Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
1,800 questions
44
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answers
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Book on mathematical "rigorous" String Theory?
I've been looking high and low for a mathematical book on String Theory. The only book I could find was "A Mathematical Introduction to String Theory" by Albeverio, Jost, Paycha and ...
44
votes
10
answers
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The finite subgroups of SL(2,C)
Books can be written about the finite subgroups of $\mathrm{SL}(2,\mathbb C)$ (and their immediate family, like the polyhedral groups...) I am about to start writing notes for a short course about ...
- 47.3k
37
votes
6
answers
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Billiard dynamics under gravity
Has the dynamics of billiards in a polygon subject to gravity been
studied?
What I have in mind is something like this:
Still Snell's Law ...
- 151k
37
votes
31
answers
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A Learning Roadmap request: From high-school to mid-undergraduate studies
Dear MathOverflow community,
In about a year, I think I will be starting my undergraduate studies at a Dutch university. I have decided to study mathematics. I'm not really sure why, but I'm ...
37
votes
6
answers
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The category of posets
I am trying to teach myself category theory and, as a beginner, I am looking for
examples that I have a hands-on experience with.
Almost every introductory text in category theory contains following ...
36
votes
1
answer
4k
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Special values of L-functions as periods
If $M$ is a pure motive over $\mathbb{Q}$, one cas define its $L$-function $L(M,s)$ which conjecturaly is a meromorphic function over $\mathbb{C}$ with finitely many poles.
For example, when $M=\...
- 26k
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votes
2
answers
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Kervaire invariant: Why dimension 126 especially difficult?
Is there any resource that might help non-experts gains some understanding of why
the Kervaire invariant problem remains open now only in dimension $126$? ($126 =2^7-2=2^{j+1}-2$;
whether $\theta_j=\...
- 151k
36
votes
2
answers
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Eigenvalues of the product of two symmetric matrices
This is mostly a reference request, as this must be well-known!
Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or $BA$...
- 2,600
34
votes
1
answer
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Early stabilization in the homotopy groups of spheres
Thanks to Freudenthal we know that $\pi_{n+k}(S^n)$ is independent of $n$ as soon as $n \ge k+2$. However, I was looking at the table on Wikipedia of some of the homotopy groups of spheres and noticed ...
- 13.5k
34
votes
2
answers
4k
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Functions whose gradient-descent paths are geodesics
Let $f(x,y)$ define a surface $S$
in $\mathbb{R}^3$ with a unique local minimum at $b \in S$.
Suppose gradient descent from any start point $a \in S$
follows a geodesic on $S$ from $a$ to $b$.
(Q1.)
...
- 151k
34
votes
1
answer
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Strong Whitney embedding theorem for non-compact manifolds
$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds.
The strong ...
- 6,210
34
votes
1
answer
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Freyd-Mitchell's embedding theorem
Freyd–Mitchell's embedding theorem states that: if $A$ is a small abelian category, then there exists a ring R and a full, faithful and exact functor $F\colon A \to R\text{-}\mathrm{Mod}$.
I have been ...
- 3,004
34
votes
1
answer
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Theme of Isbell duality
Let $C$ be a small category. Isbell duality provides an adjunction $\widehat{C} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}}\widehat{C^{\mathrm{op}}}^{\mathrm{op}}$....
- 63.1k
34
votes
2
answers
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What is the relation between the sphere spectrum and supersymmetry?
In this this google+ post of Urs Schreiber, he says: "Grading over the sphere spectrum is supersymmetry" and then he redirect us to the abstract idea of superalgebra (in nLab).
Are there some ...
- 1,720
33
votes
6
answers
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Most important results in 2022
Undoubtedly one of the news that attracted the most attention this year was the result of Yitang Zhang on the Landau–Siegel zeros (see Consequences resulting from Yitang Zhang's latest claimed ...
32
votes
19
answers
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Good books on theory of distributions
Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.
31
votes
7
answers
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English reference for a result of Kronecker?
Kronecker's paper Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten apparently proves the following result that I'd like to reference:
Let $f$ be a monic polynomial with integer ...
- 873
31
votes
5
answers
7k
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Verlinde's formula
"Verlinde's formula" predicts the dimension of the space of conformal blocks of a chiral CFT.
Depending on...
• which chiral CFT one considers (does one restrict to WZW models, or not?)
&...
- 43.2k
30
votes
1
answer
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How strong is this conjecture? $(Z/nZ)^*$ is generated by "small" elements
Conjecture: There are constants $c,k$ such that every $(Z/nZ)^*$ is generated by its elements smaller than $k (\log n)^c$.
Where $(Z/nZ)^*$ is the multiplicative group of integers mod $n$. My main ...
- 4,529
29
votes
5
answers
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Proof of the Reidemeister theorem
While preparing for my introduction to topology course, I've realized that I don't know where to find a detailed proof of the Reidemeister theorem (two link diagrams give isotopic links, iff they can ...
- 23.5k
28
votes
1
answer
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1-Wasserstein distance between two multivariate normal
The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
$$d_p(\nu_{1},\nu_{2})=\left(\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,...
- 1,256
28
votes
5
answers
6k
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Stokes theorem for manifolds with corners?
Maybe this is an elementary question, but I'm unable to find the appropriate reference for it. The Stokes theorem tells us that, for a $n+1$-dimensional manifold $M$ with boundary $\partial M$ and any ...
- 1,975
28
votes
2
answers
3k
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Probing a manifold with geodesics
Supposed you stand at a point $p \in M$ on a smooth 2-manifold $M$
embedded in $\mathbb{R}^3$.
You do not know anything about $M$.
You shoot off a geodesic $\gamma$ in some direction $u$,
and learn ...
- 151k
28
votes
6
answers
1k
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Online events during the quarantine
With many places on earth subjected to quarantine and large gathering prohibited, there are announcements of online seminars and talks open to people around the world. The talks can be conducted via ...
27
votes
5
answers
1k
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Is this a known question about the expression of a function on $\Bbb R^2$ as an infinite sum of products?
The question below was posted on Mathematics Stack Exchange. It received no answer, and I do not expect any direct answer to it here. However, the question seems to me a natural one. Thus I wonder ...
- 2,437
27
votes
3
answers
3k
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Where's the best place for an algebraic geometer to learn some algebraic number theory?
There are lots of introductions to number theory out there, but typically they are streamlined to assume as little prerequisite knowledge as possible. I'm looking for a text which does the opposite -- ...
- 63.9k
27
votes
3
answers
2k
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Configuration space of little disks inside a big disk
The space of configurations of $k$ distinct points in the plane
$$F(\mathbb{R}^2,k)=\lbrace(z_1,\ldots , z_k)\mid z_i\in \mathbb{R}^2, i\neq j\implies z_i\neq z_j\rbrace$$
is a well-studied object ...
- 35.9k
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5
answers
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References for "modern" proof of Newlander-Nirenberg Theorem
Hi,
I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In ...
26
votes
1
answer
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What is the status on this conjecture on arithmetic progressions of primes?
The Green-Tao theorem states that for every $n$, there is an arithmetic sequence of length $n$ consisting of primes.
For primes, $p$, let $P(p)$ be the maximum length of an arithmetic progression of ...
- 1,835
26
votes
5
answers
2k
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Complexity of random knot with vertices on sphere
Connect $n$ random points on a sphere in a cycle of
segments between succesive points:
I would like to know the growth rate, with respect to $n$, of the crossing number
(the minimal number of ...
- 151k
25
votes
1
answer
3k
views
Number of hypercube unfoldings
While writing the code for this answer, I noticed that I not only could calculate the number of unfoldings of the $4$-cube, but also the number of the $n$-cube for more values of $n$. Basically, we ...
- 10.7k
25
votes
1
answer
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What is a special parahoric subgroup?
Let me take this question again from the top.
I would like to know what a special parahoric subgroup is.
I think this is a "real" question, though not an especially good one -- it indicates my ...
24
votes
2
answers
4k
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complement of a totally disconnected closed set in the plane
While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected ...
24
votes
1
answer
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Building a genus-$n$ torus from cubes
I wonder if this has been studied:
What is the fewest number of unit cubes
from which one can build an $n$-toroid?
The cubes must be glued face-to-face,
and the boundary of the resulting object ...
- 151k
24
votes
3
answers
2k
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Real Lie groups versus real linear algebraic groups: differences in connexity and fundamental group
There are many introductory texts on real Lie groups, and many on linear algebraic groups in general, but fewer on the specific case of linear algebraic groups over the reals, and even fewer that try ...
- 32.5k
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2
answers
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Reference for exponential Vandermonde determinant identity
I recently stumbled upon the following identity, valid for any real numbers $\alpha_1,\dots,\alpha_n$ and $\lambda_{n1} \leq \dots \leq \lambda_{nn}$:
$$ \mathrm{det}( e^{\alpha_i \lambda_{nj}} )_{1 \...
- 114k
23
votes
2
answers
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Moments of Plücker coordinates on complex Grassmannian
Consider the Grassmannian $Gr(k,N)\simeq U(N)/(U(k)\times U(N-k))$ which parametrizes $k$-dimensional subspaces of $\mathbb{C}^N$. Let us put on it the $U(N)$-invariant probability measure. Let $\...
23
votes
9
answers
9k
views
The free group $F_2$ has index 12 in SL(2,$\mathbb{Z}$)
Is there someone who can give me some hints/references to the proof of this fact?
- 1,702
23
votes
3
answers
2k
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Is the analytic version of the Whitney Approximation Theorem true?
I initially asked this question on MSE but I haven't had any luck.
The Whitney Approximation Theorem states that any continuous map between smooth manifolds is homotopic to a smooth map. If the ...
- 19.3k
22
votes
5
answers
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Good reference for studying operads?
Can you, please, recommend a good text about algebraic operads?
I know the main one, namely, Loday, Vallette "Algebraic operads". But it is very big and there is no way you can read it fast. Also ...
22
votes
1
answer
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Why do homotopy theorists care whether or not $BP$ is $E_\infty$?
I have often heard that it is not known whether or not the Brown-Peterson spectrum $BP$ is an $E_\infty$-ring spectrum. Though I see that this is a somewhat natural question to ask, I have often ...
- 628
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2
answers
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Proofs of the Stallings-Swan theorem
It is a well-known and deep${}^\ast$ theorem that if a group $G$ has cohomological dimension one then it must be free. This was proved in the late 60's by Stallings (for finitely generated groups) and ...
- 35.9k
22
votes
1
answer
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Joyal's letter to Grothendieck
Mostly out of curiosity: Where do I find Joyal's letter to Grothendieck in which he defines a model structure on simplicial sheaves?
The question was already asked in this MO post, but that ...
- 1,450
21
votes
0
answers
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Cartan–Oka vanishing in one variable without $\overline{\partial}$?
This is a literature question, about possible proofs of some very basic results in complex analysis.
Some key facts about holomorphic functions are proved via reduction to smooth functions, using $\...
- 21.3k
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votes
3
answers
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Formality of classifying spaces
Let $G$ be a compact Lie group (or reductive algebraic group over $\mathbb{C}$), and let $BG$ be its classifying space. Fix a prime $p$. Let $\mathcal{A}$ denote the dg algebra of singular cochains on ...
- 6,162
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votes
2
answers
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Partitioning a polygon into convex parts
I'm looking for an algorithm to partition any simple closed polygon into convex sub-polygons--preferably as few as possible.
I know almost nothing about this subject, so I've been searching on Google ...
- 201
20
votes
6
answers
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Graph theory from a category theory perspective
Are there any textbooks on graph theory written for a category theorist?
It would probably have to be on directed graph theory, but if there's some trick we can use to talk about undirected graphs as ...
20
votes
2
answers
1k
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Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles
Some background on (compact) Belyi surfaces
$\newcommand{\Ch}{\hat{\mathbb{C}}}$
A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that $...
- 6,548
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votes
3
answers
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Non-stably trivial bundle with trivial characteristic classes
Though it's relatively clear that the characteristic classes do not characterise a vector bundle (and after looking through some books) I could not find an example of a vector bundle which is not ...
- 4,432
20
votes
7
answers
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Elementary reference for algebraic groups
I'm looking for a reference on algebraic groups which requires only knowledge of basic material on the theory of varieties which you could find in, for example, Basic Algebraic Geometry 1 by ...
- 15.4k