Newest Questions
159,026 questions
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When are Hilbert schemes smooth?
I know that Hilbert schemes can be very singular. But are there any interesting and nontrivial Hilbert schemes that are smooth? Are there any necessary conditions or sufficient conditions for a ...
- 21k
10
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5
answers
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Compact Kaehler manifolds that are isomorphic as symplectic manifolds but not as complex manifolds (and vice-versa)
What are some examples of compact Kaehler manifolds (or smooth complex projective varieties) that are not isomorphic as complex manifolds (or as varieties), but are isomorphic as symplectic manifolds (...
- 21k
19
votes
6
answers
4k
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Why and how are moduli spaces of (semi)stable vector bundles well-behaved?
The slope of a vector bundle $E$ is defined as $\mu(E) = \deg(E)/\mathrm{rank}(E)$. Then a vector bundle $E$ is called semistable if $\mu(E') \leqslant \mu(E)$ for all proper sub-bundles $E'$. It is ...
- 21k
10
votes
2
answers
924
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Has anyone tabulated 2-knots? Would anyone like to try?
I'd love to have a list of 'small' $2$-knots, for some sense of small. It's not clear what one should filter by, but there are two obvious candidates
Write a movie presentation, and count the frames.
...
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votes
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answer
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A $k$-component link defines a map $T^k\rightarrow \operatorname{Conf}_k S^3$. Does the homotopy type capture Milnor's invariants?
A $k$-component link defines a map $T^k \rightarrow \operatorname{Conf}_k S^3$. Does the homotopy type of this map capture the Milnor invariants?
Some special cases:
$k=2$, no, it's null homologous, ...
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5
votes
2
answers
598
views
How does one think about the "off-diagonal" part of the $R$-matrix?
The universal $R$-matrix of a quantized universal enveloping algebra is typically written as the product of two terms, one only involving elements of the Cartan, and one only involving elements of the ...
- 44.7k
14
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5
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Is the long line paracompact?
A manifold is usually defined as a second-countable hausdorff topological space which is locally homeomorphic to Rn. My understanding is that the reason "second-countable" is part of the definition is ...
- 24.1k
20
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Equivalent statements of the Riemann hypothesis in the Weil conjectures
In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
- 671
13
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3
answers
1k
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How to topologize X(R) when R is a topological ring?
Given a topological ring $R$, under what conditions and in what way, can one induce a topology on the $R$-points of a scheme $X$? For example, if $X$ is $P^n$ or $A^n$, one has natural topology on ...
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5
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3
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758
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Questions about Quivers
Hi,
The definition I have for a Path Algega of a quiver Q is that it is the algebra whose basis is formed by the oriented paths in Q, including the trivial ones. Apparently multiplication is given ...
- 690
28
votes
5
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9k
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Can a quotient ring R/J ever be flat over R?
If $R$ is a ring and $J\subset R$ is an ideal, can $R/J$ ever be a flat $R$-module? For algebraic geometers, the question is "can a closed immersion ever be flat?"
The answer is yes: take $J=...
- 24.1k
6
votes
2
answers
377
views
Bi-embeddability vs. isomorphism
Can anybody give me an example of a "naturally-occurring" algebraic category $C$ in which:
$C$ has two non-isomorphic objects $A$ and $B$ which are bi-embeddable via monic maps; but
$C$ does NOT have ...
- 3,966
18
votes
3
answers
2k
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Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
- 10.5k
44
votes
7
answers
22k
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How do you show that $S^{\infty}$ is contractible?
Here I mean the version with all but finitely many components zero.
- 10.5k
16
votes
2
answers
3k
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Is there an example of a formally smooth morphism which is not smooth?
A morphism of schemes is formally smooth and locally of finite presentation iff it is smooth.
What happens if we drop the finitely presented hypothesis? Of course, locally of finite presentation is ...
- 10.5k
30
votes
2
answers
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When is fiber dimension upper semi-continuous?
Suppose $f\colon X \to Y $ is a morphism of schemes. We can define a function on the topological space $Y$ by sending $y\in Y$ to the dimension of the fiber of $f$ over $y$.
When is this function ...
- 24.1k
11
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5
answers
3k
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Ribbon graph decomposition of the moduli space of curves
What is a ribbon graph? What is the ribbon graph decomposition of the moduli space of curves? What are some good references for this material?
- 21k
13
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4
answers
4k
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Definition of Hochschild (co)homology of a (dg or A-infinity) category
How do you define the Hochschild (co)homology of a dg category or an A-infinity category? I've only seen it defined when the category is equivalent to a category of modules over a dg algebra; then the ...
- 21k
7
votes
2
answers
2k
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What's the correct notion of determinant of a bilinear pairing?
By a pairing on a vector space $V$, I mean a linear map $A : V \otimes V \to R$. If $V$ is $n$-dimensional ($n < \infty$), then I can define the determinant of $A$ by considering the canonical ...
- 54.6k
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3
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18
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2
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Zeta-function regularization of determinants and traces
The short answer to my question may be a pointer to the right text. I will give all the background I know, and then ask my questions in list form.
Let A be an operator (on an infinite-dimensional ...
- 54.6k
8
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Is there a finite-index finite-depth II$_1$ subfactor which is more than $7$-super-transitive?
Background: See Noah and Emily's posts on subfactors and planar algebras on the Secret Blogging Seminar.
There are plenty of examples of $3$-super-transitive (3-ST) subfactors; Haagerup, $S_4 < ...
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2
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What's the sense in which A_\infty algebras are "deformable"?
I realise this is a very vague question! I've heard people say that A∞ algebras are the right homotopy-theoretic generalization of usual associative algebras, because you can deform them. What ...
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5
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Why is the Hochschild homology of k[t] just k[t] in degrees 0 and 1?
Background: the Hochschild homology of an associative algebra is the homology of the complex
$$ \ldots \longrightarrow A \otimes A \otimes A \longrightarrow A \otimes A \longrightarrow A$$
where ...
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10
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Resources on invariant theory
What are resources on invariant theory? Basically I've run into a need to teach myself some of the basics of invariant theory and was looking for a good place to start. I'd prefer online / freeish ...
5
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3
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Does the "continuous locus" of a function have any nice properties?
Suppose $f:\mathbf{R}\to\mathbf{R}$ is a function. Let $S=\{x\in \mathbf{R}|f\text{ is continuous at }x\}$. Does $S$ have any nice properties?
Here are some observations about what $S$ could be:
$S$ ...
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4
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Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?
An anonymous question from the 20-questions seminar:
Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?
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6
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When does the sequence of iterates of a rational function converge?
Darsh asks at the 20-questions seminar:
Let $f:P^1 \rightarrow P^1$ be rational function.
Can you say when the sequence $\{ f^n(x)\}_n=\{ x,f(x),f(f(x)),\cdots\} $ converges? What about the sequence ...
- 1,059
12
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4
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877
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Can you describe the image of the exponential map $B(H)\to B(H)$?
James Tener asks at the 20-questions seminar:
The exponential map $\exp:B(H)\to B(H)$ is just defined by its Taylor series. Can you describe its image?
- 1,059
7
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2
answers
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Is there a category in which finite limits and directed colimits *don't* commute
Andrew Critch asks at the 20-questions seminar:
In Set and AbGrp (the categories of sets and abelian groups, respectively), finite limits commute with directed colimits. As an example, if you're ...
- 1,059
27
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7
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How do you see the genus of a curve, just looking at its function field?
Yuhao asked in the 20-questions seminar:
The genus of a curve is a birational invariant; the function field of a curve determines it up to birational equivelance.
How do you see the genus directly ...
- 1,059
18
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2
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4k
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When do fibre products of smooth manifolds exist?
Harold asks what conditions on $f:M\to L$ and $g:N\to L$, both smooth maps of smooth manifolds, ensures the existence of the fibre product $M \times_L N$.
- 1,059
14
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Model category structures on categories of complexes in abelian categories
Section 2.3 of Hovey's Model Categories book defines a model category structure on Ch(R-Mod), the category of chain complexes of R-modules, where R is a ring. Lemma 2.3.6 then essentially states (I ...
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2
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are deformations of torsion modules always torsion?
Let's say I have a field $\mathbb{K}$ and a flat family of $\mathbb{K}[t]$-modules $M$ over the formal disk $Spec \mathbb{K}[[h]]$.
Now, assume that $M/hM$ is torsion as a $\mathbb{K}[t]$-module (...
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Can one check formal smoothness using only one-variable Artin rings?
Let $f:X\rightarrow Y$ be a morphism of schemes over a field $k$. Can one check that $f$ is formally smooth using only Artin rings of the form $k^{\prime}\left[t\right]/t^{n}$, where $k^{\prime}$ is ...
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Atiyah-MacDonald, exercise 2.11
Let $A$ be a commutative ring with $1$ not equal to $0$. (The ring A is not necessarily a domain, and is not necessarily Noetherian.) Assume we have an injective map of free $A$-modules $A^m \to A^n$...
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2
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Critical points on a fiber bundle
Consider a (smooth) bundle $E\to B$, and a (smooth) function $f: E\to\mathbf{R}$ on the total space. Then it makes sense to talk about the derivatives of $f$ along the fibers. Let $C$ be the ...
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How can you tell if a space is homotopy equivalent to a manifold?
Is there some criterion for whether a space has the homotopy type of a closed manifold (smooth or topological)? Poincare duality is an obvious necessary condition, but it's almost certainly not ...
- 31.2k
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Is every finite-dimensional Lie algebra the Lie algebra of an algebraic group?
Harold Williams, Pablo Solis, and I were chatting and the following question came up.
In Lie group land (where you're doing differential geometry), given a finite-dimensional Lie algebra g, you can ...
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Linearity of the inner product using the parallelogram law
A norm on a vector space comes from an inner product if and only if it satisfies the parallelogram law. Given such a norm, one can reconstruct the inner product via the formula:
$2\langle u,v\rangle ...
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is there a good computer package for working with bicomplexes?
I'm interested in working with bicomplexes of modules over polynomial rings, specifically tensoring them together, and the operation of taking cohomology in one direction, and then the other. Is ...
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Is there a good computer package for working with complexes over non-commutative rings?
I'm interested in doing computations with certain non-commutative rings, most of which involve taking derived tensor products. Does anyone know of a computer algebra package which will find ...
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What is the exact statement of "there are 27 lines on a cubic"?
I think there was a theorem, like
every cubic hypersurface in $\mathbb P^3$ has 27 lines on it.
What is the exact statement and details?
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121
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What do epimorphisms of (commutative) rings look like?
(Background: In any category, an epimorphism is a morphism $f:X\to Y$ which is "surjective" in the following sense: for any two morphisms $g,h:Y\to Z$, if $g\circ f=h\circ f$, then $g=h$. Roughly, "...
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What is the "right" hermitian structure on tensor products of quantum group representations?
This is pretty specific, but there are some experts around.
So, in Chari & Pressley, it's explained that in the standard *-structure, every irreducible, finite-dimensional representation of a ...
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27
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6
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What is a topos?
According to Higher Topos Theory math/0608040 a topos is
a category C which behaves like the
category of sets, or (more generally)
the category of sheaves of sets on a
topological space.
Could one ...
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14
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How to understand character sheaves
There's a well-known series of articles by Lusztig about Character Sheaves. They have important connections to many things in (geometric) representation theory, e.g. 0904.1247
How to understand these ...
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Is there an example of a scheme X whose reduction X_red is affine but X is not affine?
For Noetherian schemes this follows from Serre's criterion for affineness by a filtration argument.
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7
votes
2
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509
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How can you find small denominators inside triangles?
Darsh asked over at the 20 questions seminar:
Take a triangle in R^2 with coordinates at rational points. Can we find the smallest denominator point in the interior? (Consider denominator of an ...
- 7,800
5
votes
3
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467
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A name for star-graph with long "laces"
An $l$ long $k$-star is a graph with centeral vertex $o$ which is connected to $k$ line graphs of length $l$.
For example a 2-long 3-star looks like:
x1-x1-O-x2-x2
|
x3-x3
$o$ is the ...
