Recently Active Questions
159,085 questions
5
votes
2
answers
720
views
Relationship between Line Bundles with isomorphic ring of sections
Suppose two positive holomorphic line bundles $L_1 \to X_1, L_2\to X_2$ over two projective complex manifold $X_1, X_2$ have isomorphic ring of sections $R=R_1=R_2$ where $R_i=\oplus_{m=0}^\infty\...
- 44.7k
17
votes
1
answer
2k
views
Mathematical solution for a two-player single-suit trick taking game?
The question on games and mathematics that appeared recently on mathoverflow
(Which popular games are the most mathematical?)
reminded me of a problem I encountered some time ago : starting with the ...
CommunityBot
- 1
29
votes
3
answers
5k
views
finite generated group realized as fundamental group of manifolds
This is discussed in the standard textbooks on algebraic topology.
Pick a presentation of the group $G = \langle g_1,g_2,...,g_n|r_1,r_2,...r_m \rangle$
where $g_i$ are generators and $r_j$ are ...
CommunityBot
- 1
6
votes
2
answers
2k
views
Examples of one-dimensional non-Cohen Macaulay rings
Can you offer some examples of such rings, other than $\frac{k[x,y]}{(x^{2}, xy)}$. Thanks.
- 30.6k
8
votes
2
answers
2k
views
Functorial characterization of morphisms of schemes
This question is akin in spirit to this one:
Functorial characterization of open subschemes?
In the above MO question, a "functorial" characterization is given for closed immersions and open ...
CommunityBot
- 1
6
votes
2
answers
356
views
How strict can I be in the definition of "2-group"?
Recall that a group is an associative, unital monoid $G$ such that the map $(p_1,m) : G \times G \to G\times G$ is an isomorphism of sets. Here $p_1$ is the first projection and $m$ is the ...
- 27.5k
8
votes
3
answers
1k
views
Derivations of C(X)? or Why Must Supermanifolds be Smooth?
What are the derivations of the algebra of continuous functions on a topological manifold?
A supermanifold is a locally ringed space (X,O) whose underlying space is a smooth manifold X, and whose ...
- 27.5k
6
votes
1
answer
779
views
primitive of an exact differential form with special properties
We were working on a smoothing problem and ran across the apparently simple following question:
X is a triangulated smooth manifold of dimension $n$, and $\alpha$ is an exact differential form of ...
CommunityBot
- 1
2
votes
1
answer
279
views
Immersion with respect to a connection?
In the paper hep-th/9712042v2, p. 20, the following setup is given:
A complex manifold M and an n+1-dimensional vector bundle V on it. V has an underlying real bundle $V_{\mathbb{R}}$ with a flat ...
- 405
2
votes
1
answer
226
views
If a category is "monadic", is it necessarily so in a unique manner?
Just a minor curiosity that's flitted across my mind, but that's (part of) what this site's for, right?:
Is it possible for Hom(a, -) and Hom(b, -) to both be monadic functors from C to Set, for non-...
- 27.2k
25
votes
3
answers
2k
views
product of all F_p, p prime
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 ...
- 65.4k
4
votes
3
answers
1k
views
What are some conserved quantities of Poisson brackets?
Poisson brackets play the very important roles in Symplectic geometry and Dynamical system. I'm interested in some conserved quantities of Poisson brackets.
Let's say we are working on T^n x R^n (T^n ...
- 15.4k
2
votes
1
answer
289
views
ODE system question
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 ...
- 15.4k
3
votes
1
answer
1k
views
The Arnold cat map
How can I compute the SRB measure for the cat map? Also any pointers to references for obtaining Markov partitions and recurrence times would be lovely. Thanks
- 15.4k
5
votes
4
answers
809
views
Does it help to learn statistical mechanics in order to learn thermodynamic formalism?
Does it help to learn statistical mechanics or thermodynamics (as in physics or mathematical physics) in order to learn thermodynamic formalism: the study of equilibrium states, Gibbs measure, maximal ...
- 15.4k
3
votes
1
answer
403
views
Anosov diffeomorphisms and the chaotic hypothesis
There is a well-known "chaotic hypothesis" dating from 1995 or so in statistical physics that suggests that classical statistical-physical systems should be "effectively" Anosov. I won't get into the ...
- 15.4k
16
votes
2
answers
2k
views
Why is every symplectomorphism of the unit disk Hamiltonian isotopic to the identity?
That is, for any symplectomorphism $\psi: D^2 \to D^2$, there should be a time-dependent Hamiltonian Ht on D2 such that the corresponding flow at time 1 is equal to $\psi$.
I found this in claim a ...
CommunityBot
- 1
7
votes
0
answers
170
views
Unbounded energy growth in a Hamiltonian system
Does there exist an orbit with unbounded velocity in the system
$\ddot x = (-1)^{[t]+[x]}$, where $[*]$ denotes the integer part of *?
- 15.4k
17
votes
1
answer
760
views
Examples for which it is not known if Grothendieck's Standard Conjectures hold.
What is the simplest example of a projective variety for which it is not known if Grothendieck's Standard Conjectures hold?
What is the simplest example of a flag manifold for which it is not known ...
- 16.9k
2
votes
0
answers
254
views
Forgetting extra structure inducing Symmetries
This is a major edit of the original post after receiving helpful comments.
It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
CommunityBot
- 1
0
votes
2
answers
188
views
Intrinsic construction of other Galois extensions
Suppose L/F is a degree 4 Galois extension of global fields, with $Gal(L/F)=Z/2\times Z/2$. Then there are three degree 2 subextensions $K_i/F$ with $i=1,2,3$. Now start with $K_1/F$, how to give an ...
- 32.6k
2
votes
1
answer
226
views
Name of the Marshall Hall paper in which he proved that the intersection of all subgroups of a fixed finite index is again finite index?
can someone please tell me? I couldn't find a reference in the paper I was reading.
- 91
2
votes
2
answers
2k
views
What is the transcendence degree of Q_p and C over Q?
Is the tr.deg of Q_p over Q 1? and what about C over Q?
- 65.4k
11
votes
1
answer
875
views
An arithmetic highest weight theory?
I apologize if these questions seem naive or loaded.
Is there an analogous theory of highest weights for irreducible finite-dimensional representations of Lie algebras of algebraic group (or perhaps ...
- 7,108
22
votes
1
answer
3k
views
fpqc covers of stacks
Artin has a theorem (10.1 in Laumon, Moret-Bailly) that if $X$ is a stack which has separated, quasi-compact, representable diagonal and an fppf cover by a scheme, then $X$ is algebraic. Is there a ...
- 7,108
6
votes
2
answers
1k
views
Does projectiveness descend along field extensions?
Background: Properness is a much more robust notion than projectiveness. For example, properness descends along arbitrary fpqc covers (see, for example, Vistoli's Notes on Grothendieck topologies, ...
- 7,108
13
votes
3
answers
3k
views
How cavalier can I be when demanding a category have direct sums?
In my meaning, a direct sum in a category should really be called a "biproduct". If $X,Y$ are objects, then a direct sum $X \oplus Y$ is an object $Z$ along with isomorphisms $\hom(Z,A) = \hom(X,A) \...
- 25.2k
8
votes
1
answer
694
views
From symmetric groups to symplectic groups?
The question is self-contained finite group theory but the motivation requires more background.
The finite groups I am interested in are the groups $Sp(n,F_3)$. For $n$ even these are the usual ...
- 13.3k
20
votes
3
answers
3k
views
Exercises in Hodge Theory
I was wondering: is there a good place to find exercises in Hodge theory? Mostly computations and proving small (preferably nifty) theorems, is what I have in mind. Something roughly like the ...
- 28.9k
10
votes
3
answers
2k
views
Is $Sym^n (V^*) \cong Sym^n (V)^\ast$ naturally in positive characteristic?
Background/motivation
It is a classical fact that we have a natural isomorphism $Sym^n (V^*) \cong Sym^n (V) ^\ast$ for vector spaces $V$ over a field $k$ of characteristic 0. One way to see this is ...
- 65.4k
8
votes
2
answers
8k
views
What does "supersingular" mean?
Are supersingular primes and supersingular elliptic curves related?
(this was essentially a subquestion in my earlier question, but still looks sufficiently different to me to deserve a separate post)...
CommunityBot
- 1
7
votes
1
answer
1k
views
Co-Objects are better [closed]
This is a rather vague question, but perhaps we can talk about it.
There are two types of mathematical objects (which don't exclude each other):
A) There is a good description of morphisms defined ...
- 57.6k
10
votes
0
answers
845
views
What are the relationship between various definitions for quasi coherent sheaves?
It seems that there are many definitions of quasi coherent sheaves(modules). There is a nice page on nLab quasi coherent sheaves
My questions are:
Are there any other definitions of quasi coherent ...
- 47.3k
0
votes
3
answers
818
views
How do we construct (in a vector space) a chain of countable dimensional subspaces that can only be bounded by an subspace of uncountable dimension?
In more rigorous language:
" V: a vector space having an uncountable base
S: The set of subspaces of V that have countable dimension.
Can we construct explicitly a chain in the poset S (ordered by ...
- 237k
5
votes
3
answers
759
views
How to estimate the growth of a recurrence sequence
If we have a linear recurrence sequence where each term depends on all previous terms, say
$a_n = \sum_{k=0}^{n-1} \binom{n}{k} a_k, \quad a_0 = 1$
is there any way to estimate the growth of a_n in ...
- 39.9k
12
votes
4
answers
2k
views
triviality of fibre bundles
are there some general method in judging if a fible bundle is trivial?
At least,for vector bundles,there is a well-developed theory,that is Charicteristic Classes.the triviality of vector bundles is ...
- 23.5k
3
votes
0
answers
249
views
Inverse limits of finite unramified covers, and the need for formally unramified covers
This question evolves mostly from my surprise at the answer to a previous question of mine (Is every flat unramified cover of quasi-projective curves profinite?). My surprise was at that inverse ...
CommunityBot
- 1
14
votes
2
answers
2k
views
Two questions about units in Number Fields
If $K/\mathbb{Q}$ is a number field which is not $\mathbb{Q}$ or a quadratic imaginary field then, by the Dirichlet unit theorem it has a unit of infinite order. Is there a simple proof of this fact ...
- 32.6k
4
votes
2
answers
886
views
Homeomorphism onto a closed subset of a scheme that isn't a closed immersion
More precisely, is there a map of schemes $X$ --> $Y$ such that $f$ gives a homeomorphism between $X$ and a closed subset of $Y$, but the corresponding map on sheaves is not surjective?
- 57.6k
24
votes
1
answer
2k
views
How many ways are there to globalize Harish Chandra modules?
Suppose $G$ a reductive Lie group with finitely many connected components, and suppose in addition that the connected component $G^0$ of the identity can be expressed as a finite cover of a linear Lie ...
CommunityBot
- 1
3
votes
1
answer
2k
views
Infinite products of topological groups
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 $G_i$ ...
- 65.4k
4
votes
2
answers
1k
views
What's the space of smooth functions in L^2(R)?
Maybe this question is not appropriate here.
Let R be real numbers, and L^2(R) the square integrable functions, now what's the space of smooth functions in L^2(R)?
Edit:Sorry for the ambiguity. Let'...
- 25.8k
9
votes
4
answers
34k
views
limsup and liminf for a sequence of sets
how does limsup and liminf for a sequence of sets, apply to probability theory. any real world examples would be much appreciated
- 8,532
-2
votes
3
answers
1k
views
sl(2)-modules... [closed]
I'm trying to learn some Lie algebra without much knowledge of representation theory. While being asked to prove some things about an sl(2)-module, why can one assume that the module is irreducible ...
- 51
24
votes
2
answers
1k
views
Different interpretations of moduli stacks
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 ...
- 24.1k
5
votes
1
answer
283
views
how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space?
So, let's say one has an action of $GL_n$ on an algebraic variety $X$ over a field $k$, and two objects $F,G$ in the equivariant derived category (i.e., the derived category of constructible sheaves ...
- 6,162
-3
votes
2
answers
1k
views
Finite versus infinite on non-Hausdorff topologies [closed]
Question: Does there exist some real-valued function $f(x)$ where $f: \mathbb{R} \to \mathbb{R}$, for which $\lim_{x \to \infty}$ converges on a non-Hausdorff topology but does not converge on a ...
CommunityBot
- 1
3
votes
1
answer
376
views
Hyperspecial subgroup of a product of semisimple algebraic groups
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 ...
- 1,233
21
votes
1
answer
2k
views
Does formally etale imply flat for noetherian schemes?
This is a followup to an earlier question I asked: 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 ...
CommunityBot
- 1
3
votes
4
answers
385
views
Where can i find 'getting started' resources for statistical prediction
I wanted to learn prediction, forecasting etc. I also have time series data on millions of online videos. I would like to test out prediction algorithms etc on this data set, for eg. Linear Prediction,...
CommunityBot
- 1