Newest Questions
159,027 questions
11
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Are good introductory/pedagogical problems in algebraic geometry rare?
I have just started reading Elementary Algebraic Geometry by Hulek. It is a nice book but I find that it doesn't give many problems (about 10 to 15 per chapter), and that the exercises present are a ...
1
vote
1
answer
778
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Are cyclotomic Khovanov-Lauda-Rouquier algebras symmetric?
Recall that for k a field, a finite dimensional k-algebra A is called symmetric if it is isomorphic to its dual as a bimodule of itself. Which is to say, there's a trace map t:A -> k such that t(...
- 44.7k
15
votes
6
answers
2k
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Can one make Erdős's Ramsey lower bound explicit?
Erdős's 1947 probabilistic trick provided a lower exponential bound for the Ramsey number $R(k)$. Is it possible to explicitly construct 2-colourings on exponentially sized graphs without large ...
- 2,967
22
votes
5
answers
2k
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Homological algebra and calculus (as in Newton)
This question reminded me of a possibly stupid idea that I had a while back.
On page 2 of this paper, while discussing Euclid's axioms of plane geometry and spatial geometry, Manin makes an extremely ...
- 21k
16
votes
3
answers
5k
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Stalks of sheaf-hom
Let $F$ and $G$ be sheaves on $X$. Under what conditions is the natural map from the stalk at $p$ of $\mathcal{H}\kern{-1pt}\mathit{om}(F,G)$ to $\mathrm{Hom}(F_p, G_p)$ an isomorphism?
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107
votes
10
answers
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What is (co)homology, and how does a beginner gain intuition about it?
This question comes along with a lot of associated sub-questions, most of which would probably be answered by a sufficiently good introductory text. So a perfectly acceptable answer to this question ...
6
votes
3
answers
1k
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Quotient of a category by a free group action
Let Cat denote the 1-category of small categories. The functor Mor : Cat -> Set which assigns to a category its set of morphisms (aka Hom([• -> •], -)) does not commute with most colimits. ...
- 25.2k
14
votes
2
answers
882
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A complex manifold which is quasiprojective in two different ways
Does there exist a complex manifold M which is a quasiprojective variety in two "essentially" different ways? Let me be more specific. I'm looking for a complex manifold M together with two ...
- 44.8k
4
votes
3
answers
833
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Is there a "universal LYM inequality?"
This question is based on a blog post of Qiaochu Yuan.
Let P be a locally finite* graded poset with a minimal element, and w be a weight function on the elements of P. Suppose that the total weight ...
- 12.6k
5
votes
3
answers
912
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Two finite groups with the same identical relations?
An identical relation on a group G is a word w in Fr, the free group on r elements (for some r), such that evaluating w on any r-tuple of elements of G yields the identity (this just means ...
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27
votes
4
answers
3k
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What is an example of a presheaf P where P^+ is not a sheaf, only a separated presheaf?
There is a standard way to construct the sheafification of a presheaf on a Grothendieck topology which involves matching families. Details may be found here:
http://ncatlab.org/nlab/show/matching+...
- 3,918
5
votes
2
answers
703
views
Linear Algebra Over $F_{2}$
Suppose we call a subset S of $F^{n}$ ($F$ is the field with two elements) good if for any $x$ and $y$ (possibly $x=y$) we have $[x,y]=1$ where $[ , ]$ denotes the obvious bilinear form on F. What's ...
10
votes
5
answers
990
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Non-conjugate words with the same trace
Let n>=2, p a large prime, G = SL_n(Z/pZ).
If n=2, there are words that, while not conjugate in the free group, do have identical trace in G. For example, tr(g h^2 g^2 h)= tr(g^2 h^2 g h) for all g, ...
- 20.2k
16
votes
2
answers
2k
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"synthetic" reasoning applied to algebraic geometry
A hyperlinked and more detailed version of this question is at
nLab:synthetic differential geometry applied to algebraic geometry.
Repliers are kindly encouraged to copy-and-paste relevant bits of ...
- 19.8k
13
votes
2
answers
723
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Ideals in Factors
One can easily prove that factors have no nontrivial ultraweakly closed 2-sided ideals as these are equivalent to nontrivial central projections. One can also show type $I_n$, type $II_1$, and type $...
- 5,425
10
votes
2
answers
944
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Logarithmic structures on moduli of elliptic curves over Z
I've heard it stated that if you take the moduli of elliptic curves with some level structure imposed (as a moduli scheme over Spec(Z)), there is a logarithmic structure that you can impose at the ...
- 52.7k
8
votes
2
answers
1k
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Differentials in the Lyndon-Hochschild spectral sequence
The Lyndon-Hochschild(-Serre) spectral sequence applies to group extensions in a manner analogous to the Serre-Leray spectral sequence applied to a fibration.
Does anyone know of a good description (...
- 1,422
10
votes
4
answers
2k
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Reading for finite Fourier analysis
Can anyone recommend some good reading for Fourier analysis (and the Fourier transform) over finite abelian groups? I've found it given brief descriptions in both books on representation theory and on ...
- 7,013
26
votes
5
answers
2k
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Cauchy-Schwarz and pigeonhole
I've occasionally heard it stated (most notably on Terry Tao's blog) that "the Cauchy-Schwarz inequality can be viewed as a quantitative strengthening of the pigeonhole principle." I've certainly seen ...
- 12.6k
30
votes
5
answers
4k
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Deformation theory of representations of an algebraic group
For an algebraic group G and a representation V, I think it's a standard result (but I don't have a reference) that
the obstruction to deforming V as a representation of G is an element of H2(G,V&...
- 24.1k
3
votes
2
answers
884
views
Is the "diagonal" of a regular language always context-free?
That's very poor wording, so let me be more precise. Suppose $L$ is an unambiguous regular language on an alphabet $\{a_1, \dots, a_n\}$, and suppose to each letter of the alphabet we associate two ...
- 118k
8
votes
2
answers
823
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What is the affinization of M_g?
This question is inspired by What is an example of a function on M_g? . Consider Mg, the moduli space of genus g curves, NOT compactified. When g is 3 or greater, this is not affine. Does anyone know ...
- 156k
36
votes
6
answers
5k
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Does finite mathematics need the axiom of infinity?
A statement referring to an infinite set can sometimes be logically rephrased using only finite sets/objects. For example, "The set of primes is infinite" <-> "There is no largest prime". ...
- 11.3k
6
votes
1
answer
1k
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Specializations of Schur functions at consecutive integers
Given a partition λ = (λ1, λ2, ..., λn) denote with sλ the associated Schur function.
There exists a nice product formula for the principal specializations:
sλ...
- 1,412
73
votes
10
answers
22k
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Galois groups vs. fundamental groups
In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue ...
- 2,806
23
votes
5
answers
2k
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Why are subfactors interesting?
I get asked this question a lot, and am not very happy with any of the answers.
Vaguely I think of subfactor theory as a generalization of representation theory of groups. That is, if you have a ...
32
votes
4
answers
4k
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Largest hyperbolic disk embeddable in Euclidean 3-space?
Hilbert proved that there's no complete regular ($C^k$ for sufficiently large $k$) isometric embedding of the hyperbolic plane into $\mathbb{R}^3$. On the other hand, the pseudosphere is locally ...
- 13.6k
9
votes
0
answers
821
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Pimsner-Popa Bases
Let $N\subset M$ be a finite index $II_1$-subfactor. Let $B=\{b_i\}$ be a finite orthonormal (Pimsner-Popa) basis for $M$ over $N$. Let $d=[M\colon N]^{1/2}$. It is well known that $B_1=\{d b_{i_1} ...
- 5,425
20
votes
5
answers
2k
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How small can a group with an n-dimensional irreducible complex representation be?
More precisely, what is the smallest exponent e such that, for every n, there exists a group of size at most Cn^e for some absolute constant C and with an n-dimensional irreducible complex ...
- 118k
40
votes
6
answers
6k
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Linear transformation that preserves the determinant
It seems "common knowledge" that the following holds:
Let $T$ be a linear transformation on $n\times n$ matrices with complex coefficients that preserves the determinant. Then there exists ...
- 511
9
votes
1
answer
841
views
Limit Linear Series
A linear series on a curve C is a line bundle L together with a subspace V of the global sections of L. Eisenbud and Harris develeoped a theory of limit linear series which explans how (L, V) ...
- 611
8
votes
1
answer
1k
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what is the connection between D-modules and coordinate bundles?
Fix $n$ and a field $k$ of characteristic zero. Let $G$ be the pro-algebraic group of automorphims of $k[[x_1,...x_n]]$. Let $G_0$ be the subgroup of automorphisms preserving the closed point (note ...
- 1,038
21
votes
4
answers
4k
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Rings over which every module is free
We know that modules over skewfields are free. Is the converse true? In other words, is it true that a nontrivial ring over which every module is free is a skewfield?
If the ring A is commutative, ...
- 1,069
8
votes
3
answers
921
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Generic Noether normalisation
Suppose that $M$ is a finitely generated module over $A=k[X_1,\ldots,X_n]$ of Krull dimension $m$ with $k$ an infinite field. Then one version of Noether normalisation says there is an $m$-dimensional ...
- 3,545
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4
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What is interesting/useful about big Witt Vectors?
$p$-typical Witt vectors are (among other things) a canonical way of associating to a perfect ring $A$ of characteristic $p$ a complete DVR of characteristic $0$ with residue ring $A$ generalizing $\...
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24
votes
6
answers
5k
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Finite groups with the same character table
Say I have two finite groups $G$ and $H$ which aren't isomorphic but have the same character table (for example, the quaternion group and the symmetries of the square). Does this mean that the ...
- 10.7k
2
votes
1
answer
926
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Theta Functions and Cousins
So I am (barely) familiar with the construction of the theta function of an integral lattice $L$. The theta function, as I understand it, is defined as the function which takes a variable $z$ and ...
- 27.5k
5
votes
3
answers
1k
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Can the valuative criteria be checked "on a dense open"?
The valuative criterion for separatedness (resp. properness) says that a noetherian scheme X is separated (resp. proper) if and only if
for any DVR R, with fraction field K,
any map Spec(K)→...
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25
votes
7
answers
4k
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Can adjoint linear transformations be naturally realized as adjoint functors?
Last week Yan Zhang asked me the following: is there a way to realize vector spaces as categories so that adjoint functors between pairs of vector spaces become adjoint linear operators in the usual ...
- 118k
5
votes
3
answers
1k
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What is the expected number of maximal bicliques in a random bipartite graph?
Maximal Biclique: A complete bipartite subgraph, that isn't a subgraph of another complete bipartite subgraph.
Given a bipartite graph $G=(V_{1}\cup V_{2}, E)$ where $|V_{1}|=|V_{2}|$ with ...
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4
answers
1k
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An inner product that makes the R-matrix unitary
So, if you talk to the right people, they will tell you that the braiding of the category of representations of a quantum group are not unitary and that one can fix this by taking a different commutor ...
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22
votes
3
answers
1k
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Why are Dynkin diagrams characterized by their eigenvalues?
The Dynkin diagrams An, Dn, E6,
E7, E8 can be characterized among finite simple connected
graphs by the property that their eigenvalues (that is, the eigenvalues of their adjacency matrices) all have ...
- 118k
76
votes
9
answers
15k
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understanding Steenrod squares
There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from ...
- 6,069
15
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1
answer
715
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What are the Schur functions of the eigenvalues of a non-negative integer matrix counting?
Let A be a non-negative integer square matrix with eigenvalues x1, x2, ... xn. Any symmetric function of these eigenvalues with integer matrices is an integer. I'm aware of the following results ...
- 118k
12
votes
4
answers
2k
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Elements of infinite order in a profinite group
Say G is a profinite group with elements of arbitrarily large order. Do elements of infinite order exist (A) if we assume G is abelian? (B) in general?
A start for (A): we can ask the same question ...
- 11.3k
17
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10
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3k
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References for homotopy colimit
(1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...
- 12.8k
7
votes
8
answers
747
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What is an example of a function on M_g?
It feels bad talking about a space without knowing a single function on it, hah?
So what is a function on the moduli space of curves, from the geometric point of view?
From the functorial point of ...
- 5,052
7
votes
2
answers
477
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Characterizing the Radon transforms of log-concave functions
$f:\mathbf{R}^d\to \mathbf{R}_{\ge 0}$ is log-concave if $\log(f)$ is concave (and the domain of $\log(f)$ is convex).
Theorem: For all $\sigma$ on the sphere $\Bbb S^{d-1}$ and $r\in \mathbf{R}$,
$$
...
- 5,992
31
votes
7
answers
14k
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Advice on doing mathematical research [closed]
Please share any general tips or advice you have on doing mathematical research.
How do you identify good problems to work on or to think about? What do you do when you get stuck on a problem? Etc.
44
votes
10
answers
25k
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Fourier transform for dummies [closed]
So ... what is the Fourier transform? What does it do? Why is it useful (both in math and in engineering, physics, etc)?
(Answers at any level of sophistication are welcome.)
- 21k