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https://mathoverflow.net/questions/14486 | 21 | Speculation and background
--------------------------
Let $\mathcal{C}:=\mathrm{CRing}^\text{op}\_\text{Zariski}$, the affine Zariski site. Consider the category of sheaves, $\operatorname{Sh}(\mathcal{C})$.
According to [nLab](https://ncatlab.org/nlab/show/scheme#translation_between_the_two_approaches), schemes ar... | https://mathoverflow.net/users/1353 | Precise definition of a scheme (Key question: How to define an open subfunctor without resorting to classical scheme theory) | Check out the paper of Kontsevich-Rosenberg [noncommutative space.](http://www.mpim-bonn.mpg.de/preprints/send?bid=2331), they defined formally open immersion and open immersion completely functorial way. This definition is nothing to do with "noncommutative"
Definition:
Formally open immersion is formally smooth mo... | 7 | https://mathoverflow.net/users/1851 | 14566 | 9,778 |
https://mathoverflow.net/questions/14499 | 11 | Forgive the elementary nature of the question. I understand that the second order Peano Axioms are categorical in the sense that all their models are isomorphic. This equivalence class of models is taken to be the definition of natural numbers.
My question is that: Is the theory defined by those second order axioms ... | https://mathoverflow.net/users/3873 | Is any true sentence in the second-order Peano Axioms provable | Note that there are two different types of models of second-order logic: standard models, where second-order quantified variables range over all subsets of the domain; and Henkin models, where second-order quantified variables are allowed to range over a proper subset of the full power-set.
Henkin proved the complete... | 8 | https://mathoverflow.net/users/2004 | 14570 | 9,781 |
https://mathoverflow.net/questions/14568 | 42 | The Poisson summation says, roughly, that summing a smooth $L^1$-function of a real variable at integral points is the same as summing its Fourier transform at integral points(after suitable normalization). [Here](http://en.wikipedia.org/wiki/Poisson_summation_formula) is the wikipedia link.
For many years I have won... | https://mathoverflow.net/users/3885 | Truth of the Poisson summation formula | It is a special case of the trace formula. Both sides are the trace of the same operator.
| 25 | https://mathoverflow.net/users/927 | 14571 | 9,782 |
https://mathoverflow.net/questions/14586 | 4 | What properties does a distribution (in the generalized function sense) has to have in order to be a function. That is, when is $T(\varphi) = \int f \varphi$ for some $f$?
| https://mathoverflow.net/users/3887 | When can a function be recovered from a distribution? | First of all, $T$ must have order zero, i.e., $|T(\varphi)|\le C(K)\sup|\varphi|$ for any test function $\varphi$ supported on a compact set $K$. By Riesz representation theorem, $T$ is a measure. To be a locally integrable function, it must be absolutely continuous with respect to the Lebesgue measure. One way to expr... | 12 | https://mathoverflow.net/users/2912 | 14591 | 9,797 |
https://mathoverflow.net/questions/14587 | 40 | For all those who are unlikely to have answers to my questions, I provide some
Background:
===========
In some sense, pure motives are generalisations of smooth projective varieties. Every [Weil cohomology](http://en.wikipedia.org/wiki/Weil_cohomology_theory) theory factors through the embedding of smooth projectiv... | https://mathoverflow.net/users/956 | Understanding the definition of the Lefschetz (pure effective) motive | The motive $L$ is called Lefschetz because it is the cycle class of the point in ${\mathbb P}^1$, and so underlies (in a certain sense) the Lefschetz theorems about the cohomology of
projective varieties. To understand this better, you may want to read about how the hard Lefschetz theorem for varieties over finite fiel... | 39 | https://mathoverflow.net/users/2874 | 14592 | 9,798 |
https://mathoverflow.net/questions/14476 | 2 | This theorem is: let f:X--->Y be a proper morphism of noetherian schemes,and the induced morphism of sheaves f^#:O\_Y---->f\_\*O\_X is isomorphic.Then for any point y belongs to Y,f^-1(y) is nonempty and connected.
I have seen a proof by use of formal schemes.My question is：Is there any proof without this trick？
| https://mathoverflow.net/users/3866 | the proof of "theorem of connectedness" | A typical context in which one has the condition $f\_{\\*}\mathcal O\_X = \mathcal O\_Y$
in that in which $f$ is a birational morphism and $Y$ is normal. In this context, the proof
that the fibres are connected is due to Zariski (I believe that it's the original version
of his ``main theorem'') and certainly predates E... | 4 | https://mathoverflow.net/users/2874 | 14600 | 9,806 |
https://mathoverflow.net/questions/14622 | 6 | My understanding (please correct me if I'm wrong) is that if you have some transitive set M which is an $\epsilon$-model of ZFC, and you take an ultrapower of it using an approprate ultrafilter, you wind up with a new model whose membership relation is not the $\epsilon$ relation of the ambient set theory, but still sa... | https://mathoverflow.net/users/2361 | How can an ultrapower of a model of ZFC be "ill-founded" yet still satisfy ZFC? | For an ultrapower of $V$ by an ultrafilter $\mathcal{U}$ there is an exact characterization of when the ultrapower will be well-founded: precisely when $\mathcal{U}$ is closed under countable intersections.
As for how a model $M$ could possibly satisfy regularity but not be wellfounded, the problem is that there may ... | 18 | https://mathoverflow.net/users/2436 | 14626 | 9,823 |
https://mathoverflow.net/questions/14613 | 31 | Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x\_1,...,x\_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$).
Now the symmetric group $\mathfrak{S}\_n$ acts by permutation on the indeterminates.
The subring of invariant polynomials $R^{\mathfrak{S}\_n}$ has a nice descri... | https://mathoverflow.net/users/3701 | Invariant polynomials under a group action (hidden GIT) | The actions of $S\_n$ and $\mathbb Z\_n$ differ in the sense that in the first case the quotient is smooth (it is again $\mathbb C^n$) while in the second case it is singular. This is why in the fist case we have a nice presentation, but in the second not really. For example, the number of generators of the quotient ca... | 27 | https://mathoverflow.net/users/943 | 14628 | 9,824 |
https://mathoverflow.net/questions/14629 | 14 | I am away from Torsion theory in abelian category for some while. So it might be a stupid question.
The definition of a torsion pair in the category of modules is as follows:
Definition:
A pair $(\mathcal T,\mathcal F)$ of full subcategories of $A-\mathrm{mod}$ is called a torsion pair if following conditions hold... | https://mathoverflow.net/users/1851 | What is the relationship between t-structure and Torsion pair? | The two notions are related in the sense that they share a common generalization, namely the notion of torsion pair on a pre-triangulated category (this term has at least two meanings, here we mean a category which has compatible left and right triangulations - it covers several cases including triangulated categories ... | 16 | https://mathoverflow.net/users/310 | 14633 | 9,827 |
https://mathoverflow.net/questions/14619 | 8 | In the late 1970's and in the 1980's, Michael Freedman showed a relationship between the topological surgery problem in 4-dimensions, the slice problem for links, and the classification of non-simply-connected 4-manifolds. He also showed the failure of 4-dimensional homology surgery and the homology splitting theorem v... | https://mathoverflow.net/users/2051 | Freedman's work on non-simply-connected 4-manifolds | I would recommend you to look at the reference Freedman-Quinn book *Topology of 4-manifolds* , it might be helpful.
| 6 | https://mathoverflow.net/users/3895 | 14636 | 9,829 |
https://mathoverflow.net/questions/14627 | 34 | I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt correct, but he doesn't give references, and the thought of ploughing through Artin's collected works seems a bit daunting to ... | https://mathoverflow.net/users/1384 | The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures. | Peter Roquette has written four beautiful papers on the history of the zeta-function in characteristic $p$.
[The Riemann hypothesis in characteristic p, its origin and development. Part 1. The formation of the zeta functions of Artin and F.K. Schmidt.](http://www.rzuser.uni-heidelberg.de/~ci3/rv.pdf)
[The Riemann h... | 28 | https://mathoverflow.net/users/2275 | 14643 | 9,833 |
https://mathoverflow.net/questions/14639 | 2 | Usually we have axiomatic theory and the we look for model for it - this is book picture. Of course in real math usual one has a "model" that is given structure and looks for proper axiomatizing of it. SO it is interesting question:
Some structure called "model" is given by some (countable) number of axioms.
How m... | https://mathoverflow.net/users/3811 | Given is "model". How many theories may it be a model? | You are using the terms "model" and "theory" in an idiosyncractic way.
In model theory, a *model* is a first-order structure, that is, a set with some functions, relations and perhaps distinguished elements, called constants. A *theory*, in contrast, is a collection of assertions, a set of sentences in this language... | 3 | https://mathoverflow.net/users/1946 | 14646 | 9,836 |
https://mathoverflow.net/questions/14638 | 10 | From seminar on kdV equation I know that for integrable dynamical system its trajectory in phase space lays on tori. In wikipedia article You may read (<http://en.wikipedia.org/wiki/Integrable_system>):
>
> When a finite dimensional Hamiltonian
> system is completely integrable in the
> Liouville sense, and the e... | https://mathoverflow.net/users/3811 | Integrable dynamical system - relation to elliptic curves | If your system is algebraic, then you bet! More generally, you can get abelian varieties as the fibers for many interesting integrable systems. Google the following for more: algebraic complete integrable Hamiltonian system, Calogero-Moser System, Hitchin System.
As for elliptic curves, they'll only pop out in low di... | 12 | https://mathoverflow.net/users/622 | 14657 | 9,845 |
https://mathoverflow.net/questions/14624 | 4 | Take the following definition:
"A *parabolic subgroup* of a linear algebraic group defined over $\mathbb{C}$ is a subgroup, closed in the Zariski topology, for which the quotient space is a projective algebraic variety."
My questions are:
(i) Why include *closed* in the definition?
(ii) What is an example of a pr... | https://mathoverflow.net/users/1648 | Questions Suggested by the Parabolic Subgroup Definition | If $G={\mathbb C}$ is the additive group and $L\subset G$ is a lattice, then $L$ is not
Zariski closed, $G/L$ exists (in a sense of analytic geometry) and is projective. But of course
$L$ is not a parabolic subgroup of $G$.
| 7 | https://mathoverflow.net/users/4158 | 14663 | 9,849 |
https://mathoverflow.net/questions/14673 | 0 | (I've edited this question)
I'm searching for a continuously differentiable function $f:\mathbb R^2\to\mathbb R$ such that $f(x)+f(x+u+v)\neq f(x+u)+f(x+v)$ for all $x$ and all linearly independent $u$ and $v$.
My original question was about the special case $x=0, f(x)=0$ for merely continuous functions, which turn... | https://mathoverflow.net/users/814 | Existence of an "anti-additive" (or "never linear") map? | For your new question: functions that satisfy your inequality don't exist
Proof.
Suppose $f(x)+f(x+u+v)> f(x+u)+f(x+v)$ for all $x$ and all linearly independent $u$ and $v$.
Let us get a contradiction from it. Take any square insribed in a circle, and rotate it leaving insrcibed. Rotating continuously for the angle... | 9 | https://mathoverflow.net/users/943 | 14675 | 9,857 |
https://mathoverflow.net/questions/14690 | 18 | The fact that the Riemann zeta function $\zeta(s)$ and its brethren have a pole at $s=1$ is responsible for the infinitude of large classes of primes (all primes, primes in arithmetic progression; primes represented by a quadratic form). We cannot hope proving the infinitude of primes $p = a^2+1$ in this way because th... | https://mathoverflow.net/users/3503 | Primes of the form a^2+1 | Hi Franz,
Unfortunately I doubt this Euler product has very good behavior. If you believe the Hardy-Littlewood conjectures, then $\sum\_{n\leq X}\Lambda(n^2+1) \sim cX$ where $c=\prod\_{p>2}(1-\chi\_{4}(p)(p-1)^{-1})$ is some positive constant which is almost certainly transcendental. If $\zeta\_{G}(s)$ reflected this... | 12 | https://mathoverflow.net/users/1464 | 14694 | 9,868 |
https://mathoverflow.net/questions/14696 | 9 | I suspect that the answer to my question is well-known to be no. To be more precise, let
$G$ and $H$ be nonisomorphic finite groups of the same order. Let $S \subseteq G$ and $T \subseteq H$ be subsets satisfying the three properties: (1) the subsets are symmetric, that is $S = S^{-1}$ and $T = T^{-1}$; (2) they are mi... | https://mathoverflow.net/users/2677 | Does a Cayley graph on a minimal symmetric set of generators determine a finite group up to isomorphism? | The truncated cube (polyhedron with eight triangular faces and six octagonal faces) is a Cayley graph of both the symmetric group on four items (generators: transpose first two of the four items, rotate the last three) and of a different group that acts on 3-bit binary strings (generators: rotate the string, flip its f... | 16 | https://mathoverflow.net/users/440 | 14699 | 9,871 |
https://mathoverflow.net/questions/14679 | -1 | How to multiply this series:
$$(\sum\_{t=-\infty}^{\infty}a\_{t})(\sum\_{k=-\infty}^{\infty}b\_{k})$$
| https://mathoverflow.net/users/3900 | cauchy product for general case | It's not a problem to multiply the series: the product is $\sum\_{(t,k)\in\mathbb Z^2} a\_tb\_k$. The question is how to sum the double series that we have.
For series with nonnegative terms summation is not a problem either: we take the supremum of all finite sums. And since any finite sum is contained in a suffici... | 3 | https://mathoverflow.net/users/2912 | 14700 | 9,872 |
https://mathoverflow.net/questions/14714 | 107 | I know the following facts. (Don't assume I know much more than the following facts.)
* The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem.
* The Atiyah-Singer index theorem can be proven using heat kernels.
This implies that both Riemann-Roch and Gauss-Bonnet can... | https://mathoverflow.net/users/290 | What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem? | Here is how the heat kernel proof of Atiyah-Singer goes at a high level. Let $(\partial\_t - \Delta)u = 0$ and define the heat kernel (HK) or Green function via $\exp(-t\Delta):u(0,\cdot) \rightarrow u(t,\cdot)$. The HK derives from the solution of the heat equation on the circle:
$u(t,\theta) = \sum\_n a\_n(t) \exp... | 47 | https://mathoverflow.net/users/1847 | 14715 | 9,881 |
https://mathoverflow.net/questions/14717 | 19 | Why the Mittag-Leffler condition on a short exact sequence of, say, abelian groups, that ensures that the first derived functor of the inverse limit vanishes, is so named?
| https://mathoverflow.net/users/3903 | Mittag-Leffler condition: what's the origin of its name? | The wording of your question suggests that you're familiar with the "classical" [Mittag-Leffler theorem](http://en.wikipedia.org/wiki/Mittag-Leffler_theorem) from complex analysis, which assures us that meromorphic functions can be constructed with prescribed poles (as long as the specified points don't accumulate in t... | 14 | https://mathoverflow.net/users/763 | 14721 | 9,885 |
https://mathoverflow.net/questions/14667 | 35 | In complex projective geometry, we have a specified Kähler class $\omega$ and we have a Lefschetz operator $L:H^i(X,\mathbb{C})\to H^{i+2}(X,\mathbb{C})$ given by $L(\eta)=\omega\wedge \eta$. We then define primitive cohomology $P^{n-k}(X,\mathbb{C})=\ker(L^{k+1}:H^{n-k}(X)\to H^{n+k+2}(X))$, and we even have a nice th... | https://mathoverflow.net/users/622 | Intuition for Primitive Cohomology | The primitive classes are the highest weight vectors.
Hard Lefschetz says that the operator $L$ (which algebraic geometers know as intersecting with a hyperplane) is the "lowering operator" $\rho(F)$ in a representation $\rho
\colon \mathfrak{sl}\_2(\mathbb{C})\to End (H^\ast(X;\mathbb{C}))$. The raising operator $\... | 37 | https://mathoverflow.net/users/2356 | 14727 | 9,889 |
https://mathoverflow.net/questions/14730 | 3 | I feel it very unintuitive to understand what an equivariant sheaf is. In the simplest example, L/K is a finite Galois extension, G=Gal(L/K), G acts on Spec L, what are the equivariant sheaves on L?
| https://mathoverflow.net/users/2008 | Examples of Equivariant Sheaves under Group action | I don't think that the example you chose is the simplest one. It might be better to start
with say a vector bundle $\mathcal V \to X$ over some base space $X$. Suppose now that $G$
acts on $X$. A $G$-equivariant structure on $\mathcal V$ is a choice of $G$-action on
$\mathcal V$ (assuming it exists) preserving the vect... | 7 | https://mathoverflow.net/users/2874 | 14736 | 9,893 |
https://mathoverflow.net/questions/14735 | 29 | There is, of course, a complete classification for simple complex Lie algebras. Is there a good reference which lists the group of outer automorphisms for each?
| https://mathoverflow.net/users/3513 | Outer automorphisms of simple Lie Algebras | Proposition D.40 of Fulton and Harris' *Representation Theory* states Emerton's comment: the group of outer automorphisms of a simple Lie algebra are precisely the group of graph automorphisms of the associated Dynkin diagram. There is also some discussion of this in Section 16.5 of Humphrey's *Introduction to Lie Alge... | 25 | https://mathoverflow.net/users/321 | 14743 | 9,896 |
https://mathoverflow.net/questions/14739 | 36 | Given a commutative ring $R$, there is a category whose objects are epimorphisms surjective ring homomorphisms $R \to S$ and whose morphisms are commutative triangles making two such epimorphisms surjections compatible, and the skeleton of this category is a partial order that can be identified with the lattice of idea... | https://mathoverflow.net/users/290 | How can I define the product of two ideals categorically? | Nice question! The answer is that it's not possible! Let $R=\mathbb{F}\_3[x,y]/(x^2,y^2)$. The lattice of ideals consists of the eight ideals
$(1)$
$(x,y)$
$(x)$ $(y)$ $(x+y)$ $(x-y)$
$(xy)$
$(0)$,
in which each ideal contains all ideals at lower levels. In the middle level, some of the ideals have square z... | 54 | https://mathoverflow.net/users/2757 | 14745 | 9,897 |
https://mathoverflow.net/questions/14742 | 3 | I am reading a paper concerning the action of monoidal category to another category.
Let $k$ be a commutative ring, $R$ is a k-algebra. $A=R-mod$, $B=R^{e}-mod=R\bigotimes \_{k}R^{o}-mod$.
Consider the action:
$B\times A\rightarrow A,(M,N)\mapsto M\bigotimes \_{R}N$ is an action of monoidal category of $R^{e}-mod=... | https://mathoverflow.net/users/1851 | How is this action of monoidal derived category induced? | The original action takes the form of an additive functor $A \times B \to B$ with notation as in the question (and appropriate coherence conditions giving compatibility with the monoidal structure on $A$ presumably). By additivity this extends to the level of homotopy categories giving $K(A)\times K(B) \to K(B)$ where ... | 5 | https://mathoverflow.net/users/310 | 14746 | 9,898 |
https://mathoverflow.net/questions/14740 | 6 | As I understand, the lack of indication on how to obtain first integrals in Arnol'd-Liouville theory is a reason why we are interested in bi-Hamiltonian systems.
Two [Poisson brackets](https://en.wikipedia.org/wiki/Poisson_bracket)
$\{ \cdot,\cdot \} \_{1} , \{ \cdot , \cdot \} \_{2}$ on a manifold $M$ are compatible... | https://mathoverflow.net/users/2384 | Connection between bi-Hamiltonian systems and complete integrability | Your understanding is essentially correct. There are three basic (and closely related) approaches to
constructing the integrals of motion required for complete integrability: through separation of variables, through the Lax representation, and through the bi-Hamiltonian representation. The relationship among them is no... | 11 | https://mathoverflow.net/users/2149 | 14747 | 9,899 |
https://mathoverflow.net/questions/14752 | 16 | It's possible I just haven't thought hard enough about this, but I've been working at it off and on for a day or two and getting nowhere.
You can define a notion of "[covering graph](http://en.wikipedia.org/wiki/Covering_graph)" in graph theory, analogous to covering spaces. (Actually I think there's some sense -- ma... | https://mathoverflow.net/users/382 | Checking if two graphs have the same universal cover | Two finite graphs have the same universal cover iff they have a common finite cover. This surprising fact was first proved by Tom Leighton here:
Frank Thomson Leighton, Finite common coverings of graphs. 231-238 1982 33 J. Comb. Theory, Ser. B
I'm quite sure the paper also presents an algorithm for determining if t... | 31 | https://mathoverflow.net/users/25 | 14754 | 9,903 |
https://mathoverflow.net/questions/14748 | 3 | As can be guessed from some of my previous questions, I'm trying to understand, at the moment, the relationship between principal and their associated vector bundles. To this end I've been looking at $\mathbb{CP}^{n} = SU(n+1)/U(n)$ and trying to find the representation of $U(n)$ that gives $\Omega^{(1,0)}(\mathbb{CP}^... | https://mathoverflow.net/users/2612 | Principal bundles and associated vector bundles, the case of the complex projective space (1,0)-forms | In $SU(n+1)/U(n)$ there is a natural basepoint, the coset of the identity, which is fixed by
the action of $U(n)$ (thought of as acting on the quotient by virtue of being a subgroup
of $SU(n1)$). Since $U(n)$ fixed this point, it acts on the (complexified) cotangent space to this point,
and the problem is then to under... | 6 | https://mathoverflow.net/users/2874 | 14774 | 9,914 |
https://mathoverflow.net/questions/14776 | 19 | I hope this is serious enough. It is a well-known fact that $\pi\_1(SO(3)) = \mathbb{Z}/(2)$, so $SO(3)$ admits precisely one non trivial covering, which is 2-sheeted.
Another well known fact is that you can hold a dish on your hand and perform two turns (one over the elbow, one below) in the same direction and come ... | https://mathoverflow.net/users/828 | How does $\pi_1(SO(3))$ relate exactly to the waiters trick? | Spin(3) comes into the play only as the covering space of SO(3), I think. You do everything in SO(3). Draw a curve through your body from a stationary point, like your foot, up the leg and torso and out the arm, ending at the dish. Each point along the curve traces out a curve in SO(3), thus defining a homotopy. After ... | 11 | https://mathoverflow.net/users/802 | 14780 | 9,919 |
https://mathoverflow.net/questions/14781 | 2 | Considering the function $f:\mathbb{R} \to \mathbb{C}$, with $\left| f(x) \right|=1$ for all $x\in \mathbb{R}$.
Considering $g:\mathbb{R} \to \mathbb{C}$ with $\int\_{-\infty}^{\infty}{\left|g(x)\right|^2dx}=1$
I am interested on properties of the amplitude of the Fourier Transform of the product of $f$ and $g$: ... | https://mathoverflow.net/users/3913 | Constraints on the Fourier transform of a constant modulus function | If $g$ happens to be in $L^1$, then the amplitude of the Fourier transform of $fg$ is bounded by the $L^1$ norm of $g$, for any unimodular $f$. This is the only restriction from above since you can always choose $f$ so that $fg\ge 0$, thus bringing the (essential) supremum of $\widehat{fg}$ up to $\|g\|\_{L^1}$.
Ano... | 5 | https://mathoverflow.net/users/2912 | 14784 | 9,921 |
https://mathoverflow.net/questions/14719 | 7 | (Disclaimer: I'm a beginner in this area, so welcome corrections.)
Let $(X,x)$ be a germ of a complex surface (i.e. locally the zero set of some holomorphic functions) and assume that $x$ an isolated singular point. Mumford proved that if the local fundamental group of $X$ at $x$ is trivial, then in fact $x$ is smoot... | https://mathoverflow.net/users/460 | An algebraic proof of Mumford's smoothness criterion for surfaces? | I found what I think is the answer, in [a paper by Cutkosky and Srinivasan](https://gdz.sub.uni-goettingen.de/id/PPN358147735_0068?tify=%7B%22pages%22:%5B323%5D%7D "Steven Dale Cutkosky, Hema Srinivasan. Comment. Math. Helvetici 68 (1993), 319–332") called "Local fundamental groups of surface singularities in character... | 3 | https://mathoverflow.net/users/460 | 14785 | 9,922 |
https://mathoverflow.net/questions/14799 | 6 | Let $X$, $Y$ and $Z$ be Noetherian schemes.
If $f: Y \to X$ and $g: Z \to X$ are morphisms of finite type, such that at each point of $X$, at least one of the two morphisms is smooth/étale/unramified (at all points of its inverse image), can we conclude that the induced morphism $Y \times\_X Z \to X$ is smooth/étale... | https://mathoverflow.net/users/1107 | products and smooth/étale/unramified morphisms | No. As an extreme example, suppose that $g$ is the identity (which is etale everywhere), and that $f$ is not etale at some point. Then the fibre product is just $f$ again.
But in fact, this is essentially the general case. If $g$ is etale (or smooth) at a point, then it is etale (resp. smooth) in a n.h. of that point... | 7 | https://mathoverflow.net/users/2874 | 14802 | 9,934 |
https://mathoverflow.net/questions/14737 | 10 | In [Hartshorne, III.3] he proves that injective modules over $R$ give flasque sheaves over $Spec\ R$. I presume that's because they don't give injective sheaves, and flasque is the consolation prize. Is there an easy counterexample?
EDIT: in III.3 he's assuming Noetherian. And he's already proved in II.5.5 the equiva... | https://mathoverflow.net/users/391 | Can injective modules over R give non-injective sheaves over Spec R? | Let me put this here for the sake of clarity. As was noted by Emerton in a comment above, [this answer to a related Math Overflow question](https://mathoverflow.net/questions/6762/why-is-an-injective-quasi-coherent-sheafs-restriction-to-an-open-subset-still-an/6773#6773) shows that the answer is **no**, for an injectiv... | 7 | https://mathoverflow.net/users/1784 | 14806 | 9,936 |
https://mathoverflow.net/questions/14801 | 7 | Suppose $G$ is a simple (linear) algebraic group over an algebraically closed field of characteristic zero, that $n$ is a positive natural number, and that $g\in G$. Can we always find an $h\in G$ such that $h^n=g$?
(It appears to be possible to check this for the classical algebraic groups by direct computations in ... | https://mathoverflow.net/users/3513 | Taking roots in simple linear algebraic groups | A semisimple element lies in a maximal torus, so you can extract any root from it inside this torus.
| 7 | https://mathoverflow.net/users/3696 | 14816 | 9,941 |
https://mathoverflow.net/questions/14758 | 1 | Hi all,
I am wondering whether someone has considered the definition of the flux homomorphism for manifolds with boundary. More specifically, I am looking at the annulus and I want the diffeomorphism $\phi(r, \theta) = (r, \theta + \alpha)$ for $\alpha$ constant to have non-zero flux.
I was thinking of extending th... | https://mathoverflow.net/users/3909 | Flux homomorphism for manifolds with boundary | Here are some remarks about your definition.
1) $H\_1(A,\partial A)$ is just one-dimensional, it is generated by a path that joins two sides of $A$.
2) The definition that you gave works for the annulus, and for surfaces with a boundary as well. This will also work for manifolds with boundary $M^n$, in the case
w... | 2 | https://mathoverflow.net/users/943 | 14817 | 9,942 |
https://mathoverflow.net/questions/14808 | 12 | Given a symmetric monoidal category and a monoid object A in it, one can form the category of modules over this monoid object, i.e. objects are $A \otimes M \rightarrow M$ satisfying the natural properties analogous to modules over a ring and morphisms respecting this. The following seems to be true and I would like to... | https://mathoverflow.net/users/733 | Why is a monoid with closed symmetric monoidal module category commutative? | (I have rewritten parts to respond to potential objections, since someone disliked this answer. I didn't change the thrust of my discussion, but I did alter some places where I might have been slightly too glib. If I missed something, please feel free to correct me in the comments!)
The situation is even better than ... | 11 | https://mathoverflow.net/users/3049 | 14840 | 9,960 |
https://mathoverflow.net/questions/12499 | 9 | This is related to [another one of my questions on DM stacks](https://mathoverflow.net/questions/12472/is-the-inertia-stack-of-a-deligne-mumford-stack-always-finite). In Brian Conrad's article 'The Keel-Mori Theorem via Stacks', a sufficient condition on for an Artin stack to have coarse moduli space is that it has fin... | https://mathoverflow.net/users/370 | coarse moduli space of DM stacks | No, not every DM-stack has a coarse moduli space. The following is a counter-example (see my paper on geometric quotients):
Let X be two copies of the affine plane glued outside the y-axis (a non-separated scheme). Let G=Z2 act on X by y → –y and by switching the two copies. Then G acts non-freely on the locally clos... | 14 | https://mathoverflow.net/users/40 | 14855 | 9,970 |
https://mathoverflow.net/questions/14856 | 6 | Let $S^1$ act on $S^{2n+1}$ via Hopf action and $S^1$ also acts on $\mathbb{R}^2$ via rotation about the origin.
Then $S^1$ acts on $S^{2n+1}\times \mathbb{R}^2$ diagonally.
Let $M$ be the quotient of this diagonal action.
My question is why $ M$ can be viewed as the normal bundle of $\mathbb {CP}^n$ in $\mathbb {... | https://mathoverflow.net/users/3922 | Normal bundle of $CP^n$ in $CP^{n+1}$ | $S^{2n+1}$ sits in $S^{2n+3}$ with a trivial normal bundle. So the quotient map $S^{2n+3} \to \mathbb CP^{n+1}$ carries the normal bundle of $S^{2n+1}$ in $S^{2n+3}$ to the normal bundle of $\mathbb CP^n$ in $\mathbb CP^{n+1}$.
| 8 | https://mathoverflow.net/users/1465 | 14859 | 9,972 |
https://mathoverflow.net/questions/14632 | 4 | Let $B$ be a ring which is the colimit of rings $B\_\lambda$. Let $X\_\lambda$ be a stack (not necessarily algebraic) over $B\_\lambda$ such that $X\_\lambda \times\_{B\_\lambda} B\_\mu = X\_\mu$ and let $X = X\_\lambda \times\_{B\_\lambda} B$.
If $X$ is an algebraic stack, then does some $X\_\lambda$ have to be alge... | https://mathoverflow.net/users/28 | Approximation of stacks / algebraic spaces | The answer is no, even for the sheaf-case.
First of all, you would have to make some assumptions such as assume that Xλ→ Spec(Bλ) is locally of finite presentation. For simplicity also assume that the algebraic space X is of finite presentation over Spec(B). Then if U→X is an étale presentation, it descends to a *mor... | 7 | https://mathoverflow.net/users/40 | 14862 | 9,973 |
https://mathoverflow.net/questions/14842 | 18 | Given a formal power series $f \in k[[X]]$, where $k$ is a commutative field, is there any good way to tell whether or not $f\in k(X)$?
Edit: To clarify, "good way to tell" means "computable algorithm to tell".
Edit 2: I really screwed up this question, so I am recusing myself from accepting an answer. I will accep... | https://mathoverflow.net/users/1353 | When a formal power series is a rational function in disguise | Continued fractions!
To motivate this answer, first recall the continued fraction algorithm for testing whether a real number is rational. Namely, given a real number $r$, subtract its floor $\lfloor r \rfloor$, take the reciprocal, and repeat. The number $r$ is rational if and only if at some point subtracting the ... | 34 | https://mathoverflow.net/users/2757 | 14874 | 9,982 |
https://mathoverflow.net/questions/3190 | 12 | Suppose G is an algebraic group with an action G×X→X on a scheme. Does the fixed locus (the set of points x∈X fixed by all of G) have a scheme structure? You can obviously define the functor Fix(T)={t∈X(T)|t is fixed by every element of G(T)}. Is this functor always representable?
(This question was "broken off" of [... | https://mathoverflow.net/users/1 | Is the fixed locus of a group action always a scheme? | The question gives the "wrong" definition of Fix(T), hence the resulting confusion.
A more natural definition of the subfunctor X^G of "G-fixed points in X" is
(X^G)(T) = {x in X(T) | G\_T-action on X\_T fixes x}
= {x in X(T) | G(T')-action on X(T') fixes x for *all* T-schemes T'}.
(Of course... | 28 | https://mathoverflow.net/users/3927 | 14876 | 9,984 |
https://mathoverflow.net/questions/14877 | 63 | We may define a topological manifold to be a second-countable Hausdorff space such that every point has an open neighborhood homeomorphic to an open subset of $\mathbb{R}^n$. We can further define a smooth manifold to be a topological manifold equipped with a structure sheaf of rings of smooth functions by transport of... | https://mathoverflow.net/users/1353 | How much of differential geometry can be developed entirely without atlases? | There is the book by Ramanan "Global Calculus" which develops differential geometry relying heavily on sheaf theory (you should see his definition of connection algebra...).
He avoids the magic words "locally ringed space" by requiring the structure sheaf to be a subsheaf of the sheaf of continuous functions (hence max... | 32 | https://mathoverflow.net/users/3701 | 14886 | 9,992 |
https://mathoverflow.net/questions/14858 | 4 | Let $V$ be a complex vector space of dimension 6 and let $G\subset {\mathbb P}^{14}\simeq {\mathbb P}(\Lambda^2V)$ be the image of the Plucker embedding of the Grassmannian $Gr(2, V)$.
1. Why the degree of $G$ is 14? or in general, how to calculate the degree of a Plucker embedding?
Let ${\mathbb P}^8\simeq L\subse... | https://mathoverflow.net/users/2555 | K3 surface of genus 8 | To be able calculate the degree it is worth to read a bit of Griffiths-Harris about Grassmanians (chapter 1 section 5). To prove that $S$ is $K3$ one needs to caluclate the canonical bundle of $G$, use simple facts about Plucker embedding, use adjunction formula and finally the fact that a simply connected surface with... | 15 | https://mathoverflow.net/users/943 | 14899 | 10,001 |
https://mathoverflow.net/questions/14892 | 14 | I am now taking a course focusing on triangulated geometry. The professor has formulated the Beck's theorem for Karoubian triangulated category. The proof is very simple. Just using the universal homological functor(equivalent to Verdier abelianization)back to abelian settings(in particular, Frobenius abelian category)... | https://mathoverflow.net/users/1851 | Looking for reference talking about relationship between descent theory and cohomological descent | The relationship between cohomological descent and Lurie's Barr-Beck is exactly the same as the relationship between ordinary descent and ordinary Barr-Back. To put things somewhat blithely, let's say you have some category of geometric objects $\mathsf{C}$ (e.g. varieties) and some contravariant functor $\mathsf{Sh}$ ... | 18 | https://mathoverflow.net/users/3931 | 14901 | 10,003 |
https://mathoverflow.net/questions/14897 | 7 | A generic example is ${}\_2 F\_1(\frac{1}{3},\frac{2}{3},\frac{5}{6};\frac{27}{32})=\frac{8}{5}$. So my question: Is there any description of the set of rational points at which the hypergeometric function ${}\_m F\_n$ takes rational values?
The references to the literature where a lot of such examples listed are apr... | https://mathoverflow.net/users/2052 | at which rational points does the Hypergeometric function take rational values | There are theorems that give conditions for the set to be finite even when "rational" is replaced by "algebraic". See the work of Paula Cohen Tretkoff. E.g the following paper is a survey and Theorem 2 is about this:
<http://www.math.tamu.edu/~ptretkoff/martinpub_final.pdf>
| 7 | https://mathoverflow.net/users/2290 | 14905 | 10,006 |
https://mathoverflow.net/questions/14903 | 20 | I am wondering if it is true that **every compact, connected, oriented manifold is cobordant to a simply connected manifold.**
I believe that some sort of surgery will do the trick. Roughly speaking, I want to add handles so that I can kill representative loops. However, I don't know if my surgery process builds a co... | https://mathoverflow.net/users/1622 | Every Manifold Cobordant to a Simply Connected Manifold | Assume that $M^n$ has $\pi\_1$ finitely generated (**Edit:** and n>3). Choose a generator. We will construct (using surgery) a cobordism to $M'$ which kills that generator, and by induction we can kill all of $\pi\_1$. Choose an embedded loop which represents the generator, and choose a tubular neighborhood of the loop... | 20 | https://mathoverflow.net/users/184 | 14907 | 10,008 |
https://mathoverflow.net/questions/14861 | 16 | The HKR theorem for cohomology in characteristic zero says that if $R$ is a regular, commutative $k$ algebra ($char(k) = 0$) then a certain map $\bigwedge^\* Der(R) \to CH^\*(R,R)$ (where $\wedge^\* Der(R)$ has zero differential) is a quasi-isomorphism of dg vector spaces, that is, it induces an isomorphism of graded v... | https://mathoverflow.net/users/132 | Is there a refinement of the Hochschild-Kostant-Rosenberg theorem for cohomology? | Yes there is. It was noted by Kontsevich long time ago that the HKR quasi-isomorphism on cochains can be corrected to give a quasi-isomorphism of dg-algebras and thus induce an $A\_\infty$ quasi-isomorphism of minimal models. The correction is very natural - one needs to compose the HKR map it the contraction by the sq... | 20 | https://mathoverflow.net/users/439 | 14911 | 10,010 |
https://mathoverflow.net/questions/14910 | 2 | Can there exist two non-equivalent equivariant actions of a group $G$ on vector bundle over a $G$ space?
| https://mathoverflow.net/users/2612 | Can there exist two non-equivalent equivariant actions of a group on vector bundle? | Maybe I am miss understanding the question, but it seems the answer is yes.
Take your favorite G-space, mine is $S^1$ with the $\mathbb{Z}/2$-action "flip". Then consider the trivial vector bundles $S^1 \times V$, where $V$ is a $G$-representation. In my favorite example $V = \mathbb{R}$ can be either the trivial rep... | 5 | https://mathoverflow.net/users/184 | 14913 | 10,011 |
https://mathoverflow.net/questions/11633 | 14 | Suppose the continuum is larger than $\aleph\_2$. Does there exist a countably closed notion of forcing that collapses $\aleph\_2$ to $\aleph\_1$, but does not collapse the continuum to $\aleph\_1$? Moreover, does there exist such a forcing notion that is separative and has size continuum? It is known (see below) that ... | https://mathoverflow.net/users/3183 | Is it possible for countably closed forcing to collapse $\aleph_2$ to $\aleph_1$ without collapsing the continuum? | I got the answer from Stevo Todorcevic last weekend at the MAMLS conference in honor of Richard Laver in Boulder, CO. He told me that it is an unpublished result of his that any semi-proper forcing which collapses $\aleph\_2$ collapses the continuum. As countably closed forcing is semi-proper, the answer to my question... | 7 | https://mathoverflow.net/users/3183 | 14946 | 10,034 |
https://mathoverflow.net/questions/11505 | 8 | Does there exist a partial order, nontrivial for forcing, that is countably closed, but whose separative quotient is not countably closed? Supposing the answer is yes, then is there a partial order, nontrivial for forcing, that is countably closed, but is not forcing equivalent to any countably closed separative partia... | https://mathoverflow.net/users/3183 | closure of separative quotients | Stevo Todorcevic answered this question for me at the MAMLS conference in honor of Richard Laver last weekend in Boulder, CO. Apparently, the answer is that examples of forcings that are closed, whose separative quotients are not closed, come up frequently, with one particular example being forcings involving semi-sele... | 4 | https://mathoverflow.net/users/3183 | 14948 | 10,036 |
https://mathoverflow.net/questions/9822 | 5 | Supppose there are integers $a\_1,a\_2,\dots$ and a polynomial $p$ so that the integers $p(a\_1),p(a\_2)...$ satisfy some linear recurrence, i.e. $\sum p(a\_i)x^i$ is a rational function of $x$. Must integers $b\_i\in p^{-1}(p(a\_i))$ so that $\sum b\_ix^i$ is a rational function, necessarily exist?
(The answer is no... | https://mathoverflow.net/users/2384 | Does an inverse polynomial map on the taylor coefficients of a rational function preserve rationality? | If $p(x)=x^d$ then this question coincides with Pisot's d'th root conjecture. A proof is given in [this](http://www.emis.ams.org/journals/Annals/151_1/zannier.pdf) paper of Zannier, I'm not sure if it's the one that Qiaochu was referring to in the comments.
Edit: A more general question was answered in the subsequent... | 3 | https://mathoverflow.net/users/2384 | 14952 | 10,039 |
https://mathoverflow.net/questions/14950 | 4 | Let $z \in \mathbb{C}$. Consider the following statements:
1. The point $z$ can be constructed with straightedge and compass starting from the points $\{ 0,1\}$.
- There is a field extension $K / \mathbb{Q}$ which has a tower of subextensions, each one of degree 2 over the next, and such that $z \in K$
- The field exte... | https://mathoverflow.net/users/3065 | On using field extensions to prove the impossiblity of a straightedge and compass construction | I think they're equivalent. Clearly 3 ==> 2. To see 2 ==> 3, say an extension E/L of fields is 2-filtered if it has a tower of subextensions each of degree 2 over the next. Then note firstly that if E/L and E'/L are 2-filtered, then so is the compositum of E and E' in any extension of L containing both (induct on the l... | 11 | https://mathoverflow.net/users/3931 | 14954 | 10,040 |
https://mathoverflow.net/questions/14960 | 6 | This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, continuous embedding of $H$ into $X$. (This is the abstract Wiener space construction due to Gross, hence the [pr.probability] ta... | https://mathoverflow.net/users/238 | Dense inclusions of Banach spaces and their duals | Yes, if you mean that $i$ is one to one, for an operator $T:X\to Y$ is one to one if and only if $T$\* has weak\* dense range, which means $T$\* has dense range when $X$ is reflexive.
| 7 | https://mathoverflow.net/users/2554 | 14963 | 10,045 |
https://mathoverflow.net/questions/1053 | 10 | Let X be a random variable. Then E(|m-X|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-X|^2) is minimized when m is the mean of x.
A couple weeks ago in a technical stretch of a proof involving the [Lyapunov condition for the central limit theorem](http://en.wikipedia.org/wiki/Lyapunov... | https://mathoverflow.net/users/143 | What m minimizes E(|m-X|^3) for a random variable X? | The minimizer $m$ is the nearest point projection of $X$ onto the subspace of $L^p$ formed by the constant functions ($p=3$ in your case). This $m$ is sometimes called the $p$-prediction or $p$-predictor of $X$. Apparently, this terminology began with [Andô and Amemiya](http://www.ams.org/mathscinet-getitem?mr=189077).... | 5 | https://mathoverflow.net/users/2912 | 14965 | 10,046 |
https://mathoverflow.net/questions/14971 | 9 | A standard example for demonstrating the need for genuinely weak n-categories is that a weak 3-category with unique 0- and 1-cells amounts to the same thing as a braided monoidal category (by an Eckmann-Hilton argument), but were one to use a strict 3-category instead, this would automatically become a fortiori symmetr... | https://mathoverflow.net/users/3902 | Effects of "weak" vs. "strict" categories in Eckmann-Hilton arguments | [Here](http://cheng.staff.shef.ac.uk/degeneracy/eggclock.pdf) is a nice pictorial proof of the Eckmann-Hilton argument in a higher category (made by Eugenia Cheng). One way to read this is as a proof that in a weak 2-category with one 0-cell and one 1-cell, the composition of 2-cells is commutative: in this case each d... | 6 | https://mathoverflow.net/users/49 | 14974 | 10,052 |
https://mathoverflow.net/questions/14961 | 10 | **Background/Motivation**: The facts about the Brauer groups I will be using are mainly in Chapter IV of Milne's book on Etale cohomology (unfortunately it was not in his online note).
Let $R$ be a Noetherian normal domain and $K$ its quotient field. Then there is a natural map $f: Br(R) \to Br(K)$. In case $R$ is r... | https://mathoverflow.net/users/2083 | How to find examples of non-trival kernel of maps between Brauer groups Br(R) -> Br(K) | As R is normal all localizations at height one primes are DVR whence regular and so the question is really one of describing the kernel of the morphism
Br(R) ---> beta(R) = \cap\_P Br(R\_P) (intersection over all ht 1 primes)
beta(R) is called the 'reflexive Brauer group' (see for example ancient papers by Orzech a... | 6 | https://mathoverflow.net/users/2275 | 14976 | 10,053 |
https://mathoverflow.net/questions/14977 | 8 | Yes, this is yet another "foundational" question in valuation theory.
Here's the background: it is a well known classical fact that the dimension (in the purely algebraic sense) of a real Banach space cannot be countably infinite. The proof is a simple application of the Baire Category Theorem: see e.g. [PlanetMath](... | https://mathoverflow.net/users/1149 | Can the algebraic closure of a complete field be complete and of infinite degree? | No, there exists no such field (with a non-trivial norm). A proof can be found in Bosch, Güntzer, Remmert: Non-Archimedean Analysis, Lemma 1, Section 3.4.3.
| 6 | https://mathoverflow.net/users/1961 | 14981 | 10,054 |
https://mathoverflow.net/questions/14964 | 8 | I take the bus to work every day. Every bus has a serial number, but unlike in [the German Tank Problem](http://en.wikipedia.org/wiki/German_tank_problem), I don't know if they are numbered uniformly $1...n$.
Suppose the first $k$ buses are all different, but on day $k+1$ I take one I've been on before. What is the b... | https://mathoverflow.net/users/40544 | Estimate population size based on repeated observation | Maximum likelihood estimate is the smallest $n$ for which
$$\left( 1+\frac{1}{n} \right)^k \leq \frac{n}{n-k+1},$$
that gives a value of $n$ asymptotically equal to $\frac{k^2}{2}$, consistently with the Birthday Paradox. Not sure whether an unbiased estimate would be better for any practical purpose; maybe you do have... | 5 | https://mathoverflow.net/users/2368 | 14988 | 10,058 |
https://mathoverflow.net/questions/14979 | 2 | In a vector space $V$ over a field $F$, a nonempty subset $W$ of $V$ is a subspace if it is closed under addition and scalar multiplication. For a module $M$ over a ring $R$ with identity the similar result is true. Is it true in modules over a nonunital ring?
I should mention that the analogue of the following is no... | https://mathoverflow.net/users/3031 | characterization of a submodule | In Hungerford everything is defined without assuming that rings have
identity (or that they are commutative). At least according to Definition IV.1.3 on p. 171 of
Hungerford, the answer to your first question is affirmative.
Quoting:
Definition 1.3. Let $R$ be a ring, $A$ an $R$-module and $B$ a
nonempty subset of $... | 2 | https://mathoverflow.net/users/2734 | 14993 | 10,060 |
https://mathoverflow.net/questions/14987 | 4 | The following is a result by C.R. Johnson appearing every now and then in the literature.
Let $A$ be an $n \times n$ inverse $M$-matrix. Then
1. All principal minors of $A$ are positive.
2. Each Principal submatrix of $A$ is an inverse $M$-matrix.
I could verify for $3 \times 3$ matrix. But it does not give any ... | https://mathoverflow.net/users/3031 | inverse m-matrix | Let me begin by admitting that I have no knowledge of the subject area in question: the following is my answer as a googlist, not a mathematician.
Having disclosed that, it seems that the result that you want can be found in Section 2.5 of *Topics in Matrix Analysis* by Horn and [C.R.] Johnson.
The extent of my gr... | 6 | https://mathoverflow.net/users/1149 | 14994 | 10,061 |
https://mathoverflow.net/questions/14996 | 5 | Let $S=k[x\_0,...,x\_n]$ be the polynomial ring over a field $k$ and $f\in S$ non-zero and homogeneous. Is it true that $Ext^m(S/(f),S)$ is zero?
This would help me to show that $Ext^m(S/fI,S)\cong Ext^m(S/I,S)(\deg f)$ for $m\geq 2$ and a homogeneous ideal $I$ of codim $\geq 2$. I tried the following approach:
Apply... | https://mathoverflow.net/users/3950 | In what degrees does Ext(S/(f),S) vanish? | It is not true, $Ext^1(S/(f), S)\neq 0$ as Ben pointed out. However, to prove what you want $Ext^m(S/I,S)\cong Ext^m(S/fI,S)(deg(f))$ for $m\geq 2$, just note that
$$Ext^m(S/I,S) = Ext^{m-1}(I,S) $$ for any $I$, any $m\geq 2$ (using $0 \to I \to S\to S/I \to 0$) and $I(-deg(f)) \cong fI$ as $S$-modules.
| 6 | https://mathoverflow.net/users/2083 | 15000 | 10,064 |
https://mathoverflow.net/questions/15005 | 5 | Is there an easy way of remembering the direction of arrows between morphisms in Categories?
The direction of arrows so confuses me: products and co-products, (**EDIT-** Also, pull-backs, pushouts, contra/co-variant functors) and their universal mapping properties. I have to look back into Lang's Algebra and revise ... | https://mathoverflow.net/users/2720 | Remembering arrows' directions in basic Category Theory | I think that the best way to remember these things is to have a good example of each of these constructions in your head. If you remember the direction of the arrows for them, and if the example is "natural" enough the direction will be obvious, you will have the direction in the general case. Just watch out, sometimes... | 11 | https://mathoverflow.net/users/135 | 15006 | 10,066 |
https://mathoverflow.net/questions/15001 | 5 | This question is arose from the question
[Difference between equivalence relations on algebraic cycles](https://mathoverflow.net/questions/11774/difference-between-equivalence-relations-on-algebraic-cycles)
and [the example 3 in lecture 1](http://books.google.com/books?id=-5weWX_YD6sC&lpg=PP1&ots=1r3Rf2JuLx&dq=Lec... | https://mathoverflow.net/users/2348 | Algebraic equivalence VS Numerical Equivalence - An Example. | Numerical and algebraic equivalence coincide up to torsion for codimension $1$ cycles in non-singular projective varieties over $\mathbb C$. Also, the group $Alg\_{\tau}^1(X)/Alg^1(X)$ can be identified with $H^2(X,\mathbb Z)\_{tor}$ (here $Alg\_{\tau}^1(X)$ is the group of cycles who multiple is in $Alg^1(X)$). This i... | 7 | https://mathoverflow.net/users/2083 | 15009 | 10,068 |
https://mathoverflow.net/questions/15016 | 4 | So, in $R-Mod$, we have the rather short sequence
* $\mathrm{Ext}^0(A,B)\cong Hom\_R(A,B) $
* $\mathrm{Ext}^1(A,B)\cong \mathrm{ShortExact}(A,B)\mod \equiv $, equivalence classes of "good" factorizations of $0\in Hom\_R(A,B)\cong\mathrm{Ext}^0(A,B)$, with the Baer sum.
Question:
* $\mathrm{Ext}^{2+n}(A,B) \cong\ ... | https://mathoverflow.net/users/1631 | About higher Ext in R-Mod | They correspond to longer exact sequences under an equivalence relation due to Yoneda. See chapter III.3 (p. 82ff) of MacLane's Homology (or briefly on the wikipedia page for the Ext functor). There are also many online sources for "higher extension modules and yoneda", but MacLane's presentation is clear and describes... | 11 | https://mathoverflow.net/users/3710 | 15017 | 10,074 |
https://mathoverflow.net/questions/14980 | 3 | I've been reading about how hamming codes are used to 'solve' the [Hat Problem](http://en.wikipedia.org/wiki/Hat_puzzle), and I understand how it 'assigns' one person to be the speaker, and how that speaker knows the answer. Everything I read says that it worked $1-\frac1{2^n}$ times, but what I don't understand is why... | https://mathoverflow.net/users/3944 | Hat Problem/Hamming Codes | So I looked at Loepp and Wootters and it does seem to answer your question. Here's roughly how the argument goes.
**Strategy** During the strategy session prior to the game the players agree on some binary [$n,n-r$] Hamming code *C* and a check matrix *H* for *C*. The players also number themselves 1 through *n*. The... | 2 | https://mathoverflow.net/users/3639 | 15026 | 10,080 |
https://mathoverflow.net/questions/14992 | 17 | I am now learning localization theory for triangulated catgeory(actually, more general (co)suspend category) in a lecture course. I found Verdier abelianization which is equivalent to universal cohomological functor) is really powerful and useful formalism. The professor assigned many problems concerning the property o... | https://mathoverflow.net/users/1851 | Why do people "forget" Verdier abelianization functor?(Looking for application) | The problem with respect to applications of the abelianization is that the abelian categories one produces are almost uniformly horrible. More precisely they are just too big to deal with. So using them to produce results which aren't formal statements about some class of triangulated categories seems like it would be ... | 22 | https://mathoverflow.net/users/310 | 15034 | 10,085 |
https://mathoverflow.net/questions/15003 | 33 | Why is it that the vanishing of the integral first Chern class of a compact Kahler manifold is equivalent to the canonical bundle being trivial? I can see that it implies that the canonical bundle must be topologically trivial, but not necessarily holomorphically trivial. Does proving the equivalence require Yau's theo... | https://mathoverflow.net/users/3952 | Two definitions of Calabi-Yau manifolds | I have looked for a while for a proof
which does not use the Calabi-Yau theorem
and nobody seems to know it.
Also, there are plenty of non-Kaehler
manifolds with canonical bundle trivial
topologically and non-trivial as a holomorphic
bundle (the Hopf surface is an easiest
example).
The argument actually uses the Ca... | 25 | https://mathoverflow.net/users/3377 | 15048 | 10,093 |
https://mathoverflow.net/questions/15047 | 1 | I have Z^3/M = Z^3/N = Z\_k where M,N are submodules of Z^3 and Z\_k is cyclic order k.
I would like to say some SL\_3(Z) transformation takes M to N. Is this true? How to show?
| https://mathoverflow.net/users/2031 | equivalence of submodules | It is enough to show that
>
> if $M\subseteq \mathbb Z^3$ be a subgroup such that $\mathbb Z^3/M$ is a cyclic group of order $k$, then there exists $g\in\mathrm{SL}(3,\mathbb Z)$ such that $g(M)=\langle (1,0,0),(0,1,0),(0,0,k)\rangle$.
>
>
>
Let $M\subseteq \mathbb Z^3$ be a subgroup such that $\mathbb Z^3/M$ ... | 1 | https://mathoverflow.net/users/1409 | 15055 | 10,099 |
https://mathoverflow.net/questions/15041 | 15 | I am looking for a matrix $C$ so that the sequence $tr(C^n)$ is dense in the set of real numbers. Equivalently (in the $2 \times 2$ case), find a complex number $z$ so that the sequence $z^n+w^n$ is dense in $\mathbb{R}$ where $w$ is the conjugate of $z$.
| https://mathoverflow.net/users/3960 | Is there a matrix C so that the trace of C^n is dense in R? | The answer is yes, even in the $2 \times 2$ case. Let $q\_1,q\_2,\ldots$ be an enumeration of the rational numbers. Let $Q\_j$ be the closed interval $[q\_j-1/j,q\_j+1/j]$. Let $I\_0=[0,2\pi]$. Let $z=2e^{i \theta}$ for a $\theta \in I\_0$ to be determined.
By induction, we construct positive integers $n\_1 < n\_2 < ... | 21 | https://mathoverflow.net/users/2757 | 15056 | 10,100 |
https://mathoverflow.net/questions/15058 | 8 | How can one construct families of cocompact discrete subgroups of the topological group $\text{SL}\_2(\mathbb{C})$?
Here quaternion algebra's might help, I believe, but I have some difficulties with the construction.
| https://mathoverflow.net/users/1107 | cocompact discrete subgroups of SL_2 | Let $k$ be a number field with one complex place and let $B$ be a quaternion algebra defined over $k$ which ramifies at every real place of $k$ (and perhaps some finite places as well - Hilbert reciprocity implies that the total number of ramified places must be even). Let $\mathcal{O}$ be an order of $B$ and $\mathcal... | 10 | https://mathoverflow.net/users/nan | 15062 | 10,104 |
https://mathoverflow.net/questions/15063 | 5 | Let $F:R \to S$ be an étale morphism of rings. It follows with some work that $f$ is flat.
However, faithful flatness is another story. It's not hard to show that faithful + flat is weaker than being faithfully flat. An equivalent condition to being faithfully flat is being surjective on spectra.
The question:
Is... | https://mathoverflow.net/users/1353 | Weakened conditions for étale + X implies faithfully flat. | The definition of $f:R\to S$ being faithfully flat that I first saw is that $S\otimes\_R-$ is exact and faithful (meaning that $S\otimes\_R M=0$ implies $M=0$). I'm not sure exactly what your definition of "faithfully flat" is, but it looks like you're happy with "flat and surjective on spectra." You get flatness for f... | 7 | https://mathoverflow.net/users/1 | 15069 | 10,107 |
https://mathoverflow.net/questions/15068 | 31 | I've just finished reading Ash and Gross's *Fearless Symmetry*, a wonderful little pop mathematics book on, among other things, Galois representations. The book made clear a very interesting perspective that I wasn't aware of before: that a large chunk of number theory can be thought of as a quest to understand $G = \t... | https://mathoverflow.net/users/290 | Number theory textbook based on the absolute Galois group? | If you want to go further in understanding this point of view, I would advise you to begin learning class field theory. It is a deep subject, it can be understood in a vast variety of ways, from the very concrete and elementary to the very abstract, and although superficially it appears to be limited to describing abel... | 19 | https://mathoverflow.net/users/2874 | 15071 | 10,108 |
https://mathoverflow.net/questions/15075 | 5 | Is there any 'general' topological invariant to tell the difference between $M$ and $N$, where $M$ has homotopy type of a closed manifold and $N$ has homotopy type of a manifold with boundary.
I meant something like homotopy group/ homology group can 'detect' the obstruction of being homotopic to a closed manifold.
... | https://mathoverflow.net/users/3922 | Topological description of Manifold with boundary | If $M$ is a closed manifold, then you can always cross it with an interval to get a
manifold with non-empty boundary that is homotopic to $M$. So the distinction between the two possibilities (closed or non-closed) is not so well-defined in the homotopy category.
On the other hand, suppose that we fix the dimension $... | 24 | https://mathoverflow.net/users/2874 | 15080 | 10,112 |
https://mathoverflow.net/questions/15083 | 3 | One example is the Hurewicz theorem which tells us that (e.g) a CW-cx with only one 0-cell has a nontrivial fundamental group if H\_1 is nontrivial. What other examples are there? (The CW-complexes I have in mind always have exactly one 0-cell, but for the sake of a more wide discussion one could assume this is not gen... | https://mathoverflow.net/users/1161 | What can be said about the homotopy groups of a CW-complex in terms of its (co)homology? | Try looking up some references on rational homotopy theory. Rational homotopy theory studies the homotopy groups tensor Q, so basically you kill all torsion information. If we focus only on homotopy groups tensor Q, the question you ask becomes easier. As Steven Sam mentions in the comments, the homotopy groups of sphe... | 4 | https://mathoverflow.net/users/83 | 15086 | 10,116 |
https://mathoverflow.net/questions/15070 | 1 | This question has been bugging me for quite some time now.
Say we have some $\beta$ smaller than some $\gamma$ and a sequence
$\beta$$\epsilon$ : $\epsilon$ smaller than cf($\beta$) cofinal in $\beta$ and say
we have some sets $A$n$\epsilon$ and each of these $A$n$\epsilon$ has order type less than $\gamma$$n$.... | https://mathoverflow.net/users/3859 | Why is it important to have disjoint sets in a union for the union to make sense w.r.t the order types? | You didn't say so, but you are speaking of ordinals here and ordinal exponentiation.
The problem is that if you take the union of sets not in order, then you can't control the order type. Let me give an easy example to illustrate the point. Suppose that An for each n < ω consists of a single ordinal. If the ordinal o... | 2 | https://mathoverflow.net/users/1946 | 15091 | 10,120 |
https://mathoverflow.net/questions/14995 | 16 | Let $G$ be a group and $CG$ the complex group algebra over the field $C$ of complex number. The group von Neumann algebra $NG$ is the completion of $CG$ wrt weak operator norm in $B(l^2(G))$, the set of all bounded linear operators on Hilbert space $l^2(G)$. Let $f:G \to H$ be any homomorphism of groups. My question is... | https://mathoverflow.net/users/1546 | Is the group von Neumann algebra construction functorial? | By the way, here's the "correct" functorial property. If G and H are abelian, and $f:G\rightarrow H$ is a continuous group homomorphism, then we get a continuous group homomorphism $\hat f:\hat H\rightarrow \hat G$ between the dual groups. By the pull-back, we get a \*-homomorphism $\hat f\_\*:C\_0(\hat G) \rightarrow ... | 8 | https://mathoverflow.net/users/406 | 15093 | 10,122 |
https://mathoverflow.net/questions/15094 | 12 | Do You know any **kind of database** of **presentations of groups**?
It may be on-line or off-line in form of tables, ideally case would be integrated in some Computer Algebra System. I am interested the most in infinite group presentations, but feel free to put here also information about tables of presentations of... | https://mathoverflow.net/users/3811 | Database of finite presentations of used groups | [GAP](http://www.gap-system.org) has the following:
* [Finitely Presented Groups](http://www.gap-system.org/Manuals/doc/ref/chap47.html)
* [Presentations and Tietze Transformations](http://www.gap-system.org/Manuals/doc/ref/chap48.html)
An overview of all data libraries in GAP can be found [here](http://www.gap-sys... | 7 | https://mathoverflow.net/users/3502 | 15095 | 10,123 |
https://mathoverflow.net/questions/14565 | 8 | In various contexts, I have come across results referred to as "big monodromy." A standard arithmetic example is the open image theorem for the image of Galois action on non-CM elliptic curves. A general setup for such a result in algebraic geometry is:
Given a proper, generically smooth map $\pi:X \rightarrow S$ of ... | https://mathoverflow.net/users/81 | Geometric Intuition for Big Monodromy | I found this question a little vague, but let me at least remark on "other examples of things which inhibit big monodromy." Mumford gives an example in section 4 of
D. Mumford, “A note of Shimura’s paper “Discontinuous groups and abelian varieties”,” Math. Ann. 181 (1969), 345–351.
of an abelian variety A whose Ga... | 3 | https://mathoverflow.net/users/431 | 15102 | 10,128 |
https://mathoverflow.net/questions/15107 | 2 | In Bourbaki an *algebra* over a commutative ring $k$ is defined to be a $k$-module $A$ together with a $k$-bilinear map $A \times A \rightarrow A$. We then have the obvious notion of morphisms of $k$-algebras. This terminology is nice, because e.g. Lie algebras are then a special kind of algebras and so on. But in 95% ... | https://mathoverflow.net/users/717 | Algebra / unital associative algebra: better terminology? | If your work requires you to work mostly with unital assosiative $k$-algebras with unital morphisms, define somewhere prominent in your work 'algebra' and 'morphism of algebras' to mean precisely that, and say 'Lie algebras', 'not necessarily associative algebra', 'possibly non unital morphism of algebra', and so on in... | 16 | https://mathoverflow.net/users/1409 | 15108 | 10,131 |
https://mathoverflow.net/questions/15106 | 11 | I am in this, rather friendly, situation:
I have a complex projective space $\mathbb{P}^n$, and there i have a (possibly non-smooth) hypersurface $S$ defined by one irreducible polynomial $P$ of degree $d$.
What i want is to get information about the existence or not of linear subvarieties of $S$, and their maximal... | https://mathoverflow.net/users/3465 | Can projective hypersurfaces contain linear spaces? How big? | I think the key word you are looking for is Fano schemes of $S$. See this
[note](http://www.math.sunysb.edu/~jstarr/papers/appendix3.pdf) by Jason Starr for a reference. For example, if $S$ is smooth and non-degenerate, it can not contain a linear subspace of dimension bigger than half $dim \ S$.
A very interesting ... | 14 | https://mathoverflow.net/users/2083 | 15111 | 10,134 |
https://mathoverflow.net/questions/15121 | 6 | Let $G$ be a graph group and let $S$ be a finitely generated subgroup of $S$.
What torsion can $H\_1(S)$ have?
Let me put this in context: Let $Y$ be a graph, then the corresponding graph group has a generator for each vertex and for each edge we add the relation that the corresponding generators commute. Such groups... | https://mathoverflow.net/users/2985 | subgroups of graph groups | This isn't a proper answer, but here are some remarks.
* Wise's work actually purports to tell you that most (all?) hyperbolic 3-manifold groups *virtually* embed in graph groups - ie, a finite-index subgroup embeds. So you might lose a lot of torsion when you pass to this finite-index subgroup.
* [Haglund and Wise](... | 7 | https://mathoverflow.net/users/1463 | 15126 | 10,141 |
https://mathoverflow.net/questions/15097 | 6 | I was wondering, what a $N$-dimensional simplicial space $X$ should be. Of course the degeneracy maps force the spaces to be nonempty in high dimensions. Currently I have two different versions and i am wondering whether they are equivalent:
1) There are no nondegenerate simplexes above dimension $N$.
2) Let $\Del... | https://mathoverflow.net/users/3969 | Notion of finite dimensional simplicial space | Interesting question!
I'm cautiously optimistic that 1) $\Rightarrow$ 2). As you say, if $X$ satisfies 1), then $L(R(X))\to X$ is a continuous bijection. (Because it's a statement about point-sets, and the thing is true for simplicial sets.)
If $N=0$, it should be easy: $L(R(X))$ is the constant simplicial space, w... | 3 | https://mathoverflow.net/users/437 | 15135 | 10,147 |
https://mathoverflow.net/questions/14492 | 20 | Consider the series
$$ S\_f = \sum\_{x=1}^\infty \frac{f}{x^2+fx}. $$
Goldbach showed that, for integers $f \ge 1$,
$$ S\_f = 1 + \frac12 + \frac13 + \ldots + \frac1f $$
(this follows easily by writing $S\_f$ as a telescoping series).
Thus $S\_f$ is rational for all natural numbers $f \ge 1$.
Goldbach claimed that, fo... | https://mathoverflow.net/users/3503 | Irrational logs and the harmonic series | Please allow me to put my question on top of the list again by turning my comment into an answer. FC's remarks led me to the article "Transcendental values of the digamma function", J. Number Theory 125, No. 2, 298-318 (2007) by Ram Murty and N. Saradha, where Thm. 9 states that the values of S\_f are transcendental fo... | 5 | https://mathoverflow.net/users/3503 | 15140 | 10,149 |
https://mathoverflow.net/questions/15139 | 4 | Suppose
* C and D are two ∞-categories (quasi-categories),
* $F : C \to D$ and $G : C \to D$ are two functors (i.e. 0-simplices in the ∞-category of functors Fun(C,D), which is just the simplicial mapping complex),
* $a : F \to G$ and $b : F \to G$ are two natural transformations (i.e. 1-simplices in Fun(C,D)),
* at ... | https://mathoverflow.net/users/1100 | When are two natural transformations of infinity-categories equivalent? | No, not necessarily, since a transformation involves "higher structure" in addition to its 1-cell components. For example, let C be the "walking arrow" category with two objects 0 and 1 and one nonidentity morphism from 0 to 1, and let D be an abelian group regarded as a (2,1)-category with one object and one morphism,... | 7 | https://mathoverflow.net/users/49 | 15146 | 10,153 |
https://mathoverflow.net/questions/13292 | 15 | Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\sigma$-algebra, and let $\mathbb P$ be a probability measure on $(\Omega, \mathcal F)$.
Denote by $\vec x = (x,y)$ a point... | https://mathoverflow.net/users/238 | Conditional probabilities are measurable functions - when are they continuous? | Since a troll bumped this question to the front page, I might as well answer it. The technology which provides the solution is called [regular conditional probability](http://en.wikipedia.org/wiki/Regular_conditional_probability) or [disintegration](http://en.wikipedia.org/wiki/Disintegration_theorem).
| 3 | https://mathoverflow.net/users/238 | 15149 | 10,154 |
https://mathoverflow.net/questions/15141 | 3 |
>
> **Possible Duplicate:**
>
> [Is there a high-concept explanation for why characteristic 2 is special?](https://mathoverflow.net/questions/915/is-there-a-high-concept-explanation-for-why-characteristic-2-is-special)
>
>
>
There are so many results on primes that either fail for $p=2$ or are not known to b... | https://mathoverflow.net/users/3186 | Why is 2 so odd? | My take on this issue is that p=2 isn't really strange---all small primes are strange, it's just that the smaller you are, the earlier you become troublesome. Look at recent R=T results in the theory of automorphic representations. Nowadays people can prove these sorts of things for $n$-dimensional representations, but... | 17 | https://mathoverflow.net/users/1384 | 15150 | 10,155 |
https://mathoverflow.net/questions/15148 | 2 | Suppose that $F$ is a nonarchimedean local field, $G\_1$ and $G\_2$ are connected (linear) algebraic groups over $F$, and $\phi:G\_1\to G\_2$ is a surjective homomorphism of algebraic groups. Suppose $H$ is a hyperspecial maximal compact subgroup of $G\_1$. Is the image $\phi(H)$ necessarily a hyperspecial maximal comp... | https://mathoverflow.net/users/3513 | Image of a hyperspecial subgroup hyperspecial? | If by "surjective" you mean surjective in the usual sense (for example on $\overline{F}$-points) then maybe you have a problem, because $G\_1(F)$ may not surject onto $G\_2(F)$. So for example $SL(2)$ surjects onto $PGL(2)$ but if $R$ is the integers of $F$ then $SL(2,R)$ is hyperspecial max compact but its image in $P... | 4 | https://mathoverflow.net/users/1384 | 15152 | 10,156 |
https://mathoverflow.net/questions/15115 | 9 | Hi,
I'm looking for examples of groups that are both Hopfian and Co-Hopfian. I have a non trivial (and beautiful, at least to me) example: $\mathrm{SL}(n,\mathbb{Z})$ (with $n>2$).
Do you know others (non trivial)?
Thank you.
| https://mathoverflow.net/users/3958 | Hopfian and Co-Hopfian groups (examples) | Mapping class groups of closed surfaces are both Hopfian and co-Hopfian (the former follows from residual finiteness, and the latter is due to Ivanov-McCarthy).
Out(F\_n) also has both properties (residual finiteness and a theorem of Farb-Handel).
| 22 | https://mathoverflow.net/users/3977 | 15154 | 10,157 |
https://mathoverflow.net/questions/15151 | 25 | Let $R$ be the ring $$R = \prod\_{p\ \text{prime}} \mathbb{F}\_p$$ where $\mathbb{F}\_p$ is the field having $p$ elements.
Is it true that $R$ has a quotient by a maximal ideal which is a field of characteristic zero and contains $\overline{\mathbb{Q}}$?
Motivation: I like the problem and I can't solve it...
It... | https://mathoverflow.net/users/1107 | product of all F_p, p prime | The answer is Yes, and this is the [ultraproduct](http://en.wikipedia.org/wiki/Ultraproduct) construction. Let U be any nonprincipal ultrafilter on the set of primes. This is simply the dual filter to a maximal ideal on the set of primes, containing all finite sets of primes. (In other words, U contains the Frechet fil... | 16 | https://mathoverflow.net/users/1946 | 15155 | 10,158 |
https://mathoverflow.net/questions/15153 | 11 | A famous theorem of Euler is that Zeta(2n) is a rational number times pi^(2n). Work of Kummer, Herbrand, Ribet and others shows that the rational multiplier has number theoretic significance.
For more general L-functions attached to motives, the philosophy has emerged (Deligne, Beilinson, Bloch, Kato, etc.) that (in ... | https://mathoverflow.net/users/683 | Periods and L-values | The ingredient in the Beilinson and Bloch--Kato conjectures is a motive (over ${\mathbb Q}$,
say). If we take the integral cohomology of this motive (mod torsion, say) we get an integral lattice. If we take some kind of Neron model, and take the algebraic de Rham cohomology of this, we get a second integral lattice. No... | 14 | https://mathoverflow.net/users/2874 | 15158 | 10,161 |
https://mathoverflow.net/questions/15159 | 6 | Specifically, is it possible for a non-Noetherian ring $R$ to have $R[x]$ Noetherian? Every reference I've seen for the Hilbert basis theorem only states the direction "$R$ Noetherian $\Rightarrow$ $R[x]$ Noetherian", which would certainly seem to imply that the converse is false. Unfortunately, it's tough to think abo... | https://mathoverflow.net/users/1916 | Converse to Hilbert basis theorem? | If $A$ is an ideal of $R$, then $A[X]$ is an ideal of $R[X]$, right? So an ascending chain of ideals in $R$ which does not stabilize gives you an ascending chain of ideals in $R[X]$ which doesn't stabilize either?
| 14 | https://mathoverflow.net/users/1107 | 15161 | 10,162 |
https://mathoverflow.net/questions/15162 | 2 | Please consider a random walk on a finite N-dimensional lattice with vectors $(x\_1, ..., x\_N)$. We define the origin to be $(0, ..., 0)$ and the target to be at the point in the lattice furthest away from the origin - i.e. $(||x\_1||, ..., ||x\_N||)$ where $||x\_k||$ is the integer length of the lattice in the $x\_{k... | https://mathoverflow.net/users/3248 | Hitting times for an N-dimensional random walk on a lattice with (strictly positive) random integer steps | Assuming you mean Leonid Kovalev's interpretation, the distribution of the hitting time in the $N = 1$ case is the same as the distribution of the number of cycles of a random permutation of $[n]$.
To be more specific, I'll change coordinates. Let $X\_0 = (x\_0^1, \ldots, x\_0^N) = (S, S, \ldots, S)$. Let $X\_1 = (x\... | 3 | https://mathoverflow.net/users/143 | 15164 | 10,163 |
https://mathoverflow.net/questions/15181 | 11 | Can you divide one square paper into five equal squares?
You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper.
| https://mathoverflow.net/users/3056 | Dividing a square into 5 equal squares | The Wallace-Bolyai-Gerwien Theorem theorem says:
Any two simple polygons of equal area are equidecomposable
(where simple means no self intersections and equidecomposable means finitely cut and glued).
For your problem you can take the first polygon to be a unit square and the second to be a sqrt(5) by 1/sqrt(5... | 34 | https://mathoverflow.net/users/3623 | 15184 | 10,175 |
https://mathoverflow.net/questions/15147 | 4 | I am trying to optimize a function of the following form:
$L = \int\_{t=0}^{T}(AR-x)dt$, where A is a system parameter
i.e. I am trying to find the optimum x(t) that minimizes L over all admissible x(t)s. R is related to x using a relation:
$\frac{dR}{dt} = axRY - bR$, where a and b are system parameters and $R(0... | https://mathoverflow.net/users/3560 | Minimizing a function containing an integral | I don't think the problem as posed has an optimal solution. This is a problem in optimal control, typically dealt with by solving the [Hamilton–Jacobi–Bellman equation](http://en.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Bellman%5Fequation). Your problem here is quite general, namely, a *linear* and *unconstra... | 1 | https://mathoverflow.net/users/802 | 15187 | 10,177 |
https://mathoverflow.net/questions/15180 | 12 | Suppose I have the group presentation $G=\langle x,y\ |\ x^3=y^5=(yx)^2\rangle$. Now, $G$ is isomorphic to $SL(2,5)$ (see my proof [here](http://www.artofproblemsolving.com/Forum/viewtopic.php?t=328702)). This means the relation $x^6=1$ should hold in $G$. I was wondering if anyone knows how to derive that simply from ... | https://mathoverflow.net/users/1446 | Deriving a relation in a group based on a presentation | The theory (and practice) of automatic groups is the most generally useful systematic way to deal with these things. There is a nice package written by Derek Holt and his associates called kbmag (available for download here: <http://www.warwick.ac.uk/~mareg/download/kbmag2/> ). There is a book "Word Processing in Group... | 13 | https://mathoverflow.net/users/2784 | 15191 | 10,179 |
https://mathoverflow.net/questions/15186 | 3 | Suppose that $F$ is a nonarchimedean local field, and that $G\_1$, $G\_2$ are connected, simply connected algebraic groups over $F$. Suppose moreover $G\_1$ and $G\_2$ are semisimple. Suppose $H$ is a hyperspecial maximal compact subgroup of $G\_1\times G\_2$. Is $H$ necessarily a product $H\_1\times H\_2$ where the $H... | https://mathoverflow.net/users/3513 | Hyperspecial subgroup of a product of semisimple algebraic groups | To allow all characteristics, the semisimplicity requirement on the Lie algebras should be replaced with the requirement that the $G\_i$ are semisimple as $F$-groups. (In positive characteristic the Lie algebras of connected semisimple groups can often fail to be semisimple.) In fact, the simply connected and semisimpl... | 9 | https://mathoverflow.net/users/3927 | 15192 | 10,180 |
https://mathoverflow.net/questions/15123 | 1 | This question may be trivial, I did not think hard about it.
A friend of mine is looking for an irreducible (reduced) analytic subspace $C \subset \mathbb{C}^2$ with the following property. Let $f \colon C \rightarrow \mathbb{C}$ be the projection on the first factor. He wants that
1) All singular points of $C$ an... | https://mathoverflow.net/users/828 | Riemann surface disconnected at infinity | This is an extended version of my comment. Suppose we stay on the surface $z^2+w^2=1$ but away from the origin. The identity
$|z^2+w^2|^2=|z|^4+|w|^4+2Re((z \bar w)^2)$
tells us that the square of $z \bar w$ has negative real part. The set of complex numbers $\zeta$ such that $Re(\zeta^2)<0$ has two connected componen... | 6 | https://mathoverflow.net/users/2912 | 15194 | 10,182 |
https://mathoverflow.net/questions/15116 | 21 | This is a followup to an earlier question I asked: [Does formally etale imply flat](https://mathoverflow.net/questions/11868/does-formally-etale-imply-flat/)? After some remarks I received on MO I noticed that this was answered to the negative by an answer to an earlier question [Is there an example of a formally smoot... | https://mathoverflow.net/users/917 | Does formally etale imply flat for noetherian schemes? | Every formally smooth morphism between locally noetherian schemes is flat; this is a deep result of Grothendieck. Indeed, the formal smoothness is preserved by localization on target and then likewise on the source, so we can assume we're deal with a local map between local noetherian rings. By EGA 0$\_{\rm{IV}}$, 19.7... | 41 | https://mathoverflow.net/users/3927 | 15200 | 10,186 |
https://mathoverflow.net/questions/15199 | 2 | While studying for a topological groups course, I wondered if we could define the product of uncountably many topological groups such that the product is still a topological group. That is: let $G\_i$ be a topological group with product law $p\_i$ for each $i \in I$ (with $I$ uncountable). We can give $G = \prod\_{i \i... | https://mathoverflow.net/users/3887 | Infinite products of topological groups | You can define the product of an arbitrary family $(G\_i)\_{i \in I}$ of topological groups $G\_i$ by equipping the group-theoretic product $G = \prod\_{i \in I} G\_i$ with the product topology; the product topology is indeed compatible with the group structure (confer Bourbaki, General topology, III.2.9, but it's pret... | 8 | https://mathoverflow.net/users/717 | 15201 | 10,187 |
https://mathoverflow.net/questions/15202 | 24 | I'm taking my first steps in the language of stacks, and would like something cleared up. The intuitive idea of moduli spaces is that each point corresponds to an object of what we're trying to classify (smooth curves of genus g over ℂ, for example). Fine moduli spaces are defined to be the objects that represent the f... | https://mathoverflow.net/users/3238 | Different interpretations of moduli stacks | I'll assure you that you're not crazy. Not only does the idea go through for stacks, but it's impossible (or at least very hard) to make sense of stacks without that idea.
If you're trying to parameterize wigits, you can build a functor F(T)={flat families of wigits over T}. If there is a space M that deserves to be ... | 28 | https://mathoverflow.net/users/1 | 15207 | 10,190 |
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