All Questions
16 questions
3
votes
1
answer
529
views
Zeros of a function defined on $\mathbb{S}^2 \times \mathbb{S}^2$
Let $u$ be a smooth function on the sphere, and for each $y \in \mathbb{S}^2$, let $R_y$ be the $180^\circ$ rotation about the vector $y$. For each pair $(x, y) \in \mathbb{S}^2 \times \mathbb{S}^2$, ...
- 434
0
votes
0
answers
70
views
Example of DS with a dense trajectory in the whole state space
Let $U \subset \mathbb{R}^n$ be an open and connected set. We assume there is a vector field $F \in \mathcal{C}^1(\overline{U})$ giving rise to a DS ($\overline{U}$ denotes the closure)
$$\dot{\mathbf{...
- 247
36
votes
1
answer
3k
views
Is there a general theory of "compactification"?
In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of ...
- 63.9k
6
votes
0
answers
376
views
Topological Singularities in Affine Varieties
Let $X$ be an affine variety over $\mathbb{C}$. Let $x\in X$.
If $x$ is non-singular, then $x$ is locally holomorphic (in the Euclidean topology). See here for a relevant MO post.
By results of ...
- 8,529
8
votes
1
answer
595
views
Question about taking the Zariski closure in $\mathbb{A}_{\mathbb{R}}^n$
Let $\mathbb{A}_{\mathbb{R}}^n$ be $\mathbb{R}^n$ endowed with the Zariski topology, where closed sets are algebraic sets (in $\mathbb{R}^n$) defined by real polynomials.
Suppose $V \subseteq \mathbb{...
- 3,625
1
vote
0
answers
112
views
Question regarding the image of a polynomial map containing a small box
I have the following question, which intuitively seems it should be true but I wasn't sure how to prove it rigorously.
Let $\delta, \varepsilon > 0$.
Let $\Psi_i(w_1, w_2, \mathbf{v})$ be a ...
- 3,625
2
votes
0
answers
35
views
If the set of non-0 stalks of F is relatively open, is the same true of its Verdier dual?
Let $X$ be a complex manifold, $F$ a bounded complex of $\Bbb C_X$-modules with constructible cohomology. If the set $\{x: F_x\neq0\}$ is relatively open (i.e. open in its closure), is the same true ...
- 3,079
16
votes
3
answers
3k
views
Physical interpretations/meanings of the notion of a sheaf?
I fairly understand the fiber bundles, both the mathematical concept of fiber bundles and the physics use of fiber bundles. Because the fiber bundles are tightly connected to the gauge field theory in ...
- 10.5k
18
votes
3
answers
3k
views
What are parabolic bundles good for?
The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate ...
- 2,751
-3
votes
1
answer
330
views
Loop space of manifold [closed]
Question A: The free loop space of a manifold is also a manifold?
Question B: The free loop space of an algebraic variety is also a algebraic variety ?
Are these questions asked or answered anywhere ...
- 189
3
votes
2
answers
459
views
Smooth paths on affine varieties
I have the following question which is in some way related to an application of Randell Isotopy Theorem to complex hyperplane arrangements.
Let $h,k\geq1$ be integer numbers and let $F_{1},\ldots,F_{...
- 741
3
votes
2
answers
517
views
Connected complement manifold
I'm working on some problem in algebraic geometry. I need a reference to the following result:
Let $h\in\mathbb{N}$ with $h\geq1$ and let $F\in\mathbb{C}\left[x_{1},\ldots,x_{h}\right]$
be a non ...
- 741
17
votes
5
answers
2k
views
What abstract nonsense is necessary to say the word "submersion"?
This question is closely related to these two, but the former doesn't go far enough and the latter didn't attract much attention, and anyway I want to ask the question slightly differently.
Recall ...
- 54.6k
2
votes
2
answers
657
views
Varieties, Frechet Completions, and Regular Functions
Take an algebraic variety $V$, and its set of smooth functions $C^{\infty}(V)$. One can endow $C^{\infty}(V)$ with a canonical locally convex topology (the seminorms are defined using the local ...
- 1,462
13
votes
11
answers
4k
views
Are nets and filters useful in geometry and topology?
Many results in topology can be restated using the concepts of nets and ultrafilters. This seems to be of interest for set theorists, maybe even logicians. But for geometers and topologists, those who ...
- 15.4k
8
votes
1
answer
688
views
Universal covers of domains in complex projective space
The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain ...
- 4,485