Newest Questions
159,026 questions
5
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answers
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$E_\infty$ spectrum corresponding to $\Bbb Z_p$
First of the questions about derived algebraic geometry from a noobie.
The way I understand it, every discrete ring $R$ corresponds to some ring spectrum whose $\pi_0$ is $R$. Now consider $p$-adic ...
- 15.1k
4
votes
3
answers
473
views
What's the best reference for actual formulas for RT invariants?
If one really wants to understand the formulas for how to construct the Reshetikhin-Turaev 3-manifold invariants coming from quantum groups in terms of R-matrices and such, what's the best reference ...
- 44.7k
27
votes
13
answers
4k
views
Homological algebra for commutative monoids?
Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of ...
- 27.5k
5
votes
1
answer
468
views
When does a transitive action of a profinite group have an infinite orbit?
That is: suppose G is a profinite group acting 1-transitively (but maybe not regularly) on a set X. Is there a reasonable criterion for when there is a g in G and a point a in X such that the g-orbit ...
- 3,966
9
votes
2
answers
2k
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What is an example of a smooth variety over a finite field F_p which does not lift to Z_p?
Somebody answered this question instead of the question here, so I am asking this with the hope that they will cut and paste their solution.
- 10.5k
1
vote
1
answer
322
views
Request for info on the space of commuting matrices preserving a flag.
Fix a flag of subspaces V1 in V2 in V3, etc. all in Cn.
Consider the space of pairs of commuting linear transformations A and B such that:
A preserves the flag (i.e. A(Vi) is in Vi), and
B strictly ...
- 44.7k
15
votes
5
answers
3k
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Existence of (smooth) models
Hi everyone,
let X be a variety over a field k, S an integral scheme such that the function field K of S is contained in k. An S-scheme X is called model of X/k if X x_S k = X, i.e. if the generic ...
- 4,450
66
votes
5
answers
8k
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Does homology have a coproduct?
Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor ...
- 661
8
votes
1
answer
998
views
Generalized Teichmuller representatives
Fix a prime $p$. The Teichmuller representative associated to a $p$-adic integer $a$ is the unique root of $x^p - x$ in $Z_p$ congruent to $a$ mod $p$. One can identify this representative with the ...
- 118k
15
votes
2
answers
2k
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What is an example of a smooth variety over a finite field F_p which does not embed into a smooth scheme over Z_p?
Such an example of course could not be projective and would not itself lift to Z_p. The context is that one can compute p-adic cohomology of a variety X over a finite field F_p via the cohomology of ...
- 10.5k
34
votes
4
answers
2k
views
How is tropicalization like taking the classical limit?
There is a folk — I can't call it a theorem — "fact" that the mathematical relationship between Complex and Tropical geometry is analogous to the physical relationship between Quantum and ...
- 54.6k
39
votes
4
answers
3k
views
Does a scheme have a "separification"?
Background:
(1) If C and D are categories and there is a forgetful functor U:C→D, then a C-ification functor F:D→C is an adjoint to U. For example, the (left) adjoint to the forgetful ...
- 24.1k
43
votes
6
answers
9k
views
"A gentleman never chooses a basis."
Around these parts, the aphorism "A gentleman never chooses a basis," has become popular.
Question. Is there a gentlemanly way to prove that the natural map from $V$ to $V^{**}$ is surjective if $V$...
- 5,275
85
votes
19
answers
119k
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Reading list for basic differential geometry?
I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is ...
19
votes
4
answers
3k
views
What's the right way to think about "anomalies" in 3d TQFTs?
3d TQFTs constructed from modular tensor categories don't in general give an honest 3d TQFT, instead they have an "anomaly." My vague understanding from Kevin Walker's talks and from skimming Freed-...
- 28.1k
23
votes
5
answers
3k
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Do all 3D TQFTs come from Reshetikhin-Turaev?
The Reshetikhin-Turaev construction take as input a Modular Tensor Category (MTC) and spits out a 3D TQFT. I've been told that the other main construction of 3D TQFTs, the Turaev-Viro State sum ...
- 27.5k
38
votes
6
answers
9k
views
Deformation theory and differential graded Lie algebras
There is supposed to be a philosophy that, at least over a field of characteristic zero, every "deformation problem" is somehow "governed" or "controlled" by a differential graded Lie algebra. See for ...
- 21k
74
votes
9
answers
26k
views
Motivating the Laplace transform definition
In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem.
The Laplace transform of a function $f(t)$, ...
- 891
16
votes
4
answers
2k
views
Deligne's conjecture (the little discs operad one)
Deligne's conjecture states that the Hochschild cochain complex of an A-infinity algebra is an algebra over the operad of chains on the topological little discs operad.
Of course the conjecture has ...
- 21k
19
votes
5
answers
4k
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Derived categories and homotopy categories
There are two constructions that look quite similar to me: the derived category of an abelian category, and the homotopy category of a model category. Is there any explicit relationship between these ...
- 21k
2
votes
3
answers
1k
views
What is the base change in number theory?
I'm somewhat familiar with base change in scheme theory: sometimes a property of a morphism X \to Y survives a base change ...
- 15.1k
104
votes
10
answers
24k
views
Motivation for algebraic K-theory?
I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...
67
votes
14
answers
23k
views
A reading list for topological quantum field theory?
Can you suggest a reading list, or at least a few papers that you think would be useful, for a beginner in topological quantum field theory? I know what the curvature of a connection is, know basic ...
200
votes
89
answers
54k
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Examples of great mathematical writing
This question is basically from Ravi Vakil's web page, but modified for Math Overflow.
How do I write mathematics well? Learning by example is more helpful than being told what to do, so let's try to ...
4
votes
3
answers
751
views
What is a formula for the "group-like Drinfeld element"?
Any quantized universal enveloping algebra (in fact, any toplogically quasi-triangular Hopf algebra) has an (in its completion) an element u called the Drinfeld element which gives an isomorphism from ...
- 44.7k
4
votes
3
answers
1k
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When and how is a group of order n isomorphic to a regular subgroup of equal order?
In "Group Theory and Its Application to Physical Problems" by Morton Hamermesh, Morton states Cayley's theorem: Every group G of order n is isomorphic with a subgroup of the symmetric group Sn, which ...
10
votes
1
answer
1k
views
Why are torsion points dense in an abelian variety?
Hi everyone,
let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$ of characteristic $p\geqslant 0$. I'm trying to prove that the subgroup $A'$ which is the union of ...
- 4,450
25
votes
2
answers
2k
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Is every functor a composition of adjoint functors?
My understanding of Ben's answer to this question is that even though associated graded is not an adjoint functor, it's not too bad because it is a composition of a right adjoint and a left adjoint.
...
- 24.1k
7
votes
6
answers
1k
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Does the space of $n \times n$, positive-definite, self-adjoint, real matrices have a better name?
This is also the space of real, symmetric bilinear forms in $\Bbb R^n$.
- 8,512
73
votes
3
answers
18k
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What is Koszul duality?
Okay, let's make sure I'm on the same page with those who know homological algebra.
What is Koszul duality in general?
What does it mean that categories are Koszul dual (I guess representations of ...
- 15.1k
2
votes
2
answers
3k
views
How to attack this diophantine equation in 3 variables?
From link:
Find integers a, b and c such that:
987654321a + 123456789b + c = (a + b +
c)³
- 121
32
votes
4
answers
3k
views
Spectrum of the Grothendieck ring of varieties
Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my ...
- 15.1k
7
votes
1
answer
2k
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Dualizing sheaf on singular curves
I am trying to understand the stabilization map, which takes a prestable curve (a curve with some marked points, and at worst nodal singularities) and returns a stable curve (a curve with some marked ...
- 21k
6
votes
2
answers
2k
views
Eigenvalues of Laplacian
What's the most natural way to establish the asymptotics of $\Delta$ on a compact Riemannian manifold $M$ of dimension $N$? The asymptotics should be
$$ \#\{v < A^2\} = \mathrm{const}\ast\mathrm{...
- 15.1k
4
votes
3
answers
568
views
Functions on hyperbolic space and modular curves
The decomposition of $L^{2}\left(S^{2}\right)$ under $SO\left(3,\mathbb{R}\right)$ is well-known.
Focus now on the hyperbolic plane $H$ presented as the quotient $SL\left(2,\mathbb{R}\right)/SO\left(...
- 15.1k
8
votes
2
answers
819
views
Are there interesting monoidal structures on representations of quantum affine algebras?
Is there a good monoidal structure on a category of integrable representations of a quantum affine algebra? In the ordinary affine Kac-Moody case, there is the usual tensor product (symmetric, adds ...
- 45.7k
7
votes
3
answers
3k
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Beilinson-Bernstein and Koszul duality
For geometric representation theorists down here.
Consider the Beilinson-Bernstein theorem:
Functor of global sections establishes
the correspondence between twisted
D-modules with fixed ...
- 15.1k
11
votes
2
answers
1k
views
Elliptic curve over spectra?
Filling the gaps in my knowledge to understand the tmf question.
So, what is the analogue of elliptic curve over the category of spectra?
- 15.1k
10
votes
8
answers
3k
views
Which came first: the Fibonacci Numbers or the Golden Ratio?
I know that the Fibonacci numbers converge to a ratio of .618, and that this ratio is found all throughout nature, etc. I suppose the best way to ask my question is: where was this .618 value first ...
- 175
13
votes
3
answers
3k
views
What is the Theorem of the Cube?
What is the "theorem of the cube" for abelian varieties? What is the statement and how should I think about it?
- 27.5k
3
votes
2
answers
1k
views
What is the difference between the Power Law and Zipf's Law?
I am new to statistics. Could somebody tell me what is the difference between a Power Law and Zipf's Law. The latter could be just for texts but I cant see any difference in their essence.
- 31
22
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3
answers
2k
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What is a TMF in topology?
What is a topological modular form? How are they related to 'normal' (number-theoretic) modular forms?
- 15.1k
12
votes
5
answers
1k
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Questions about ordering of reals and irrationals
Three problems from G.Rosenstein "Linear orderings" (from the end of Chapter 2 and beginning of Chapter 4):
1) Is there a nondecreasing function from irrationals onto reals?
2) Is there a ...
- 918
7
votes
2
answers
1k
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Non-zero sheaf cohomology
Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is non-zero?...
- 47.5k
11
votes
2
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1k
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Finding monochromatic rectangles in a countable coloring of $R^{2}$
Given a countable coloring of the plane, is it always possible to find a monochromatic set of points $\left\{ \left(x,y\right),\left(x+w,y\right),\left(x,y+h\right),\left(x+w,y+h\right)\right\} $ (the ...
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votes
2
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164
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Are irregular points of an action necessarily in the closure of a larger orbit?
Suppose $G$ is an affine algebraic group acting linearly on a vector space $V$. A point $v \in V$ is stable if the orbit $Gv$ is closed and $v$ is regular (the dimension of the stabilizer of $v$ is ...
- 24.1k
33
votes
4
answers
4k
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What is the universal property of associated graded?
Given a filtered vector space (or module over a ring) $0=V_{0}\subseteq V_{1}\subseteq\cdots\subseteq V$, you can construct the associated graded vector space $\mathrm{gr}\left(V\right)=\oplus_{i}V_{i+...
- 24.1k
-1
votes
1
answer
924
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Fundamental: Division by Zero [closed]
All the articles I've read regarding "Division by Zero" the main argument for it being an undefined operation, because all proofs lead to contradictions.
...
20
votes
2
answers
10k
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does a line bundle always have a degree
For curves there is a very simple notion of degree of a line bundle or equivalently of a Weil or Cartier divisor. Even in any projective space $\mathbb P(V)$ divisors are cut out by hypersurfaces ...
- 3,968
7
votes
4
answers
1k
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Are abelian non-degenerate tensor categories semisimple?
A pivotal monoidal category is called non-degenerate if the inner product $\left(x,y\right) = Tr\left(xy^{*}\right)$ (where $y^{*}$ is the dual map) is non-degenerate. As a rule of thumb non-...
- 28.1k