Recently Active Questions
159,041 questions
9
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Is there a subfactor construction involving 2-groups?
I seem to recall that there is a straightforward subfactor construction that yields fusion categories given by G-graded vector spaces and representations of G, for finite groups G. Is there an ...
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7
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3
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Symmetric Powers, Tableau and Wreath Products
Let V and W be irreducible representations of $S_n$ and $S_m$ over a field of characteristic 0. Then the Littlewood-Richardson coefficients allow us to compute the isomorphism type of the induced $S_{...
- 1,680
7
votes
1
answer
380
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Relation between dendroidal and opetopic sets
To my shame I have to admit that I have as yet not looked much into opetopes and opetopic sets.
I am in the process of writing nLab entries on dendroidal sets and noticed that some remarks on the ...
- 27.7k
27
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3
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"Dirty" proof that Eilenberg-MacLane spaces represent cohomology?
The standard approach to proving that $H^n(X; G)$ is represented by $K(G, n)$ seems to be to prove that $\text{Hom}(X, K(G, n))$ defines a cohomology theory and then use Eilenberg-Steenrod uniqueness. ...
- 141
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1
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Group structure on an interval in Z[1/p]
Is there any natural group structure on the set $I_p = \{x \in \mathbb{Z}[1/p] \mid |x| < p/2\}$?
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13
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1
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What are tame and wild hereditary algebras?
What are tame and wild hereditary algebras?
Are they related to hereditary rings? (Those are rings for which every left (resp. right) ideal is projective, equivalently, for which every left (resp. ...
- 47.3k
7
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2
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What are the fibres of a representable simplicial sheaf (in the Nisnevich topology)
Let $k$ be a field and $X$ a smooth $k$-scheme. We consider now the pointed (constant) simplical Nisnevich-sheaf $X_{+}$ that is represented by $X$. Let now $\nu\in U$ be a point of a smooth scheme, ...
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Is there a name for this topology?
Let $X$ be a set and let $f: X\longrightarrow X$ be a function on $X$. Introduce a topology on $X$ by the following basis of open sets: for any subset $S$ of $X$, let $B_S$ be the set of forward ...
- 56.6k
2
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1
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Is there an agreed name for partial ordering based on Pareto Dominance relation?
What's the correct mathematical name for the partial ordering on vectors based on what is sometimes called "Pareto Dominance"?
Does Pareto Dominance have an alternative name in fields other than ...
- 2,992
4
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3
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1k
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Quantum Frobenius
In what sense does Lusztig's quantum Frobenius, defined on a quantum enveloping algebra, generalise the classical Frobenius mapping on a variety over a finite field?
- 3,131
4
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1
answer
437
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Does the non-commutative Chern class depend on the choice of connection?
In classical geometry the calculation of the Chern classes of a vector bundle using a connection is independent of the choice of connection. Does any such result hold for projective modules in non-...
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1
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(n+1,r+1)-Theta space of (n,r)-Theta spaces?
I started writing nLab:Theta space. Not done yet, but while I am working on it:
is there a good proposal for what the "$(n+1,r+1)$-$\Theta$-space of all $(n,r)$-$\Theta$-spaces" would be?
- 27.2k
2
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1
answer
142
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Classical Calculi as Universal Quotients
As is well known, every differential calculus $(\Omega,d)$ over an algebra $A$ is a quotient of the universal calculus $(\Omega_A,d)$, by some ideal $I$. In the classical case, when $A$ is the ...
- 47.3k
2
votes
1
answer
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Spectra of rings that are projective module over a subring
This question is motivated from my last question here. I wonder if one has a ring A and an over-ring of this ring say B, and if we know that B is a projective A-module can we have a particular idea of ...
CommunityBot
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2
votes
1
answer
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Basis for Universal Calculus
Can anyone give an explicit basis of the universal (noncommutative) differential calculus over an algebra $A$ with basis ${e_i}$. (The universal calculus over $A$ is the kernel of the multiplication ...
- 47.3k
2
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2
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Examples of random variables
I'm looking for a list of examples of random variables to use in teaching a measure-theoretic probability course. For example, the Rademacher functions are an explicit construction of independent ...
- 11.4k
28
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4
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4k
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(∞, 1)-categorical description of equivariant homotopy theory
I'm trying to learn a bit about equivariant homotopy theory. Let G be a compact Lie group. I guess there is a cofibrantly generated model category whose objects are (compactly generated weak ...
- 3,685
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1
answer
146
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Is there a natural way to give a bisimplicial structure on a 2-category?
I mean by the nerve functor.
Given a 2-category $\mathcal{C}$, if we forget the 2-category structure (just view $\mathcal{C}$ as a category), the nerve functor will give us a simplicial set $N\...
- 19.8k
8
votes
1
answer
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Hodge Index Theorem for Gr(n,k)
I'm trying to understand the Hodge-Index Theorem at the moment. What does it say explicitly for the case of the complex Grassmannian Gr($n,k$), and can this be established without recourse to the ...
- 56.6k
35
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5
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The Relationship between Complex and Algebraic Geomety
I have recently begun to study algebraic geometry, coming from a differential geometry background. It seems that there is a deep link between complex manifolds and complex varieties. For example, one ...
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0
votes
1
answer
685
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On algebraic field extensions
Let $L:K$ be a field extension. Let $A$ be a set of elements in $L$, all of which are algebraic over $K$. Construct the field extension $M=K(A)$. I have two questions:
[1] Is $M:K$ an algebraic field ...
- 56.6k
6
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2
answers
1k
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Explicit Direct Summands in the Decomposition Theorem
Let f:X→Y be a semismall resolution of singularities. Then the pushforward of the constant sheaf on X is a semisimple perverse sheaf on Y. Under these conditions, I know how to calculate the ...
- 15.1k
5
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3
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230
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Is the Fell-Doran problem trivial in a topological setting?
The Fell-Doran problem is a problem in functional analysis. It goes as follows: Let $A$ be a complex unital algebra, $X$ a locally convex space, and $L(X)$ the algebra of all continuous endomorphisms ...
- 26.8k
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3
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What proportion of math papers are collaborative?
In this 2005 Notices article, Jerold Grossman tracks the proportion of papers in Math Reviews with 1, 2, 3, and >3 authors over time. His data set ends in 1999. I seem to recall reading that in 200k,...
- 7,800
3
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1
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392
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Hodge-Index Theorem for $\mathbb{C}P^2$
I'm trying to understand the Hodge-Index Theorem at the moment. What does it say explicitly for the case of $\mathbb{CP}^2$?
- 1,462
5
votes
3
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2k
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Atiyah-MacDonald: exercise 5.29 - "local ring of a valuation ring"
The exercise is the following:
Let $A$ be a valuation ring, $K$ its field of fractions. Show that every subring of $K$ which contains $A$ is a local ring of $A$.
Does anyone know what is meant by "...
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11
votes
2
answers
6k
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Picard-Fuchs equations
If I have the periods $$\pi_1(\lambda)=\int_0^1\frac{dx}{\sqrt{x(x-1)(x-\lambda)}}$$ and $\pi_2(\lambda)$ similarly defined of the cubic curve $$y^2z=z(x-z)(x-\lambda z)$$ Such functions will be ...
- 21k
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7
answers
2k
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ubiquity, importance of path algebras
I work in planar algebras and subfactors, where the idea of path algebras on a graph (alternately known as graph algebras, graph planar algebras, etc.) is quite useful. The particular result I'm ...
CommunityBot
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3
votes
4
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Is there a use for a Hilbert space that uses a different norm than the one induced by the inner product?
$l_1$ minimization / compressed sensing comes to mind. Does anyone have any concrete examples? Or is such a construct completely useless?
- 24.1k
3
votes
2
answers
1k
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Castelnuovo Positivity (Rewrite of: Weil's original proof for FP^2)
Weil's proof of the Riemann Hypothesis for projective curves relies upon the following positivity result: Let $\mathbb{F}q$ be the finite field with $q$ elements, $\overline{\mathbb{F}q}$ its closure, ...
- 156k
11
votes
3
answers
2k
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$\omega$-topos theory?
I've been reading through Lurie's book on higher topos theory, where he develops the theory of $(\infty,1)$-toposes, which leads me to the following question: Is there any sort of higher topos theory ...
- 19.8k
2
votes
1
answer
552
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An application of the Künneth formula in the proof of the theorem of the cube
For the question, everything is over an algebraically closed field and by a scheme we mean a scheme of finite type. The theorem of the cube is the following:
Let $X$ be a complete variety, $Y$ a ...
- 8,681
14
votes
6
answers
1k
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How to smootly interpolate between möbius transformations?
If you have two Möbius transformations represented as:
$f(z) = \frac{az + b}{cz + d}$
$g(z) = \frac{pz + q}{rz + s}$
where $a, b, c, d, p, q, r, s, z \in \mathbb{C}$
Is it possible to derive a ...
- 56.6k
10
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1
answer
698
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Ramified covers of 3-torus
It is known that every orientable 3-manfiold can be obtained as a ramified cover of S3 with a ramification (of some order) at a link in S3. I am curious if there is a reasonable characterization of 3-...
- 56.6k
12
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4
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2k
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Mystery of the Monstrous Moonshine
There's a very famous group, the largest sporadic simple finite group, sometimes called a monster whose size is quoted below. What's the explanation that the primes appearing in it,
...
CommunityBot
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16
votes
6
answers
1k
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Solving polynomial equations when you know in which number field the solutions live
Suppose I have a bunch of polynomial equations with coefficients in a number field, and suppose further that I'm guaranteed a priori that they have a solution in that number field. Can I leverage ...
CommunityBot
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28
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5
answers
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How does one handle two-body job searches?
One thing I've heard conflicting advice on is how to handle a job search for two people simultaneously; especially the strategy of things like: when do you mention your situation to the departments in ...
- 19.2k
3
votes
1
answer
422
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Categorifying the group representations
I've heard about this construction on the lecture about higher representation theory:
Given a Lie algebra $g$, one constructs $\mathcal A$, a category whose $K_0$ is the universal enveloping ...
- 15.1k
2
votes
3
answers
184
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References for Relationships amongst the Number-theoretic Functions
Do you know of any on-line references regarding relationships among the elementary number-theoretic functions?
The sort of thing I'm interested in is as at the Wikipedia page on Arithmetic Functions.
...
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2
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Is the maximum domain to which a Dirichlet series can be continued always a halfplane?
Let $f(s)=\sum_n a_n n^{-s}$ be a Dirichlet series whose coefficients satisfy $\lvert a_n\rvert\leq n^{C}$. Then $f(s)$ converges absolutely in some halfplanes, and is conditionally convergent in (...
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4
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Wick rotation in mathematics
In physics, esp. quantum field theory, Wick rotation (i.e. putting $t \mapsto i\tau$, imaginary time) is often used to simplify calculations, make things convergent or make connections between ...
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2
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235
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What is a cograph of an n-functor?
I'm trying to get my head around what a cograph of an n-functor is. We (some n-Lab people) are discussing it here. As a start, I'd be happy to understand what the cograph of a 0-functor, i.e. function ...
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2
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Evaluate a fair game [closed]
I'm not a mathematician, so my question may be not so clear, sorry.
Let's say we toss up an ideal coin and win 1 dollar on heads and lose 1 dollar on tails. So, expected value is M = 1×0.5 &...
3
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1
answer
242
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Are mapping spaces paracompact?
Let X be a (finite dimensional) manifold. Consider smooth mapping space $$PX = C^\infty(I, X)$$ where I = [0,1] is the closed interval. Is this space paracompact? What if we fix a point x in X and ...
- 47.3k
9
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1
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695
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Asymptotics of Power Series With Branch Singularities
I am wondering if there are analytic tools to find asymptotic formulae for the coefficients of a complex power series of a function with branch singularities. For example, it is possible to show ...
- 61.9k
5
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5
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5k
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Notions of Matrix Differentiation
There are a few standard notions of matrix derivatives, e.g.
If f is a function defined on the entries of a matrix A, then one can talk about the matrix of partial derivatives of f.
If the entries of ...
- 2,284
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2
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1k
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Abelianization of Lie groups
If G is a group, its abelianization is the abelian group A and the map G → A such that any map G → B with B abelian factors through A. Abelianization is a functor, and in general a very ...
CommunityBot
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4
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Is it possible to write a research statement which is too short?
We all know that it is possible to write a research statement which is so long it becomes counterproductive. At some point people will give up on reading it.
But can you write one (let's say for job ...
- 44.7k
9
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1
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395
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Is there a coalgebraic characterisation of the hyperfinite II_1 factor?
Peter Freyd showed that the real interval [0, 1] is a final coalgebra for a functor on sets equipped with two points, which sends such a set to the 'wedge' of two copies of itself, identifying the ...
- 56.6k
7
votes
3
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562
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Abstract Relation between Presehaves and Simplicial Sets
Every presheaf (let's say on a topological space) comes with restriction maps. The open sets of a topological space are ordered by inclusion and these inclusions yield the restrictions. Now a sheaf ...
- 1,926