Recently Active Questions
159,054 questions
22
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3
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7k
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Chevalley Eilenberg complex definitions?
In Weibel's An Introduction to Homological Algebra, the Chevalley-Eilenberg complex of a Lie algebra $g$ is defined as $\Lambda^*(g) \otimes Ug$ where $Ug$ is the universal enveloping algebra of $g$. ...
- 47.3k
3
votes
1
answer
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A bijection between "symplectic" partitions and bi-partitions via Springer correspondance
The following is from this talk: http://www.maths.usyd.edu.au/u/anthonyh/piecestalk.pdf, Slide 14.
The Springer correspondence gives bijections
SO2n+1 \ N(so2n+1) ↔ {(μ; ν) | μi ≥ νi − 2, νi ≥ μi+1}...
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0
votes
3
answers
3k
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min cut and max cut
If you want to maximize a function f(x), you can do this by minimizing -f(x). Naively it seems like an analogous trick could convert a max cut problem into a min cut problem, however this is ...
- 4,462
1
vote
2
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604
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Homomorphism between exterior powers of a free module of finite rank
I´m looking for homomorphisms between exterior powers of a free module M of rank m
ΛmR M → Λm-1R M
Exactly, I´m looking for an explicit isomorphism
M → Hom R (ΛmR M , Λm-1R M)
I compare the ranks ...
- 47.3k
5
votes
3
answers
369
views
Are injective Omega-spectra the S-local objects of symmetric spectra for some class S?
I am trying to read the Hovey-Shipley-Smith article as defining the stable model structure on symmetric spectra as a left Bousfield localization (as explained on nLab) of the projective level model ...
- 8,167
5
votes
2
answers
766
views
Can we distinguish the algebraic and continuous duals of a Banach space without choice (or HBT)?
The algebraic dual of a normed vector space is the space of all linear functionals to the ground field (either $\mathbb{R}$ or $\mathbb{C}$ for this question). The continuous dual is the subspace of ...
CommunityBot
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8
votes
4
answers
748
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Tensored Over Abelian Groups?
Suppose I have an category additive category C (i.e. the hom sets are enriched in abelian groups and there are finite direct sums). Suppose further that C has cokernels. Then I can make C tensored ...
- 26.8k
4
votes
1
answer
1k
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alternating sums of terms of the Vandermonde identity
Using Vandermonde's identity we know:
$\sum_{i=0}^k \binom{k}{i}\binom{n-k}{n/2-i} = \binom{n}{n/2}$.
I'm interested in how close the alternating sum is to 0 when k << n. I.e.,
$\sum_{i=0}^k (...
- 85.6k
2
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2
answers
1k
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Why (and whether) is any smooth embedded torus in R^4 isotopic to an embedded Lagrangian torus?
The question is pretty self-explanatory; we are dealing with the standard symplectic structure on ℝ4.
Some background: I'm reading the thesis "Lagrangian Unknottedness of Tori in Certain ...
- 2,927
8
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1
answer
655
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Coherent spaces
In Proofs and Types, Girard discusses coherent (or coherence) spaces, which is defined as a set family which is closed downward ($a\in A,b\subseteq a\Rightarrow b\in A$), and binary complete (If $M\...
- 236k
7
votes
1
answer
1k
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N=(2,2) supersymmetry in two-dimensional Euclidean space
Can anyone describe (or give a reference for) the 2d superspace formulation of N=(2,2) SUSY in Euclidean signature?
I'm reading Hori's excellent introduction to QFT in the book 'Mirror symmetry', and ...
- 33.3k
4
votes
1
answer
988
views
Torsion line bundles with non-vanishing cohomology on smooth ACM surfaces
I am looking for an example of a smooth surface $X$ with a fixed very ample $\mathcal O_X(1)$ such that $H^1(\mathcal O(k))=0$ for all $k$
(such thing is called an ACM surface, I think) and a globally ...
- 30.6k
16
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2
answers
819
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Spin structures on 7-dimensional spherical space forms
Background
Let $M$ be a spin manifold and let $\Gamma$ be a finite group acting freely and isometrically on $M$ in such a way that $M/\Gamma$ is a smooth riemannian manifold. The quotient will be ...
- 33.3k
5
votes
2
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1k
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Elementary theory of finite fields
I read on Ax's article that the elementary theory of finite fields is decidable if one assumes the continuum hypothesis to be true. What about if one assumes the hypothesis to be false?
- 236k
14
votes
4
answers
2k
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Is a polynomial with 1 very large coefficient irreducible?
I am asking for some sort of generalization to Perron's criterion which is not dependent on the index of the "large" coefficient. (the criterion says that for a polynomial $x^n+\sum_{k=0}^{n-1} a_kx^k\...
- 16.9k
3
votes
4
answers
1k
views
Examples of divisors on an analytical manifold
I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. I have browsed through Hartshone's book but everything is ...
- 2,132
3
votes
3
answers
687
views
Nature of Invertible Sheaves in which there are no global sections.
EDIT: Let me try to make the question clearer.
Consider the invertible sheaves $\mathcal{O}(d)$ over the projective space $\mathbb{P}^n$ where $d\in \mathbb{Z}$. Now, if $d>0$, among many ...
- 2,132
24
votes
2
answers
4k
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Why is the decomposition theorem awesome?
I saw the statement of the decomposition theorem for perverse sheaves sometime ago. I know that (modulo most of the details) it implies some big theorems in algebraic geometry and gives new proofs for ...
CommunityBot
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7
votes
1
answer
718
views
Ways to characterize supersingular primes?
I've read the definition, and it basically says p is a supersingular prime iff
the fundamental domain of a group generated by \Gamma(p) and a matrix ((0, 1), (-p, 0)) is rational.
And there's a ...
CommunityBot
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15
votes
5
answers
3k
views
Can we count isogeny classes of abelian varieties?
Let's fix a finite field F and consider abelian varieties of dimension g over F. Can we say how many isogeny classes there are? Is it even clear that there's more than one isogeny class? For g=1, ...
CommunityBot
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10
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2
answers
393
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Counting points on varieties of low codimension
The graduate students here at MIT have been thinking about questions like the following: Over $\mathbb{F}\_q$, how many symmetric matrices are there with nonzero determinant and $0$'s on the diagonal? ...
- 65.4k
3
votes
2
answers
536
views
Broken Symmetry
I have a tangled web of ideas about natural transformations, vector spaces, equivalence classes, local coordinates, etc. in my head that I'm trying to unravel. So here are some of the questions I ...
CommunityBot
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5
votes
5
answers
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Cardinality of Equivalence Classes of Cauchy Sequences
What's the cardinality of a single equivalence class of Cauchy sequences in ℚ?
To clarify, I'm not asking for the cardinality of the real numbers, but for the cardinality of the set of Cauchy ...
- 236k
16
votes
5
answers
2k
views
Elliptic Curves over F_1?
Is there an notion of elliptic curve over the field with one element? As I learned from a previous question, there are several different versions of what the field with one element and what schemes ...
CommunityBot
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7
votes
1
answer
1k
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Encoding fuzzy logic with the topos of set-valued sheaves
One of the canonical examples used by Barr & Wells in order to motivate the use of topoi is that we can construct a theory for fuzzy logic and fuzzy set theory as set-valued sheaves on a poset (...
- 1,292
6
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3
answers
3k
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Existence of projective resolutions in abelian categories
It is a standard result of elementary homological algebra that to every R-module $A$ there exists a projective resolution. It is often said that the category of R-modules has "enough projectives." ...
CommunityBot
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6
votes
1
answer
875
views
What should Spec Z[\sqrt{D}] x_{F_1} Spec \bar{F_1} be?
What should be $\text{Spec } \mathbb{Z}[\sqrt{D}] \times_{\mathbb{F}_1} \text{Spec } \overline{\mathbb{F}}_{1}$?
Sure, there's more than one definition.
I'm looking for any answer that uses at least ...
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1
vote
1
answer
336
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Systems of conics
It seems well-known that the system of conics given by $\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1$ for $c>0$ fixed and $a \in (0,c)\cup(c,\infty)$ varying is orthogonal: whenever two of these curves ...
- 1,618
16
votes
2
answers
489
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Are there piecewise-linear unknots that are not metrically unknottable?
A stick knot is a just a piecewise linear knot. We could define "stick isotopy" as isotopy that preserves the length of each linear piece.
Are there stick knots which are topologically trival, but ...
- 21k
4
votes
2
answers
934
views
a question about Gromov-Witten invariant
Do the Gromov-Witten invariants count the morphisms from a curve to a variety over $\mathbb{C}$?
- 21k
4
votes
5
answers
3k
views
Generalize Fourier transform to other basis than trigonometric function
The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions.
In how far can this ...
- 25.6k
3
votes
3
answers
293
views
Expressing field inclusions by polynomial equalities on coefficients
Let $A$ be the set of all quadruples $(a_0,a_1,a_2,a_3) \in {\mathbb Q}^4$ such that
the polynomial $P=X^4+a_3X^3+a_2X^2+a_1X+a_0$ is irreducible and if $z$ is any root
of $P$, then ${\mathbb Q}(z)$ ...
- 1,618
6
votes
3
answers
976
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examples of admissible representations of $GL_{n}(K)$ over p-adic field
I've been reading about the Langlands program (the paper by Torsten Wedhorn "Local langlands correspondence for GL(n) over p-adic fields, to be precise), and I want to get my hands dirty with examples....
- 4,487
20
votes
3
answers
2k
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Simple example of a ring which is normal but not CM
I try to keep a list of standard ring examples in my head to test commutative algebra conjectures against. I would therefore like to have an example of a ring which is normal but not Cohen-Macaulay. I'...
5
votes
1
answer
503
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On the materials about cohomological induction
I am now learning induction problems in representation theory. I know David Vogan's book cohomological induction and unitary representation theory might be good references,but it is too thick.
I ...
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4
votes
1
answer
755
views
Coordinates on Teichmuller space
We know that every surface of genus ($g\geq 2$) admits a pair of pants decomposition. And there is the Fenchel Nielsen Coordinates on the Teichmuller space associated to such a decomposition where we ...
- 16k
2
votes
1
answer
617
views
Representations of reductive groups over finite rings
What results are known about representations of reductive groups over finite rings in general? Here by finite rings I usually mean an algebra over $F_q$, I guess.
I know Lusztig has a paper ...
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4
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1
answer
731
views
Canonical basis for the extended quantum enveloping algebras
I am trying to understand some construction done by Lusztig in his book on quantum groups. Given some Cartan datum, let $U=U_q(\mathfrak{g})$ the standard quantized enveloping algebra of the Kac-Moody ...
- 3,131
0
votes
2
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351
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name for "solid" subset of a partially ordered set?
For P a partially ordered set, let S be a subset of P such that if:
a,c\in S and b\in P and a<=b<=c then b\in S
Is there a name for a subset with this property? The term "dense" subset is ...
- 2,537
12
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1
answer
827
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Graphs of Tangent Spheres
The Koebe–Andreev–Thurston theorem gives a characterization of planar graphs in terms of disjoint circles being tangent. For every planar graph $G$ there is a circle packing whose graph is G. What ...
- 6,523
11
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4
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12k
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How to find all integer points on an elliptic curve?
How can I determine the integer points of a given elliptic curve if I know its rank and its torsion group?
I read same basic books on elliptic curves but as a non-professional I didn't understand ...
- 1,209
7
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2
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659
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Chebyshev-like polynomials with integral roots
Chebyshev polynomials have real roots and satisfy a recurrence relation. I was wondering if one can find a sequence of polynomials with integral or rational roots with similar properties. More ...
CommunityBot
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2
votes
1
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131
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Some equivalent statements about primitive algebras
I was reading a paper, and it said that the following were equivalent using the Axiom of Choice, but I tried working it out, and I wasn't sure how: an algebra $A$ is primitive; $A$ has a proper left ...
- 2,992
24
votes
4
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6k
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What is a section?
This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...
CommunityBot
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5
votes
3
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3k
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Endomorphisms of vector bundles
I'm a bit stuck, and I'm hoping someone can help me out. I have a vector bundle $E$ on an algebraic curve (the ones I am interested in are holomorphic, but I'm sure that doesn't matter so much for ...
- 27.5k
4
votes
1
answer
2k
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Intersection cohomology of flag varieties/Schubert varieties
How do you compute in characteristic $0$, intersection cohomology of partial flag varieties (corresponding to a fixed partition $\lambda$)? I understand the answer involves Kazhdan-Lusztig polynomials;...
- 3,093
6
votes
4
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409
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Is tensoring with a module representable iff it is locally free of finite rank?
Motivation:
It's nice when you can think of the elements of an $A$-module $M$ as sections some $A$-scheme $Y\to Spec(A)$. That is, maps $Spec(A)\to Y$ such that $Spec(A)\to Y \to Spec(A)$ is the ...
CommunityBot
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11
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3
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2k
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Why is a variety of general type hyperbolic?
I heard people mentioned this in one sentence, but don't see the reason.
Why a (smooth) variety of general type, i.e. an algebraic variety X with K_X big, is hyperbolic, i.e. has no non-constant map ...
- 28.9k
1
vote
1
answer
336
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A probabilistic inequality [closed]
Suppose $x_1,x_2,...,x_6$ are non-negative Independent and identically-distributed random variables, is it true that $P(x_1+x_2+x_3+x_4+x_5+x_6 \lt 3\delta) \lt 2P(x_1 \lt \delta)$ for any $\delta \...
- 1,588
2
votes
1
answer
651
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Splitting matrix of rank one
Let R a normal domain, that is an integrally closed noetherian domain, like Dedekind domains, UFD, etc
Let A=(a i j ) a matrix with elements in R and dimension n x m.
Suppose
rank A=1 ↔ all 2 x ...
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