 17 | I've been reading about D-modules, and have seen a proof that D\_X, the ring of differential operators on a variety, is "almost commutative", that is, that its associated graded ring is commutative. Now, I know that with the Ore conditions, we can localize almost commutative rings, and so we get a legitimate sheaf D to... | https://mathoverflow.net/users/622 | How much theory works out for "almost commutative" rings? | Don't get too excited about the theory of algebraic geometry for almost commutative algebras. A ring can be almost commutative and still have some very weird behavior. The Weyl algebras (the differential operators on affine n-space) are a great example, since they are almost commutative, and yet:
* They are simple ri... | 23 | https://mathoverflow.net/users/750 | 2253 | 1,446 |
https://mathoverflow.net/questions/2270 | 16 | I have a basic working knowledge of category thoery since I do research in programming languages and typed lambda-calculus. Indeed, I have refereed many papers in my area based on category theory.
But, in doing this refereeing, and in reading many important categorial papers in my area, I simply find the terminology ... | https://mathoverflow.net/users/1086 | Why do I find Category Theory mostly just a way to make simple things difficult? | Although to you, category theory is merely an inefficient framework for data about logic and programming languages, to mathematicians working in areas like algebraic geometry and algebraic topology, categories are truly essential. For us, some of our most basic notions make no sense and look extremely awkward (in fact,... | 21 | https://mathoverflow.net/users/622 | 2284 | 1,463 |
https://mathoverflow.net/questions/2281 | 15 | Hello, apparently finite groups which are n-transitive with n>5 are only the permutation groups Sn or the alternating groups An+2, see e.g. page 226 this book by Isaacs <http://books.google.fr/books?id=pCLhYaMUg8IC&pg=PA226>
Is this characterization useful at all? For instance, are there famous proofs (maybe in a geo... | https://mathoverflow.net/users/469 | Use of n-transitivity in finite group theory | This fact is used in a nice way by [Dunfield and Thurston](http://arxiv.org/abs/math/0502567) to show that, for any finite simple group Q, the number of Q-covers of a "random" 3-manifold in their sense follows a Poisson distribution. (The multiple transitivity appears in Thm 7.4.)
Also: I don't have a reference for t... | 16 | https://mathoverflow.net/users/431 | 2285 | 1,464 |
https://mathoverflow.net/questions/2264 | 8 | A rig is an algebraic object with multiplication and addition, such that multiplication distributes over addition and addition is commutative. However, instead of requiring that the set forms an abelian group under addition, we require only that it forms an abelian monoid. A commutative rig is a rig in which multiplica... | https://mathoverflow.net/users/382 | Which commutative rigs arise from a distributive category? | No commutative rig with a nontrivial solution to x + y = 0 (in particular, no nonzero ring) can arise from a distributive category. I believe Schanuel observes this near the beginning of one of his papers on Euler characteristics.
The proof is simple. Suppose C is a distributive category and R is its associated commu... | 12 | https://mathoverflow.net/users/126667 | 2291 | 1,468 |
https://mathoverflow.net/questions/2302 | 6 | So, I've been running in both stacky circles and logarithmic circles and I've been wondering: is there a definition of log stack that is "useful"? I can imagine two such definitions:
1) A log stack is a stack along with an effective divisor (could be useful for studying moduli of smooth curves, then)
2) A log stack... | https://mathoverflow.net/users/622 | What are Log Stacks | This question was answered in Martin Olsson's thesis ( <http://math.berkeley.edu/~molsson/thesis.ps> ). He gives sufficient conditions for a fibered category on the category of log schemes to arise from "a stack with a log structure", which I think is more or less what Charles intended by Option 1.
| 6 | https://mathoverflow.net/users/35508 | 2307 | 1,479 |
https://mathoverflow.net/questions/2293 | 12 | I've recently started to look at elliptic curves and have three basic questions:
1. Is it correct to say that elliptic curves $E$ in the projective plane are in **bijective** correspondence with lattices $L$ in the complex plane via $E$ <--> $C/L$.
2. If so, is there an explicit expression of the lattice generators ... | https://mathoverflow.net/users/1095 | Elliptic Curves, Lattices, Lie Algebras | What you want in terms of the relation between lattices and elliptic curves over C is proposition I.4.4 of Silverman's Advanced topics in the arithmetic of elliptic curves. Additionally, to go from a lattice to the equation of the elliptic curve (explicitly), you use Eisenstein series as in Corollary I.4.3 of that book... | 10 | https://mathoverflow.net/users/1021 | 2311 | 1,483 |
https://mathoverflow.net/questions/2185 | 35 | I've read about model categories from an Appendix to one of Lurie's papers.
What are the examples of **model categories**? What should be my intuition about them?
E.g. I understand the typical examples come from taking homotopy of something β but are all model categories homotopy categories?
| https://mathoverflow.net/users/65 | How to think about model categories? | Model categories are 1-categorical presentations of (β,1)-categories, which you can just think of as categories enriched in topological spaces, such as the category of spaces itself. (Actually, there are conditions on (β,1)-categories that come from model categories--most importantly they must have all homotopy limits ... | 17 | https://mathoverflow.net/users/126667 | 2317 | 1,488 |
https://mathoverflow.net/questions/2300 | 106 | I've heard of this many times, but I don't know anything about it.
What I do know is that it is supposed to solve the problem of the fact that the final object in the category of schemes is one-dimensional, namely $\mathop{\text{Spec}}\mathbb Z$.
So, what is the field with one element? And, what are typical geometr... | https://mathoverflow.net/users/100 | What is the field with one element? | As other have mentioned, F\_1 does not exist of a field. Tits conjectured the existence of a "field of characteristic one" F\_1 for which one would have the equality G(F\_1) = W, where G is any Chevalley group scheme and W its corresponding Weyl group.
Later on Manin suggested that the "absolute point" proposed in De... | 63 | https://mathoverflow.net/users/914 | 2327 | 1,497 |
https://mathoverflow.net/questions/2269 | 12 | Manin [stressed](http://www.ihes.fr/document?id=1704&id_attribute=48 "Manin talk") that every projective scheme should have a quantum-cohomology structure. I'd like to know more about that. And since the varieties considered in texts about monodromy resp. vanishing cycles which I have read are projective, I am curious ... | https://mathoverflow.net/users/451 | ubiquitous quantum cohomology | I'm not sure what you mean by "every projective scheme should have a quantum cohomology structure". In the talk abstract that you link to, it does not say "projective scheme" but "smooth projective variety". I don't know whether the theory generalizes to non-smooth things or to things that are not varieties.
Quantum ... | 13 | https://mathoverflow.net/users/83 | 2345 | 1,510 |
https://mathoverflow.net/questions/2333 | 8 | Toeplitz operators provide a natural language with which to do geometric quantization. I don't want to really understand them, and I don't need them in full generality. I'm looking for some references that will provide formulas with which to compute products of Toeplitz operators, and specifically formulas for the asym... | https://mathoverflow.net/users/78 | Where can I learn about (the asymptotics of) Toeplitz operators? | You probably need some conditions to restrict the kind of Schroedinger operators you want to work with. Have a look at the work of Charles and Vu-Ngoc on Toeplitz operators and the semiclassical limit, in particular theorem 1 of this <http://people.math.jussieu.fr/~charles/Articles/BerToep.pdf> and section 1.3 of this ... | 2 | https://mathoverflow.net/users/469 | 2365 | 1,527 |
https://mathoverflow.net/questions/2380 | 5 | Suppose I have a regular n-gon. I want to draw some noncrossing diagonals to subdivide it into smaller polygons. In how many ways can I do this? The vertices are unlabeled, so I don't distinguish between rotations or reflections of a given subdivision.
A triangle has 1 subdivision (do nothing!); a square has 2, a pen... | https://mathoverflow.net/users/913 | Number of subdivisions of an n-gon | I think this could be [A001004](http://www.research.att.com/~njas/sequences/A001004) in Sloane's Encyclopedia. It's hard to be sure without checking the references given there; the sequence is defined as "Number of symmetric dissections of a polygon.", which may or may not be what you mean. (In particular, the OEIS cla... | 4 | https://mathoverflow.net/users/143 | 2384 | 1,542 |
https://mathoverflow.net/questions/2372 | 30 | Suppose I put an ant in a tiny racecar on every face of a soccer ball. Each ant then drives around the edges of her face counterclockwise. The goal is to prove that two of the ants will eventually collide (provided they aren't allowed to stand still or go arbitrarily slow).
My brother told me about this result, but I... | https://mathoverflow.net/users/1 | The ants-on-a-ball problem | This is known as Klyachko's Car Crash Theorem. It was proved in order to prove a theorem about finitely presented groups. In fact, the result allows the ants to move at arbitrary nonzero speeds so long as they make infinitely many loops around their 2-cell. The conclusion is that there's either a collision between ants... | 26 | https://mathoverflow.net/users/1109 | 2387 | 1,544 |
https://mathoverflow.net/questions/2211 | 6 | So, there's a construction of Reshetikhin and Turaev which extracts knot invariants from ribbon monoidal categories, which are (usually) the representation category a Hopf algebra with a choice of ribbon element.
How do these knot invariants change if I pick a different ribbon element in the same Hopf algebra? In par... | https://mathoverflow.net/users/66 | How do quantum knot invariants change when I pick a funny ribbon element? | Did you look at prop 5.21 in the [paper with Peter](http://arxiv.org/PS_cache/arxiv/pdf/0810/0810.0084v2.pdf)? I think that should answer your question.
There are two slightly different questions you could ask. First how does the framing-dependent invariant change. Here it is just (\pm 1)^#L where # is the number of ... | 2 | https://mathoverflow.net/users/22 | 2394 | 1,549 |
https://mathoverflow.net/questions/2329 | 1 | Am I right in assuming that one cannot define an antipode for $M\_q(n)$ the bi-algebra of $nXn$ quantum matrices? If so, does anyone know a proof?
| https://mathoverflow.net/users/1095 | Antipode for quantum matrices. | I think a proof could be provided like this.
Claim: Let $H$ be a bi-algebra. Then an antipode S on H, if it exists, is unique.
Proof: If H is a bialgebra, then the set of linear maps from H to H inherits an algebra structure given by f\*g(x)=f(x\_1)g(x\_2), where \Delta(x)=x\_1\ot x\_2 is Sweedler's notation. This ... | 1 | https://mathoverflow.net/users/1040 | 2400 | 1,553 |
https://mathoverflow.net/questions/2019 | 14 | Let M be a smooth manifold. The double tangent bundle, TTM,can be viewed as a fibre bundle over TM in two ways, with the projection maps given by T\_ΟM (i.e. the derivative of the projection from TM to M) and Ο\_TM (i.e. the standard projection onto TM). There is also a canonical involution K:TTM->TTM, which basically ... | https://mathoverflow.net/users/855 | In how many ways can an iterated tangent bundle (T^k)M be viewed as a fibre bundle over (T^(k-1))M? | If we use the notation $(TM, p\_M, M)$ for the tangent bundle of any manifold $M$, then you are right to think that $T^{\ k}M$ has $k$ natural vector bundle structures over $T^{\ k-1}M$ and so on down to $M$, making a diagram which is a $k$-dimensional cube. Such a structure is a $k$-fold vector bundle (See articles by... | 15 | https://mathoverflow.net/users/1116 | 2413 | 1,563 |
https://mathoverflow.net/questions/2414 | 29 | Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free $\mathcal{O}\_X$-modules of rank $n$ and vector bundles of rank $n$. So, equivalently, principal $\mathrm{GL}(n,\mathbb{C})$-bundles are given by locally free sheaves of rank $n$.
So...what about other groups?... | https://mathoverflow.net/users/622 | Sheaf description of $G$-bundles | $\newcommand{\O}{\mathcal{O}}$
$\newcommand{\F}{\mathcal{F}}$
The way you get a locally free sheaf of rank $n$ from a $GL(n)$-torsor $P$ is by twisting the trivial rank $n$ bundle $\O^n$ (which has a natural $GL(n)$-action) by the torsor. Explicitly, the locally free sheaf is $\F=\O^n\times^{GL(n)}P$, whose (scheme-t... | 17 | https://mathoverflow.net/users/1 | 2425 | 1,572 |
https://mathoverflow.net/questions/1043 | 4 | Hi all,
The short-time fourier transform decomposes a signal window into a sin/cosine series.
How would one approximate a signal in the same way, but using a set of arbitrary basis functions instead of sin/cos? These arbitrary basis functions are likely in my case to be very small discrete chunks of a 1-dimensional... | https://mathoverflow.net/users/663 | Decomposing a 1-d signal into arbitary basis functions | Wavelets are generally used for nonperiodic signals. They're often used in earthquake detection and things like that. There are many books on the subject, a quick look for "Wavelets" in amazon.com should reveal many.
The Haar Wavelet and Daubchies Wavelet might be good choices. Haar may be better if you don't need a ... | 4 | https://mathoverflow.net/users/429 | 2426 | 1,573 |
https://mathoverflow.net/questions/2402 | 16 | Abel's identity for the dilogarithm (see the wikipedia page about polylogarithms)
plays a role in web geometry as it is one of the abelian relations of the
first example of exceptional web (Bol's 5-web) to appear in the literature.
I have heard it is important in other domains (cohomology of SL(3,C), algebraic K-the... | https://mathoverflow.net/users/605 | Abel's equation for the dilog | One basic answer is given by hyperbolic geometry.
Ideal tetrahedra in hyperbolic 3-space $\mathbb{H}^3$ are equivalent (under the action of the automorphism group $PGL\_2(C))$ to tetrahedra with vertices $\{0,1,\infty,z\}$, and their volume is given by $D(z)$, where $D(z)$ is the Bloch-Wigner dilogarithm, which is a ... | 13 | https://mathoverflow.net/users/nan | 2427 | 1,574 |
https://mathoverflow.net/questions/2433 | 12 | If we think of the l1 distance as a grid-distance between points, then we can think of l2 distance as what we get when we "shortcut" the grid by going "inside" a cell.
Making the grid finer doesn't change the l1 distance, so there's no obvious sense in which the l2 distance can be seen as a limiting version of the l... | https://mathoverflow.net/users/972 | Making an l_2 distance out of l_1 distance | L\_2 can be viewed as the limit of a sequence of metrics whose metric balls are (in the plane) regular polygons with 2n sides, where n goes to infinity. In higher dimensions one doesn't have enough regular polytopes to make a limit out of them, but it works just as well to use irregular ones. This technique is sometime... | 5 | https://mathoverflow.net/users/440 | 2436 | 1,581 |
https://mathoverflow.net/questions/2283 | 7 | Consider the tome of Bruhat and Tits: Groupes rΓ©ductifs sur un corps local : I. DonnΓ©es radicielles valuΓ©es. Publications MathΓ©matiques de l'IHΓS, 41 (1972), p. 5-251. (available on [NUMDAM](http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1972__41__5_0)). I am interested primarily in the statements of Propositions 4.4... | https://mathoverflow.net/users/425 | Iwasawa and Cartan Decompositions. | I think the key point is the proposition 4.4.2, where "good" subgroups are caracterised geometrically as stabilisers of special subgroups (ie, stabilisers of a point o such that the Weyl group W is the semidirect product of its translations and of the stabiliser of o in W).
Then the group G is the product of B (the s... | 6 | https://mathoverflow.net/users/915 | 2442 | 1,585 |
https://mathoverflow.net/questions/2459 | 1 | Varieties decompose uniquely into finitely many irreducibles, and each variety is generated by only finitely polynomials. These two finiteness properties make varieties seemingly "manageable" objects, and leads me to the question:
Can a computer, given a variety (finite set of polynomials) produce a list of its irred... | https://mathoverflow.net/users/416 | Does automatic decomposition of varieties into irreducibles exist? | This is just about primary decomposition! There are several CAS which can do that, for example Singular.
| 2 | https://mathoverflow.net/users/717 | 2463 | 1,601 |
https://mathoverflow.net/questions/2461 | 10 | This is perhaps too broad or vague (or silly) a question, but here it is anyway: why should I care about constructing line bundles *on* a moduli space? This comes up all of the time, but I seem to be missing the (probably obvious) motivation.
Ideally, it would be nice to attach to this question a particular moduli sp... | https://mathoverflow.net/users/1124 | Line bundles on moduli spaces | The reasons for caring which occur to me seem to fall into two broad categories.
First, line bundles are useful for identifying interesting substacks of your moduli stack. You usually first encounter this idea in GIT theory, but the philosophy applies more generally. See, for example, Teleman's papers on the stack o... | 11 | https://mathoverflow.net/users/35508 | 2466 | 1,603 |
https://mathoverflow.net/questions/2464 | 6 | I would like to know what Drinfeld's [scanned manuscript](http://math.uchicago.edu/~drinfeld/bestdream.pdf "low res, high res on his website") "Best Dream" is about: the title makes me curious.
It's in Russian.
| https://mathoverflow.net/users/451 | What is Drinfeld's manuscript "Best Dream" (in Russian!) about? | From the first line it appears to about **D-modules on stacks**.
The next topic is an attempt to construct a "duality" (called F) for derived category of (quasicoherent) D-modules on `Bun_G` (the space/stack of G-bundles). The problem is to find a "reverse Langlands transform" (?) by defining a suitable scalar produc... | 7 | https://mathoverflow.net/users/65 | 2498 | 1,627 |
https://mathoverflow.net/questions/2484 | 2 | I just saw this paper recently which mentioned that the optimization on a Grassmanian Manifold can be used to get an achieve an best approximation of a multilinear rank of a tensor (in the sense of a multidimensional matrix, also called a hypermatrix). Does anyone happen to know what Grassmanians have to do with tensor... | https://mathoverflow.net/users/429 | Algebraic Geometry in an applied setting? | You might also want to check out [this](http://www.math.tamu.edu/~jml/rankvsbrank.pdf) paper by Landsberg and Teitler on getting bounds for the Waring rank (and border rank) of symmetric tensors using geometry.
The point here is that the projective bundle on a vector space captures the geometry of its symmetric ten... | 4 | https://mathoverflow.net/users/310 | 2508 | 1,633 |
https://mathoverflow.net/questions/183 | 18 | The short answer to my question may be a pointer to the right text. I will give all the background I know, and then ask my questions in list form.
Let *A* be an operator (on an infinite-dimensional vector space). You might as well assume that its spectrum is all real and positive. In fact, I only care when the spectr... | https://mathoverflow.net/users/78 | Zeta-function regularization of determinants and traces | I will answer some of my questions in the negative.
**3.**
First consider the case of rescaling an operator *A* by some (positive) number Ξ». Then ΞΆΞ»*A*(s) = Ξ»-sΞΆ*A*(s), and so TR Ξ»*A* = Ξ» TR *A*. This is all well and good. How does the determinant behave? Define the "perceived dimension" DIM *A* to be logΞ»[ (DET Ξ»*A*... | 7 | https://mathoverflow.net/users/78 | 2519 | 1,640 |
https://mathoverflow.net/questions/2518 | 5 | What's the background I need to know to understand the conjectural
```
D (Bun_G) =?= O(LocSys)
```
from [this question](https://mathoverflow.net/questions/2464/what-is-drinfelds-manuscript-best-dream-in-russian-about). I know the LHS is about the derived category of D-modules on the space (stack?) of some (stab... | https://mathoverflow.net/users/65 | Sheaves on Bun_G | Don't need stable. It's the stack of all G bundles on a curve. The right hand side is the derived category of local systems, which are vector bundles with flat connection (G-bundles, for the Langlands dual of G in this equation). As for background, that depends on how deep an understanding you want.
For a lot of the ... | 3 | https://mathoverflow.net/users/622 | 2527 | 1,644 |
https://mathoverflow.net/questions/2522 | 4 | What os the meaning of `a reverse Langlands transform` to which Drinfeld [seems to refer](https://mathoverflow.net/questions/2464/what-is-drinfelds-manuscript-best-dream-in-russian-about)?
| https://mathoverflow.net/users/65 | Reverse Langlands transform | My guess is that, traditionally, the Geometric Langlands program seems to be looking for a functor from D\_{coh}(Loc,O)\to D\_{coh}(^L Bun,D), that is, from the derived category on the space of local systems to the derived category of D-modules on the space of bundles for the Langlands dual. So, by reverse Langlands tr... | 1 | https://mathoverflow.net/users/622 | 2528 | 1,645 |
https://mathoverflow.net/questions/2517 | 2 | Consider a system of the form: dx/dt = f(x,y) , dy/dt=g(x,y), with the property that the associated ODE dy/dx = g(x,y)/f(x,y) has a unique solution to IVP y(0)=0.
Also, f(x,y) is smooth every *except* the point (0,0), at which it has an infinite discontinuity, and g(x,y) is continuous everywhere. Does it follow that... | https://mathoverflow.net/users/1142 | ODE system question | I may be missing some subtle point here, but it seems to me that if you let Y(x) be your presumed solution to Y'(x)=g(x,Y)/f(x,Y) and then let x(t) solve dx/dt=f(x,Y(x)) and put y(t)=Y(x(t)), you have your answer. The only way this could fail to tend to (0,0) for some value of t is if f(x,Y(x))=0 for arbitrarily small ... | 2 | https://mathoverflow.net/users/802 | 2531 | 1,646 |
https://mathoverflow.net/questions/2525 | 15 | I know of three ways to define the dimension of a finitely-generated commutative algebra A over a field F:
* The Gelfand-Kirillov (GK) dimension, based on the growth of the Hilbert function.
* The Krull dimension, based on chains of prime ideals.
* The transcendence degree of the fraction field of A over F.
Accordi... | https://mathoverflow.net/users/290 | Different definitions of the dimension of an algebra | In a non-commutative ring, you need to be careful with what you even mean by a prime ideal, and usually there are very few two-sided ideals you might call prime. Oh, and even in the cases when there is a nice ring of fractions, it won't be a field, and so transcedence degree is still bad.
My personal favorite notion ... | 10 | https://mathoverflow.net/users/750 | 2532 | 1,647 |
https://mathoverflow.net/questions/2503 | 15 | Given a curve C, and a reductive group G, there is a moduli stack Loc\_G(C), the stack of G-local systems. I keep reading that there's a substack of "opers" but am having trouble locating a definition. So what's an oper, and how should I think about them?
| https://mathoverflow.net/users/622 | What is an Oper? | Look at <http://arxiv.org/abs/math/0501398> (*Opers*, Beilinson and Drinfeld, 1993/2005)
| 7 | https://mathoverflow.net/users/439 | 2535 | 1,649 |
https://mathoverflow.net/questions/2520 | 22 | Is it possible give an example of (or explain) how the Voevodsky et al.'s homotopy theory of schemes computes higher Chow groups?
| https://mathoverflow.net/users/65 | Homotopy theory of schemes examples | To keep things simple, let us assume we work over a perfect field.
The easiest part of motivic cohomology which we can get is the Picard group (i.e. the Chow group in degree 1). This works essentially like in topology: in the (model) category of simplicial Nisnevich sheaves (over smooth k-schemes), the classifying spac... | 29 | https://mathoverflow.net/users/1017 | 2546 | 1,658 |
https://mathoverflow.net/questions/2566 | 8 | Given a group G, we can define Gop to be a new group whose underlying set is the same as that of G but with the new multiplication g.h = hg, i.e. multiply as if you were in G but reverse the order.
Is there anything interesting to say about this construction?
When is a group isomorphic to its opposite group?
On... | https://mathoverflow.net/users/493 | Silly Question about "opposite groups" | Every group is isomorphic to its opposite group, the map $g \mapsto g^{-1}$ being an isomorphism.
In ring theory this is not the case, and the opposite ring is an important construction. For instance, if $A$ is a central simple algebra over a field (or an Azumaya algebra over a ring), then the classes of $A$ and $A^{... | 23 | https://mathoverflow.net/users/1149 | 2568 | 1,673 |
https://mathoverflow.net/questions/2557 | 36 | In "Quantum field theory and the Jones polynomial" (Comm. Math. Phys. 1989 vol. 121 (3) pp. 351-399), Witten writes:
>
> A representation *Ri* of a group *G* should be seen as a quantum object. This representation should be obtained by quantizing a classical theory. The Borel-Weil-Bott theorem gives a canonical way... | https://mathoverflow.net/users/78 | Examples of applications of the Borel-Weil-Bott theorem? | Look at the orbits of `G` on `g*` the dual of the Lie algebra. These 'coadjoint orbits' have a canonical symplectic structure. Each of these orbits intersects the positive Weyl chamber exactly once; consider those intersecting it at a 'positive weight' (i.e. the the elements of `g*` that lift to characters on `T`, the ... | 15 | https://mathoverflow.net/users/3 | 2570 | 1,675 |
https://mathoverflow.net/questions/2015 | 12 | If not, are there any interesting subcategories that can be concertized? If I am not mistaken, the category of reduced finite type varieties over the complex numbers would be an example, where the forgetful functor to sets would be given by looking at the underlying map of points.
| https://mathoverflow.net/users/788 | Can the Category of Schemes be Concretized? | The category of schemes is not small-concrete.
Let $S$ be a generating set. Let $U$ be the set of all rings $A \neq 0$ such that $\mathrm{Spec}(A)$ is an open subscheme of a scheme in $S$. Let $X$ be a set whose cardinality is larger than any element of $U$, for example, $2^{\bigsqcup\_{A \in U} A}$. Let $K$ be the f... | 22 | https://mathoverflow.net/users/297 | 2595 | 1,695 |
https://mathoverflow.net/questions/2597 | 8 | In the answer to [this](https://mathoverflow.net/questions/2071/non-finitely-generated-ring-of-regular-functions/) question we saw that [there exists](http://math.stanford.edu/~vakil/files/nonfg.pdf) a nonsingular quasi-projective threefold over a field with non-finitely generated global sections.
I was talking a... | https://mathoverflow.net/users/310 | Can any countably generated k-algebra occur as the ring of global sections of some variety? | No. Take k[t] and invert countably many relatively prime polynomials. This obeys all of your adjectives (localization preserves noetherianness, the others are obvious.)
However, a ring of global sections must be a subring of some finitely generated k-algebra. (I pointed this out in the last discussion.) Hence, its un... | 10 | https://mathoverflow.net/users/297 | 2601 | 1,697 |
https://mathoverflow.net/questions/2409 | 11 | Let Xi be a sequence of identically-distributed random variables with finite-range dependence (i.e. there exists I such that if |i-i'| β₯ I, then Xi and Xi' are independent), and a finite moment-generating function (i.e. EerXi < β for all r β R).
It's not too hard to show that Xi satisfies a strong law of large number... | https://mathoverflow.net/users/238 | Strong Law of Large Numbers for weakly dependent random variables | This question sounds like an exercise: Split the sequence into I sequences of iid random variables. Apply the classical SLLN to each sequence. Recombine.
Tom:
Of course it is true with exponential decay of the corellation function, but it is not easy.
The essential difficulty is that one wants to reduce the SLLN... | 7 | https://mathoverflow.net/users/1158 | 2605 | 1,700 |
https://mathoverflow.net/questions/2548 | 13 | For a variety V, its [Albenese variety](http://en.wikipedia.org/wiki/Albanese_variety) Alb(V) is a variety with a map V β Alb(V) which factors uniquely into any map from V to an abelian variety. Can we say something similar for an arbitrary scheme? When do we know there exists an "Albanese" scheme Alb(X)? That is,
>... | https://mathoverflow.net/users/84526 | "Albanese" schemes: When does an "initial abelian scheme" exist under a given scheme? | The construction of an Albanese scheme and an Albanese map for proper and geometrically irreducible schemes over a perfect field goes back to the work of Chevalley, to [this talk](http://archive.numdam.org/article/SCC_1958-1959__4__A10_0.pdf) of Serre, and to Grothendieck, e.g. [Theorem 3.3. in FGA Exp.VI](http://www.... | 16 | https://mathoverflow.net/users/439 | 2606 | 1,701 |
https://mathoverflow.net/questions/69 | 14 | For $n$ an integer greater than $2$, Can one always get a complete theory over a finite language with exactly $n$ models (up to isomorphism)?
Thereβs a theorem that says that $2$ is impossible.
My understanding is this should be doable in a finite language, but I donβt know how.
If you switch to a countable langu... | https://mathoverflow.net/users/27 | Complete theory with exactly n countable models? | You can refine Ehrenfeuchtβs example getting rid of the constants.
Here is what John Baldwin suggested:
Consider the theory in the language $L=\{\le\}$, saying
* $\le$ is a total preorder (transitive, total [hence reflexive], not necessarily anti-symmetric) without least or last element. (Notice that the binary ... | 17 | https://mathoverflow.net/users/1152 | 2612 | 1,704 |
https://mathoverflow.net/questions/2615 | 31 | Hello, I'm wondering if there is a standard reference discussing the least number of charts in an atlas of a given manifold required to describe it.
E.g. a circle requires at least two charts, and so on (I couldn't manage to get anything relevant neither on wikipedia nor on google, so I guess I'm lacking the correct... | https://mathoverflow.net/users/469 | Least number of charts to describe a given manifold | It's not quite the same thing, but a related object is the LyusternikβSchnirelmann category of a topological space. See
<http://en.wikipedia.org/wiki/Lyusternik-Schnirelmann_category>
| 15 | https://mathoverflow.net/users/317 | 2616 | 1,707 |
https://mathoverflow.net/questions/2305 | 4 | Be done the triangle ABC, it is known the method to finding the point Q that minimises the sum QA+QB+QC among all points Q in the plane (The Fermat point).
I want a hint for solving this problem using the construction of the tangent to ellipse.
(Hadamard, Lesson de Geometrie Elementaire, II, problem no. 745).
| https://mathoverflow.net/users/1093 | How to find the Fermat Point using the construction of the tangent to ellipse? | I have the vague idea that Hadamard is referring to the construction where you erect equilateral triangles BCA', CAB' and ABC' on the sides of the triangle, as described [here](http://www.cut-the-knot.org/Generalization/fermat_point.shtml). The Fermat point is the intersection of the cevians AA', BB' and CC'. It can al... | 1 | https://mathoverflow.net/users/296 | 2625 | 1,715 |
https://mathoverflow.net/questions/1497 | 9 | If (C,tensor,1) is a symmetric monoidal category and f:A-->B is a morphism of PROPs (or monoidal cats = colored PROPs), one gets a forgetful functor f^\*:B-Alg(C)-->A-Alg(C) (where B-Alg(C)=tensor-preserving functors from B to C) defined by precomposing with f.
Does anyone conditions on A,B,C under which this functor... | https://mathoverflow.net/users/733 | When do PROP-morphisms induce adjunctions? | Paul-André Melliès has quite an interesting paper on this topic:
<http://hal.archives-ouvertes.fr/docs/00/33/93/31/PDF/free-models.pdf>
...but phrased in the more general terms of T-algebras of a pseudomonad. The idea is that a pseudomonad on a 2-category (especially Cat), let you put algebraic structures on catego... | 3 | https://mathoverflow.net/users/800 | 2631 | 1,719 |
https://mathoverflow.net/questions/2610 | 1 | This question may be too detailed but perhaps somebody knows the answer: Neukirch proofs in his algebraic number theory book in Chapter IV, Proposition 6.2, that his class field axiom implies that the Tate cohomology groups $H^n(G(L|K),U\_L)$ for $n=0,-1$ vanish for finite unramified extensions $L|K$, where $U\_L$ is t... | https://mathoverflow.net/users/717 | Neukirch's class field axiom and cohomology of units for unramified extension | You don't need to make sense out of $\pi\_K^m$ for a general $m$ in $\hat{\mathbb{Z}}$. All you really need to know for his argument is that $v\_K(A\_K) = v\_L(A\_L)$ as subgroups of $\hat{\mathbb{Z}}$. I didn't think this through but I think it should be pretty easy to establish from the fact that $\pi\_K$ is prime fo... | 2 | https://mathoverflow.net/users/493 | 2632 | 1,720 |
https://mathoverflow.net/questions/2640 | 9 | I know that the Weyl groups of affine Lie algebras don't have a longest element, but are there any good substitutes for w\_0. In particular, is there any good substitute for a reduced decomposition of the longest element?
| https://mathoverflow.net/users/788 | Longest Element of an Affine Weyl Group | If you look at section 4 of Thomas Lam and Pavlo Pylyavskyy's [recent preprint](https://arxiv.org/abs/0906.0610), they study infinite reduced words in W, modulo braid and commutation relations. This is a good substitute for a reduced word for the long word in the context of total positivity, as they explain. I think it... | 9 | https://mathoverflow.net/users/297 | 2642 | 1,728 |
https://mathoverflow.net/questions/2521 | 6 | Consider a random walk on a two-dimensional surface with circular reflecting boundary conditions (say, of radius 'R'). Here, for a fixed-size area, one finds a larger fraction of the probability density (for the position of the walker) near the midpoint of the circle than near its contour.
Given this example, my que... | https://mathoverflow.net/users/774 | 'Focusing' the mass of the Probability Density Function for a Random Walk | Hmm, you're asking for concentration for heat kernels. Over long periods of time, these kernels are dominated by the low-energy eigenfunctions, so basically one needs to construct domains which have concentrated low-energy eigenfunctions.
Generally one expects in fact that heat kernels become smoother and disperse o... | 6 | https://mathoverflow.net/users/766 | 2648 | 1,734 |
https://mathoverflow.net/questions/2117 | 4 | When it comes to solving the heat diffusion equation u\_t=u\_xx the two most important solutions are
a) a combination (sum) of sin-terms to resemble the function of the initial condition (that is essentially a fourier series)
b) a convolution-integral of the function of the initial condition with the Gauss-curve
In m... | https://mathoverflow.net/users/1047 | Solutions to the diffusion equation | The two solutions solve different problems for the same equation.
The Fourier series solution solves the heat equation u\_t=u\_xx on a bounded interval [a,b] with an initial condition at t=0 of the form u(x,0)=f(x), a <=x<=b, and boundary conditions at both ends of the interval. These conditions can be of different t... | 6 | https://mathoverflow.net/users/1168 | 2652 | 1,737 |
https://mathoverflow.net/questions/2653 | 9 | I heard this from Haskell Rosenthal many years ago.
If V is a complex vector space, say the **opposite** of V is the complex vector space with the same elements, the same operations except switch scalar multiplication to scalar multiplication by the complex conjugate scalar. Of course this definition applies in parti... | https://mathoverflow.net/users/454 | opposite Banach space | Does [this paper of Kalton](http://arxiv.org/abs/math/9402207) do the trick? (disclaimer: I haven't read through the details)
| 9 | https://mathoverflow.net/users/763 | 2667 | 1,747 |
https://mathoverflow.net/questions/2659 | 3 | Is there a lower bound on the volume of a delta-ball in the orthogonal group O(n) of the type
f(n) \* delta^{n(n-1)/2}? For which f(n)? How can it be proven?
n(n-1)/2 is the number of degrees of freedom in the orthogonal group.
The volume in the orthogonal group is measured by the Haar measure, which is the up to s... | https://mathoverflow.net/users/1170 | Lower bound on the volume of a delta-ball in the orthogonal group O(n) of the type f(n)*delta^{n(n-1)/2} | Well, there is certainly a bound of that form. Here is a crude proof:
There is an exponential map from skew-symmetric matrices to O(n). This is differentiable with nonsingular Jacobian near the origin. The preimage of the delta ball, if I understand your notations, is the log(1+delta) ball. So the volume of this prei... | 2 | https://mathoverflow.net/users/297 | 2669 | 1,749 |
https://mathoverflow.net/questions/2671 | 4 | Hello,
homotopic maps induce isomorphic pullback bundles, and so isomorphism classes of vector bundles over X correspond to homotopy classes of maps from the grassmannian to X. But I think, in the case of X = S^1 (the 1-sphere), the isomorphism classes of 1-bundles correspond to (the generators of) Ο\_\_\_{1}(S^1), s... | https://mathoverflow.net/users/1057 | (how) are vector bundles and homotopy groups related? | The map goes the other way: vector bundles over X correspond to homotopy classes of maps from X into a grassmannian.
Let BO(n) be the grassmannian of n-plane bundles in R-infinity. Then, if you want to know about n-dimensional real vector bundles over Sk you are led to study the homotopy classes of maps from Sk to BO... | 9 | https://mathoverflow.net/users/910 | 2674 | 1,751 |
https://mathoverflow.net/questions/2672 | 21 | I made the following claim over at the [Secret Blogging Seminar](http://sbseminar.wordpress.com/2009/10/26/concrete-categories/), and now I'm not sure it's true:
Let $f: X \to Y$ and $g: X \to Y$ be two maps between finite CW complexes. If f and g induce the same map on $\pi\_k$, for all k, then f and g are homotopic... | https://mathoverflow.net/users/297 | Whitehead for maps | This is not true. Consider, for example, a degree 1 map from a torus $S^1 \times S^1$ to $S^2$ (concretely, realize the torus as a square with identifications, and then collapse the boundary of the square to a point). This map is trivial on all homotopy groups (since for any $n>0, \pi\_n$ is 0 for either the domain or ... | 39 | https://mathoverflow.net/users/75 | 2678 | 1,753 |
https://mathoverflow.net/questions/2677 | 7 | If *G* is a group, its **abelianization** is the abelian group *A* and the map *G* β *A* such that any map *G* β *B* with *B* abelian factors through *A*. Abelianization is a functor, and in general a very lossy operation. The map *G* β *A* is always a surjection/quotient, because we can construct *A* by dividing *G* b... | https://mathoverflow.net/users/78 | Abelianization of Lie groups | I don't have anything to say about specific examples, but here are some general remarks. A way to construct the abelianization of any compact group is to consider its image under the product of all its 1-dimensional unitary representations. This is because a compact abelian group is characterized by its set of characte... | 4 | https://mathoverflow.net/users/75 | 2679 | 1,754 |
https://mathoverflow.net/questions/2682 | 1 | This is inspired by [*The Whitehead for maps*](https://mathoverflow.net/questions/2672/whitehead-for-maps) question.
Consider two maps `f, g: X\to Y` which happen to induce the same maps (of discrete spaces) `[Z, X] \to [Z, Y]` for every Z. Does this mean `f` and `g` are homotopic?
And what would be the lessons fro... | https://mathoverflow.net/users/65 | Something like Yoneda's lemma | Yes, this is a special case of Yoneda. Let Z=X and consider the identity map in [X,X]; the hypothesis says that f1=f and g1=g are then equal as elements of [X,Y].
| 2 | https://mathoverflow.net/users/75 | 2684 | 1,756 |
https://mathoverflow.net/questions/2696 | 14 | I'm a little bit hesitant to ask this here, so please notice the tag. My hope is that someone will have a more satisfying answer than what I've heard before...
A long time ago I read (perhaps 'browsed' is a better word) Wolfram's "A New Kind of Science". There are many many references to the "Rule 30" CA - <http://ma... | https://mathoverflow.net/users/774 | An intuitive reason why the "Rule 30" CA is random/pseudorandom? | If you look at the [results of elementary cellular automata](http://mathworld.wolfram.com/ElementaryCellularAutomaton.html) from mathworld, most of them seem to have some kind of symmetry. (I don't want to make "symmetry" formal here; what I mean is that you get nice patterns of some sort.)
But I suspect that in gene... | 5 | https://mathoverflow.net/users/143 | 2705 | 1,768 |
https://mathoverflow.net/questions/2681 | 9 | This is related to [Theo's question](https://mathoverflow.net/questions/2677/abelianization-of-lie-groups) about the abelianizations of finite dimensionsal Lie groups.
I am interested in a specific (infinite-dimensional) case of the above question. Let H be an infinite-dimensional Hilbert space and GL(H) represent th... | https://mathoverflow.net/users/792 | Abelianization of GL(H) | The solution to Problem 240 in Halmos is: "On an infinite-dimensional Hilbert space, the commutator subgroup of the full linear group is the full linear group itself."
It is mentioned, although the details are not given, that every invertible operator is the product of two(!) commutators. Reference: A. Brown and C. P... | 10 | https://mathoverflow.net/users/430 | 2709 | 1,771 |
https://mathoverflow.net/questions/2692 | 12 | I was thinking about quivers recently, and the following idea came to me.
Let ei,j denote the matrix unit in Mn for 1 β€ i,j β€ n. Let Ξ denote the complete quiver on vertices {1, β¦, n}: one directed edge Ei,j for each ordered pair (i, j), including self-loops Ei,i.
Mn(k) is then the quotient of the path algebra PΞ b... | https://mathoverflow.net/users/813 | Matrices into path algebras | There's a slightly different equivalence that is also useful. Consider the quiver with n elements, and an arrow `E_i` from `i` to `i+1` and another `F_i` from `i+1` to `i` for all `i`. The relations are then that `E_i F_i = e_i` and `F_i E_i = e_{i+1}`, where `e_j` is the `j`-th simple idempotent. This gets the same pa... | 4 | https://mathoverflow.net/users/750 | 2716 | 1,775 |
https://mathoverflow.net/questions/2713 | 3 | I believe there was an old conjecture that there's **always a prime number between `N` and `2N`**.
What's the history and how is this proven is the easiest/elementary/deepest ways?
| https://mathoverflow.net/users/65 | Bertrand's postulate | Proven. This is called Bertrand's postulate. Here is [Erdos' elementary proof](http://www.nd.edu/~dgalvin1/pdf/bertrand.pdf); the original proof is due to Chebyshev.
| 13 | https://mathoverflow.net/users/143 | 2717 | 1,776 |
https://mathoverflow.net/questions/2708 | 15 | Is there known to be an $x$ such that for all positive integers $N$ there exists some $n>N$ such that $p\_{n+1}-p\_n \leq x$, where $p\_n$ is the $n$th prime? Or, in other words, is it known that limit as $n$ goes infinity of $p\_{n+1}-p\_n$ is not infinity? If such an $x$ is known to exist, what is the current best kn... | https://mathoverflow.net/users/597 | Is there a known bound on prime gaps? | (**Edit**: things have happened since the original post, changing the short answer to yes. See for example <http://arxiv.org/abs/1410.8400> for the status in 2014 where $x \leq 600$ unconditionally. GRP **End Edit**)
The short answer is no, though if one assumes the Elliot-Halberstam conjecture then one can take x=16... | 27 | https://mathoverflow.net/users/766 | 2718 | 1,777 |
https://mathoverflow.net/questions/2710 | 6 | Let (A,m) be a complete local Noetherian ring and let X and Y be two schemes of finite type over A (and flat over A). Let Xn and Yn be the reductions of X and Y mod mn+1.
>
> **Question**: Suppose there is a compatible system of isomorphisms between Xn and Yn (for all n). Does there necessarily exist an isomorphism... | https://mathoverflow.net/users/2 | Can isomorphisms of schemes be constructed on formal neighborhoods? | No. Let A be k[[t]]. Let X be A^1 \setminus {-1,0,1} and Y be A^1 \setminus {1,t,-1}. In explicit equations, X = Spec k[[t]][x, y]/y(x-1)x(x+1)-1 and Y = Spec k[[t]][x, y]/y(x-1)(x-t)(x+1)-1.
Over k[[t]]/t^{n+1}, the reductions of X and Y are isomorphic because all infinitesimal deformations of a smooth affine scheme... | 7 | https://mathoverflow.net/users/297 | 2720 | 1,779 |
https://mathoverflow.net/questions/2704 | 32 | I'm taking introductory courses in both Riemann surfaces and algebraic geometry this term. I was surprised to hear that any compact Riemann surface is a projective variety. Apparently deeper links exist.
What is, in basic terms, the relationship between Riemann surfaces and algebraic geometry?
| https://mathoverflow.net/users/416 | Links between Riemann surfaces and algebraic geometry | For simplicity, I'll just talk about varieties that are sitting in projective space or affine space. In algebraic geometry, you study varieties over a base field k. For our purposes, "over" just means that the variety is cut out by polynomials (affine) or homogeneous polynomials (projective) whose coefficients are in k... | 38 | https://mathoverflow.net/users/1018 | 2722 | 1,781 |
https://mathoverflow.net/questions/2724 | 7 | Is it known that for every epsilon there is `N_0` such that all intervals of the form [N, (1+\epsilon)\*N], where N > `N_0`, contain prime numbers?
| https://mathoverflow.net/users/65 | Strong Bertrand postulate | This follows from the Prime Number Theorem. Let Ο(n) be the number of primes less than n. Then Ο(n) ~ n/log(n); it follows Ο((1+Ξ΅)n)-Ο(n) -> β as n -> β.
| 8 | https://mathoverflow.net/users/143 | 2725 | 1,783 |
https://mathoverflow.net/questions/2748 | 26 | This is somewhat related to [Greg's question](https://mathoverflow.net/questions/2551/why-do-groups-and-abelian-groups-feel-so-different) about groups and abelian groups. Suppose you met someone who was well-acquainted with groups, but who was unwilling to accept rings as a meaningful object of study. How would you des... | https://mathoverflow.net/users/290 | What is the "right" definition of a ring? | Well rings are naturally the objects which act on abelian groups - indeed composition always endows the endomorphisms of an abelian group with the structure of a ring. So if one is interested in the endomorphisms of groups one is actually interested in rings.
One can make this analogy more precise especially if one p... | 28 | https://mathoverflow.net/users/310 | 2751 | 1,803 |
https://mathoverflow.net/questions/2660 | 4 | Suppose I uniformly sample matrices X from the Gaussian Unitary Ensemble (GUE) with variance \sigma^2. Consider the Ky-Fan d norm, i.e. the sum of the singular values, of X. Let's call this Z=||X||\_1. What is the distribution of Z as a function of the dimension d and the variance \sigma^2? Really, all I want are estim... | https://mathoverflow.net/users/1171 | Distribution of 1-norm for Gaussian Unitary Ensemble | Let's normalise the variance of the entries to be $1$. Then GUE asymptotically obeys the semicircular law, i.e., the eigenvalues (which equal the singular values, as GUE is Hermitian), after dividing by $\sqrt{n}$, are distributed according to the law $\frac{1}{2 \pi} (4 - x^2)^{1/2}\_+ dx$. So the Schatten $1$-norm (K... | 6 | https://mathoverflow.net/users/766 | 2760 | 1,808 |
https://mathoverflow.net/questions/2755 | 74 | As Akhil had great success with his [question](https://mathoverflow.net/questions/1291/a-learning-roadmap-for-algebraic-geometry), I'm going to ask one in a similar vein. So representation theory has kind of an intimidating feel to it for an outsider. Say someone is familiar with algebraic geometry enough to care about... | https://mathoverflow.net/users/622 | A learning roadmap for Representation Theory | I second the suggestion of Fulton and Harris. It's a funny book, and definitely you want to keep going after you finish it, but it's a good introduction to the basic ideas.
You specifically might be happier reading a book on algebraic groups.
While I third the suggestion of Ginzburg and Chriss, I wouldn't call it a... | 25 | https://mathoverflow.net/users/66 | 2763 | 1,811 |
https://mathoverflow.net/questions/2779 | 22 | The projective curve $3x^3+4y^3+5z^3=0$ is often cited as an example (given by Selmer) of a **failure of the [Hasse Principle](http://en.wikipedia.org/wiki/Hasse_principle)**: the equation has solutions in any completion of the rationals $\mathbb Q$, but not in $\mathbb Q$ itself.
I don't think I've ever seen a proo... | https://mathoverflow.net/users/25 | Proof of no rational point on Selmer's Curve $3x^3+4y^3+5z^3=0$ | My friend has written an introduction to algebraic number theory before, which contains a short proof of this statement, but I didn't check its validity.
**Edit:** updated the link of the document, <http://www.2shared.com/document/2d6M7kNU/Introduction_to_Algebraic_Numb.html>
p. 41 of the document, or p. 45 of the PD... | 7 | https://mathoverflow.net/users/nan | 2785 | 1,827 |
https://mathoverflow.net/questions/2734 | 2 | Brown defines the classifying space of a crossed complex in the following way.
Given a filtration X\* of a space X, define the fundamental crossed complex by:
C\_0 = X\_0, C\_1=\pi(X\_1,X\_0) (the fundamental groupoid), C\_n = the family of groups \pi(X\_n,X\_n-1,p) for all p in X\_0.
Now let Ξ^n be the cell comple... | https://mathoverflow.net/users/343 | Classifying space of a crossed complex | It is indeed (weakly equivalent to) the trivial one, in the [model structure on crossed complexes](http://ncatlab.org/nlab/show/folk+model+structure).
Your question seems to indicate that you think this is a problem. But it is not: each cell of NC is contractible (as it is for each cell of a space!) but there may st... | 2 | https://mathoverflow.net/users/381 | 2787 | 1,829 |
https://mathoverflow.net/questions/2596 | 18 | Illusie [mentions](http://www.math.uchicago.edu/~mitya/langlands/Illusie.wav) tape recordings of Grothendieck explaining his trace formula and more. Are they or similar recordings online? I guess, even if (what I doubt) everything he thought about that is somewhere in print, it would give an interesting insight in his ... | https://mathoverflow.net/users/451 | Are there any recordings of Grothendieck online? | Illusie told me last year that these tapes of his meetings with Grothendieck are somewhere in his basement where they're very hard to find, so they're certainly not online.
The recording of Illusie's reminisces of Grothendieck to which both posts link has been transcribed [here](http://math.uchicago.edu/~tp/main.dvi... | 7 | https://mathoverflow.net/users/307 | 2788 | 1,830 |
https://mathoverflow.net/questions/1814 | 11 | It took me some effort to work out Gerashenko's nice simple example [Can a singular Deligne-Mumford stack have a smooth coarse space?](https://mathoverflow.net/questions/1565/can-a-singular-deligne-mumford-stack-have-a-smooth-coarse-space/1584#1584) of a DM stack non-equisingular with its coarse moduli space, which mea... | https://mathoverflow.net/users/307 | What are some examples of coarse moduli spaces? | Thank you all for your very kind answers! It was silly of mine to suggest we rate the examples, since it's unclear if any examples are better than others. To atone for my mistake, let me offer a summary of the proposed examples in the form of a table.
stack : coarse moduli space
---------------------------
$\mathca... | 2 | https://mathoverflow.net/users/307 | 2792 | 1,833 |
https://mathoverflow.net/questions/2607 | 63 | Does anyone have an answer to the question "What does the cotangent complex measure?"
Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as geometric ones (like "homology detects holes"), as are intuitions which do not exactly answer the above question.
In particul... | https://mathoverflow.net/users/733 | Intuition about the cotangent complex? | One thing the cotangent complex measures is what kind of deformations a scheme has. The precise statements are in Remark 5.30 and Theorem 5.31 in Illusie's article in "FGA explained". Here's the short simplified version in the absolute case:
If you have a scheme $X$ over $k$, a first order deformation is a space $\ma... | 29 | https://mathoverflow.net/users/473 | 2796 | 1,835 |
https://mathoverflow.net/questions/2765 | 6 | SGA 7, tome 1, exp. IX, contains in its introduction and in section 13.4 remarks about ideas and conjectures of Deligne on a βthΓ©orie de NΓ©ron pour motifs de poids quelconqueβ. Would someone please give an epitome of that theory or references? (This is Thomas Riepe's request <http://sbseminar.wordpress.com/requests/#co... | https://mathoverflow.net/users/307 | NΓ©ron theory for motives of arbitrary weight | Deligne commented last year:
""Neron model" is perhaps misleading. It is only the case of unipotent (rather than quasi-unipotent) local monodromy I want to consider. The questions I had in mind were :
-What can one say about a motive about the field of fraction of a discrete valuation ring ; what objects over the r... | 4 | https://mathoverflow.net/users/451 | 2799 | 1,837 |
https://mathoverflow.net/questions/2795 | 76 | Last year I attended a first course in the representation theory of finite groups, where everything was over C. I was struck, and somewhat puzzled, by the inexplicable perfection of characters as a tool for studying representations of a group; they classify modules up to isomorphism, the characters of irreducible modul... | https://mathoverflow.net/users/1202 | Why are characters so well-behaved? | Orthogonality makes sense without character theory. There's an inner product on the space of representations given by $\dim \operatorname {Hom}(V, W)$. By Schur's lemma the irreps are an orthonormal basis. This is "character orthogonality" but without the characters.
How to recover the usual version from this concept... | 70 | https://mathoverflow.net/users/22 | 2808 | 1,843 |
https://mathoverflow.net/questions/2809 | 47 | It is well known that the operations of differentiation and integration are reduced to multiplication and division after being transformed by an integral transform (like e.g. Fourier or Laplace Transforms).
My question: Is there any intuition why this is so? It can be proved, ok - but can somebody please explain the ... | https://mathoverflow.net/users/1047 | Intuition for Integral Transforms | It might help you to think about a discrete model: consider complex valued functions on $Z/n$. The discrete Fourier transform takes $f(k)$ to
$g(j) :=\sum\_{k=1}^n f(k) \zeta^{jk}$ where $\zeta=e^{2 \pi i/n}$. It is pretty easy to see that, if we change $f(k)$ to $f(k+1)$, we change $g(j)$ to $g(j)\*\zeta^j$.
Simila... | 23 | https://mathoverflow.net/users/297 | 2823 | 1,855 |
https://mathoverflow.net/questions/2848 | 6 | Suppose you have N symbols (e.g. "1","2",...,"N" or "a","b",...,"$") and a string of these symbols (say, the first trillion digits of pi). Then does there exist a prime number whose N-ary representation contains that string of digits?
| https://mathoverflow.net/users/1132 | Prime numbers and strings of symbols | Yes. This follows from the strong version of [Bertrand's postulate](https://mathoverflow.net/questions/2724/strong-bertrand-postulate). For example, to see that there is a prime which contains the digits 314159, use the fact that there is a prime between 314159\*10^N and
314159\*10^N\*(1.000001) for N sufficiently lar... | 9 | https://mathoverflow.net/users/297 | 2850 | 1,875 |
https://mathoverflow.net/questions/2665 | 5 | The following is a result I feel like I've seen some form of before, but can't figure out how to prove or find a reference for. Suppose you have a map p:E \to B, with B paracompact, and suppose that every point in B has a neighborhood U such that there is a map p^{-1}(U) \to U \times F over U which is a fiber homotopy ... | https://mathoverflow.net/users/75 | Is a map that is locally fiberwise equivalent to a product a Hurewicz fibration? | The answer is no; Allen Hatcher sent me the following:
An example where this fails is the projection of the letter L onto its horizontal base, which I'll call B. The deformation retraction of L onto B is a fiberwise homotopy equivalence. The homotopy lifting property fails: Map a point to the left endpoint of B, then... | 4 | https://mathoverflow.net/users/75 | 2855 | 1,878 |
https://mathoverflow.net/questions/2857 | 6 | Can you build a probabilistic scheme for colouring each edge (independently of all other edges) of the complete graph G on the positive integers such that the probability that G contains an infinite monochromatic complete subgraph is neither 0 nor 1?
| https://mathoverflow.net/users/416 | Non trivial colouring of the edges of an infinite complete graph | It seems like this would go against the [Kolmogorov 0-1 law.](http://en.wikipedia.org/wiki/Kolmogorov%27s_zero-one_law). If we let Xi denote the coloring of all of the edges from i to integers larger than i, wouldn't the existence of an infinite monochromatic subgraph be a tail event?
| 9 | https://mathoverflow.net/users/405 | 2859 | 1,880 |
https://mathoverflow.net/questions/2776 | 6 | If β# exists then why is cof(ΞΈL(β)) = Ο? Also I have the same question for the L(VΞ»+1) generalization (if it's actually a different proof; I presume it isn't), i.e. if ΞΈ is defined as the sup of the surjections in L(VΞ»+1) of VΞ»+1 onto an ordinal, then if VΞ»+1# exists why is cof(ΞΈL(VΞ»+1)) = Ο?
| https://mathoverflow.net/users/1178 | Cofinality of Theta if sharps exist | This is because the pieces of the sharp singularize Theta. Let s\_n be the sequence of the first n cardinals above continuum and let a\_n be the nth cardinal above continuum. Then the theory of reals with a parameter s\_n in L\_{a\_n+1}(R) is a set of reals A\_n. They are Wadge cofinal in Theta, another words the seque... | 5 | https://mathoverflow.net/users/20584 | 2862 | 1,882 |
https://mathoverflow.net/questions/2861 | 14 | Given a high precision real number, how should I go about guessing an algebraic integer that it's close to?
Of course, this is extremely poorly defined -- every real number is close to a rational number, of course! But I'd like to keep both the coefficients and the degree relatively small. Obviously we can make trade... | https://mathoverflow.net/users/3 | How should I approximate real numbers by algebraic ones? | The Lenstra-Lenstra-Lovasz lattice basis reduction algorithm is what you need. Suppose that your real number is a and you want a quadratic equation with as small coefficients as possible, of which a is nearly a root. Then calculate 1,a,a^2 (to some precision), find a nontrivial integer relation between them, and use th... | 29 | https://mathoverflow.net/users/3304 | 2864 | 1,884 |
https://mathoverflow.net/questions/2142 | 6 | I've read an interesting article, [math.NT/0409456](http://arxiv.org/abs/math.NT/0409456) where you're just trying to solve a simple problem:
>
> For a given (finite) set of primes S find all solutions to an equation `a + b = c` with the condition that all prime divisiors of integers a, b, c must be in S.
>
>
>
... | https://mathoverflow.net/users/65 | Solving "a, b, a+b have given divisors" problem | This is bound to be pretty hard! Knowing the prime divisors of a,b,c implies knowing the radical rad(abc) (the product of the distinct primes dividing abc), and then knowing the solutions a,b,c of a + b = c in positive integers, you would know the largest c with a,b,c coprime (pairwise or not, doesn't matter). So you w... | 1 | https://mathoverflow.net/users/3304 | 2878 | 1,895 |
https://mathoverflow.net/questions/2876 | 12 | Let X ---> Y be a finite surjective morphism of smooth, projective, connected varieties over a finite field F\_q. Can one describe the zeta function Z(X, t) in terms of the zeta-function Z(Y,t) of Y? Can one say anything at all about how they are related? What if we assume the morphism is finite etale?
| https://mathoverflow.net/users/81 | Behaviour of Zeta-function under Finite Morphism | I don't think you can say much of anything of consequence unless you have better control over things. As an example, consider the kth power map A^1 -> A^1. This is finite, and surjective and even etale if you throw out 0, and has degree k, but the zeta functions are the same.
Another way of rerephrasing Ilya's commen... | 2 | https://mathoverflow.net/users/66 | 2884 | 1,899 |
https://mathoverflow.net/questions/2890 | 18 | This question is related to [this question](https://mathoverflow.net/questions/1346/representablity-of-cohomology-ring) from Dinakar, which I found interesting but don't yet have the background to understand at that level.
Unless I'm mistaken, the rough statement is that $H^n(X;G)$ (the $n$-dimensional cohomology of ... | https://mathoverflow.net/users/303 | Cohomology and Eilenberg-MacLane spaces | 1. We are working in the **homotopy** category of topological spaces where morphisms are homotopy classes of continuous maps. More accurately, we tend to work in the based category where each object has a distinguished base point and everything is required to preserve that base point. The non-based category can be embe... | 17 | https://mathoverflow.net/users/45 | 2898 | 1,907 |
https://mathoverflow.net/questions/2900 | 17 | This came up in the [question about Eilenberg-MacLane spaces](https://mathoverflow.net/questions/2890/cohomology-and-eilenberg-maclane-spaces). Given the definition of `K(G, n)`, it's easy to prove that there is a map `K(G,n) x K(G,n) --> K(G,n)` that endows cohomology with an additive structure.
>
> **Question:** ... | https://mathoverflow.net/users/65 | How to get product on cohomology using the K(G, n)? | If you form the smash product $X = K(A,p) \wedge K(B,q)$ of two Eilenberg-MacLane spaces, then the resulting space is $(p+q-1)$-connected, and the first non-trivial homotopy group in dimension $p+q$ is $A \otimes B$.
To see this "geometrically", I would model the EM spaces as CW-complexes, whose first non-basepoint c... | 21 | https://mathoverflow.net/users/437 | 2906 | 1,910 |
https://mathoverflow.net/questions/2904 | 27 | Inspired by [this question](https://mathoverflow.net/questions/2857/non-trivial-colouring-of-the-edges-of-an-infinite-complete-graph), I was curious about a comment in [this article](http://en.wikipedia.org/wiki/Kolmogorov%27s_zero-one_law):
>
> In many situations, it can be easy to
> apply Kolmogorov's zero-one l... | https://mathoverflow.net/users/441 | Examples where Kolmogorov's zero-one law gives probability 0 or 1 but hard to determine which? | There's a set of good examples from percolation theory:
<http://en.wikipedia.org/wiki/Percolation_theory>
If you create a "random network" with a certain probability p of edges between nodes (see article above for precise definitions) then there is an infinite cluster with probability either zero or one. But for a gi... | 26 | https://mathoverflow.net/users/1227 | 2911 | 1,915 |
https://mathoverflow.net/questions/2913 | 9 | There was a previous post on the correspondence between Riemann surfaces and algebraic geometry. I want to ask a related but more detailed question.
BACKGROUND:
Engelbrekt gave an overview of how you start with a compact Riemann surfaces and map them into projective space
[Links between Riemann surfaces and algebr... | https://mathoverflow.net/users/7 | Analogues of the Weierstrass p function for higher genus compact Riemann surfaces | I think Hunter and Greg's answers make it hard to see the forest for the trees. Let X be a compact Riem. surface of genus >= g. Let Y be the universal cover of X equipped with the complex structure pulled back from X. As a complex manifold, Y is isomorphic to the upper half plane, and the deck transformations form a su... | 9 | https://mathoverflow.net/users/297 | 2923 | 1,922 |
https://mathoverflow.net/questions/2875 | 19 | I've heard that irreducible unitary representations of noncompact forms of simple Lie groups, the first example of such a group `G` being `SL(2, R)`, can be completely described and that there is a discrete and continuous part of the spectrum of `L^2(G)`.
1. How are those representations described?
2. Do all unitary ... | https://mathoverflow.net/users/65 | Unitary representations of SL(2, R) | I strongly recommend you read the article "Representations of semisimple Lie groups" by Knapp and Trapa in the park city/ias proceedings "Representation theory of Lie groups". It's a very nice introduction to the problem of describing the "unitary dual" (which is what you are asking about) which focusses on SL(2,R). Fo... | 19 | https://mathoverflow.net/users/1021 | 2940 | 1,936 |
https://mathoverflow.net/questions/2944 | 23 | Let $\{a\_n\}$ be a sequence of complex numbers indexed by the positive integers. Does there always exist an analytic function $f$ such that $f(n) = \{a\_n\}$ for $n=1,2,...$? If not, are there any simple necessary or sufficient conditions for the existence of such $f$? This analytic function should be defined on some ... | https://mathoverflow.net/users/302 | Which sequences can be extended to analytic functions? (e. g., Ackermann's function) | It's a standard theorem in complex analysis that if $z\_n$ is a sequence that goes to infinity, there is an entire function taking any prescribed values at the $z\_n$. [There is a function $f$ vanishing to order 1 at each $z\_n$](http://en.wikipedia.org/wiki/Weierstrass_factorization#Existence_of_entire_function_with_s... | 26 | https://mathoverflow.net/users/75 | 2947 | 1,941 |
https://mathoverflow.net/questions/2905 | 25 | Sometimes people say that the Fukaya category is "not yet defined" in general.
What is meant by such a statement? (If it simplifies things, let's just stick with Fukaya categories of compact symplectic manifolds, not Fukaya-Seidel categories or wrapped Fukaya categories or whatever else might be out there)
What sho... | https://mathoverflow.net/users/83 | Is the Fukaya category "defined"? | From what I understand, the most fundamental issue obstructing the definition of the Fukaya category in general is the fact that the boundaries of the relevant moduli spaces typically have codimension-one pieces arising from the bubbling off of pseudoholomorphic discs. In situations where there are no bubbles (for inst... | 19 | https://mathoverflow.net/users/424 | 2948 | 1,942 |
https://mathoverflow.net/questions/2903 | 9 | I've heard that there are different theories providing knot invariants in form of homologies. My understanding is that if you embed knot in a special way into a space, there is a special homology theory called Floer homology.
>
> **Question:** what's the definition and properties of a Floer homology of a knot? How ... | https://mathoverflow.net/users/65 | What is Floer homology of a knot? | I can say something about this for Heegaard Floer homology. Given a 3-manifold Y, you can take a Heegaard splitting, i.e. a decomposition of Y into two genus g handlebodies joined along their boundary. This can be represented by drawing g disjoint curves a1,...,ag and g disjoint curves b1,...,bg on a surface S of genus... | 16 | https://mathoverflow.net/users/428 | 2955 | 1,948 |
https://mathoverflow.net/questions/2945 | 5 | If f is a weight 2 newform on $\Gamma\_1(N)$ then there exists an abelian variety Af whose endomorphism algebra is isomorphic to the field generated by the coefficients of f.
I've seen this proven in Shimura's book, but was wondering if anyone knows of a different reference (perhaps one that is a bit more readable...... | https://mathoverflow.net/users/nan | Modular forms reference | Have a look at Section 6.6 of Diamond and Shurman, *A First Course in Modular Forms*:
As an aside, the theorem states a bit more than you have said: for instance, when the field of Fourier coefficients is $\mathbb{Q}$, you are just asserting the existence of an elliptic curve $E\_{/\mathbb{Q}}$ with $\operatorname{En... | 4 | https://mathoverflow.net/users/1149 | 2962 | 1,954 |
https://mathoverflow.net/questions/2971 | 13 | I am reading some papers which involve D-modules on a Lie algebra g, which are supported on the nilpotent cone n. They are equivariant for the action of G. (In particular, I consider g=sl\_n).
It was explained to me (the statement, not the proof) that the category of such D-modules is semisimple, and that the simple ... | https://mathoverflow.net/users/1040 | D-modules supported on the nilpotent cone | Watch out that there are more simple objects than it looks like at first glance, even for sl\_n. Although the orbits are parameterized by partitions, they can carry nontrivial local systems whose intermediate extensions to the nilpotent cone will be new simple D-modules.
I have heard that the general problem of writ... | 6 | https://mathoverflow.net/users/1048 | 2992 | 1,973 |
https://mathoverflow.net/questions/2991 | 3 | The group scheme G\_a here is the one-dimensional additive group.
| https://mathoverflow.net/users/425 | Over which schemes can there exist non-trivial G_a bundles? | Principal Ga-bundles on a scheme X, in any of the Zariski, etale, or flat topologies, are classified by the coherent cohomology group H^1(X,OX). For a smooth complex projective variety, this is the antiholomorphic component of the de Rham group H^1(X,C), which is a topological invariant. So (in this smooth Kahler setti... | 5 | https://mathoverflow.net/users/1048 | 2995 | 1,975 |
https://mathoverflow.net/questions/3008 | 15 | What are some number theoretic sequences that you know of that occur as (or are closely related to) the sequence of Fourier coefficients of some sort of automorphic function/form or the sequence of Hecke eigenvalues attached to a Hecke eigenform?
I know many such sequences, but am always looking for more.
Some exam... | https://mathoverflow.net/users/683 | Number theoretic sequences and Hecke eigenvalues | Characters of rational vertex operator algebras tend to yield modular functions. This is due to the space of torus partition functions in a chiral conformal field theory being a complex moduli invariant. The standard example is the monster vertex algebra, whose character is j-744. Other examples come from lattice CFTs ... | 6 | https://mathoverflow.net/users/121 | 3012 | 1,987 |
https://mathoverflow.net/questions/2638 | 4 | I am looking for some good references on singularity theory. I'm interested in singularity theory in the context of mirror symmetry, so this means I'm interested in things like Picard-Lefschetz theory, oscillating integrals, Frobenius manifolds (Saito theory). I have looked at the book by Arnold, Gusein-Zade, Varchenko... | https://mathoverflow.net/users/83 | Singularity theory references | Some of the topics you are interested in are covered (very nicely) in C.Sabbah, Isomonodromic deformations and Frobenius manifolds. An introduction. A more specialised book is C.Hertling, Frobenius manifolds and moduli spaces for singularities.
| 2 | https://mathoverflow.net/users/1220 | 3028 | 1,998 |
https://mathoverflow.net/questions/3007 | 8 | If I'm given a division algebra D with Z(D)=F, then how can I view Dx as an algebraic group defined over F? I'd like to see first how Dx can be given the structure of a variety defined over F, and then to see how the group law on Dx is defined over F.
| https://mathoverflow.net/users/493 | Division Algebras as Algebraic Groups | Choose an F-basis of D. The multiplication is described by certain quadratic functions, with respect to this basis; D\* is given by the nonvanishing of a polynomial function (the norm).
So the multiplication can be understood as defining an algebraic group structure on the complement of a hypersurface in an affine spa... | 11 | https://mathoverflow.net/users/513 | 3035 | 2,003 |
https://mathoverflow.net/questions/2969 | 9 | In Signal Processing books, a fundamental theorem is that linear time invariant systems can be represented as a convolution with a distribution. Could you give a mathematically rigorous statement of this theorem, or refer a book that includes it?
Edit: For example, would the following be a correct statement?
"Let S... | https://mathoverflow.net/users/1229 | What is a rigorous statement for "linear time-invariant systems can be represented as convolutions"? | I think the result you are looking for is the following:
Let T be a linear continuous and translation invariant operator mapping S into S' (rather than S' into S').
Then there exists a distribution K s.t. Tf = f\*K, for every f in S.
The continuity of T is referred to the usual Frechet topology on S and the weak dua... | 5 | https://mathoverflow.net/users/1049 | 3054 | 2,016 |
https://mathoverflow.net/questions/3024 | 8 | Can someone verify this for me.. or tell me what reference shows me this... is this true:
>
> Let $k$ be a field. Then a field extension $K$ of $k$ is separable over $k$ iff for any field extension $L \supseteq k$ the Jacobson radical of the tensor product $K\otimes\_k L$ is trivial.
>
>
>
I got this idea by l... | https://mathoverflow.net/users/1245 | Characterisation for separable extension of a field | You should take a look at Theorem 3.4 (p 85) of Farb and Dennis' book *Noncommutative Algebra*. The statement is:
Let $L/k$ be a finite extension of fields. Then $K\otimes \_k L$ is semisimple for every field $K\supseteq k$ if and only if $L/k$ is a separable extension.
That the tensor product is semisimple implie... | 6 | https://mathoverflow.net/users/nan | 3062 | 2,021 |
https://mathoverflow.net/questions/2985 | 19 | I would like to understand the relationship between the derived category definition of a right derived functor $Rf$ (which involves an initial natural transformation $n: Qf \rightarrow (Rf)Q$, where $Q$ is the map to the derived category) and the "universal delta functor" definition given in Hartshorne III.1.
I alrea... | https://mathoverflow.net/users/84526 | Derived functors vs universal delta functors | I haven't checked all the details, but I think the story could go like this. (I have to apologize: it's a bit long.)
(1) Let $F:\mathsf A\rightarrow \mathsf B$ be an additive left exact functor between two abelian categories. Take an injective resolution of an object $A$ in $\mathsf A$:
$$0\rightarrow A \rightarrow... | 10 | https://mathoverflow.net/users/1246 | 3065 | 2,023 |
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