All Questions
23,892 questions
40
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Reference on Persistent Homology
I will be teaching a course on algebraic topology for MSc students and this semester, unlike previous ones where I used to begin with the fundamental group, I would like to start with ideas of ...
- 3,173
40
votes
5
answers
8k
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Is $\zeta(3)/\pi^3$ rational?
Apery proved in his paper from 1979 that $\zeta(3)$ is irrational, and we know that for all integers $n$,
$\zeta(2n)=\alpha \pi^{2n}$
for some $\alpha\in \mathbb{Q}$. Given these facts, it seems ...
- 7,013
40
votes
5
answers
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views
"Entropy" proof of Brunn-Minkowski Inequality?
I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality.
The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then
$$ m(...
- 22.8k
40
votes
4
answers
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Pre-triangulated category that isn't triangulated
I've been working through some of the early parts of Neeman's book on triangulated categories, and he mentions that he does not know of a pre-triangulated category that is not triangulated. Is this ...
- 13.5k
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5
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Is there a natural measures on the space of measurable functions?
Given a set Ω and a σ-algebra F of subsets, is there some natural way to assign something like a "uniform" measure on the space of all measurable functions on this space? (I suppose first ...
- 1,475
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6
answers
8k
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Doing geometry using Feynman Path Integral?
I have often heard in the folk-lore that Feynman Path Integral can be used to compute geometric invariants of a space.
Coming from a background of studying Quantum Field Theory from the books like ...
- 3,541
40
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2
answers
2k
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Can the nth projective space be covered by n charts?
That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$?
I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t ...
- 10.6k
40
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3
answers
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Good user manuals for technical topics?
This question is motivated by this (highly recommended) comment by Emerton on Terry Tao's post "Learn and relearn your field". In particular, the following paragraphs:
In particular, the ...
39
votes
10
answers
7k
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Dimensional Analysis in Mathematics
Is there a sensible and useful definition of units in mathematics? In other words, is there a theory of dimensional analysis for mathematics?
In physics, an extremely useful tool is the Buckingham Pi ...
39
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10
answers
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Are there some other notions of "curvature" which measure how space curves?
I am learning differential geometry and have a few questions on curvature. -- Background:
Gauss invented "Gauss curvature" to measure how surface curves.
Riemann gives an ingenious generalization of ...
39
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5
answers
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The letter $\wp$; Name & origin?
Do you think the letter $\wp$ has a name? It may depend on community - the language, region, speciality, etc, so if you don't mind, please be specific about yours. (Mainly I'd like to know the English ...
- 665
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4
answers
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On critical reviews of Hawking's lecture "Gödel and the end of the universe"
The search for a neat Theory of Everything (ToE) which unifies the entire set of fundamental forces of the universe (as well as the rules which govern dark energy, dark matter and anti-matter realms) ...
39
votes
3
answers
14k
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Is the Invariant Subspace Problem interesting?
There's an amusing comment in Peter Lax's Functional Analysis book. After a brief description of the Invariant Subspace Problem, he says (paraphrasing) "...this question is still open. It is also an ...
- 1,241
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votes
8
answers
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Good introductory book to type theory?
I don't know anything about type theory and I would like to learn it.
I'm interested to know how we can found mathematics on it.
So, I would be interested by any text about type theory whose angle ...
39
votes
2
answers
3k
views
Volume of the unitary group
I saw a very remarkable asymptotic formula (or a conjecture?) for the volume of of the unitary group $ U(n)$ which is the following:
$$\log[\mathrm{Volume}(U(n))] \sim_{n\rightarrow \infty} \frac{n^...
- 1,607
39
votes
2
answers
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Euler's Master's Thesis
At the age of 16, Leonhard Euler defended his Master's Thesis, where he discussed and compared Descartes' and Newton's approaches to planet motion. I don't know anything else about it. In particular, ...
- 52.3k
39
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3
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Manifold of probability measures: connections between two types of metrics
The space of probability measures could be viewed as an infinite-dimensional manifold, equipped with two possible types of metrics — (1) Wasserstein and (2) Fisher-Rao. Metric (1) is connected with ...
- 1,127
39
votes
2
answers
6k
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What is Serre's condition (S_n) for sheaves?
The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...
- 30.5k
38
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5
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Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?
The Diophantine equation
$$x^3+y^3+z^3=3$$
has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
- 2,350
38
votes
10
answers
18k
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Fast matrix multiplication
Suppose we have two $n$ by $n$ matrices over particular ring. We want to multiply them as fast as possible. According to wikipedia there is an algorithm of Coppersmith and Winograd that can do it in $...
- 1,791
38
votes
2
answers
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What, exactly, has Louis de Branges proved about the Riemann Hypothesis?
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis ...
- 1,990
38
votes
6
answers
7k
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Companion to theoretical physics for working mathematicians
In the Princeton Companion to Mathematics one reads that even pure mathematicians should know some theoretical physics and applied mathematics. What are some well-organized comprehensive companions to ...
38
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1
answer
4k
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Updates to Stanley's 1999 survey of positivity problems in algebraic combinatorics?
In 1999, Richard Stanley wrote a very nice survey on open problems in algebraic combinatorics, with a specific focus on positivity, called "Positivity problems and conjectures in algebraic ...
38
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4
answers
3k
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Is there research on human-oriented theorem proving?
I know there is already a research community that is working on automatic theorem proving mostly using logic (and things like Coq and ACL2). However, I came across a lecture from a fields medalist W.T....
- 524
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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 ...
37
votes
6
answers
3k
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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
3
answers
3k
views
On Mathematical Analysis of MathSciNet & MathOverflow
This question has two original motivations: mathematical and social.
The mathematical motivation is mainly based on what I have seen about Zipf's law here and there. The Zipf's law simply states ...
37
votes
11
answers
8k
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"Must read" papers in numerical analysis
In 1993, Prof. L.N. Trefethen published a NA-net posting with a list of thirteen paper he used for teaching the seminar Classic Papers in Numerical Analysis.
In Trefethen's words, ... this course ...
37
votes
2
answers
4k
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How to find Erdős' treasure trove?
The renowned mathematician, Paul Erdős, has published more than 1500 papers in various branches of mathematics including discrete mathematics, graph theory, number theory, mathematical analysis, ...
37
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2
answers
3k
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Building algebraic geometry without prime ideals
$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\ev{ev}$Teaching algebraic geometry, in particular schemes, I am struggling to provide intuitive proofs. In particular, I find it counter-intuitive ...
- 3,772
37
votes
4
answers
4k
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Which differential equations allow for a variational formulation?
Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation
$$
\frac{d}{dt}\frac{\...
- 7,583
37
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1
answer
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What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
Variants have been asked here before (e.g. Which small finite ...
- 2,959
37
votes
2
answers
2k
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Moving one family of commuting self-adjoint operators to another without losing commutativity on the way
This is actually not a question of mine, so I'll be short on motivation and say nothing beyond that if this were true, a few fancy harmonic analysis techniques that a colleague of mine used in proving ...
- 61.9k
36
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5
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What is the equivariant cohomology of a group acting on itself by conjugation?
This question makes sense for any topological group $G$, but I'd particularly like to know the answer for $G$ a compact, connected Lie group.
$G$ acts on itself by conjugation. One has the equivariant ...
- 13.2k
36
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4
answers
4k
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How far is Lindelöf from compactness?
A while ago I heard of a nice characterization of compactness but I have never seen a written source of it, so I'm starting to doubt it. I'm looking for a reference, or counterexample, for the ...
- 2,396
36
votes
2
answers
5k
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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
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10
answers
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Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature
A curve in the plane is determined, up to orientation-preserving
Euclidean
motions, by its curvature function, $\kappa(s)$.
Here is one of my favorite examples, from
Alfred Gray's book,
Modern ...
- 151k
36
votes
3
answers
5k
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What are D-branes, really?
In the past couple years, I've read many words pertaining to "D-branes" without feeling I have fully comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as ...
- 1,415
36
votes
2
answers
32k
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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
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1
answer
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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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The coupon collector's earworm
[EDITED mostly to report on the answer by Kevin Costello
(and to improve the gp code at the end)]
I thank Nicolas Dupont for the following question
(and for permission to disseminate it further):
...
- 79.9k
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0
answers
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Grothendieck's "List of classes of structures"
In Lawvere's article Comments on the Development of Topos Theory, the author writes:
Similarly, Grothendieck and others unerringly recognized which kinds of mathematical structures are 'preserved ...
- 10.5k
35
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7
answers
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Fraktur symbols for Lie algebras
Does anyone know when and why the Fraktur script was introduced for Lie and other algebras—$\mathfrak{g}$, $\mathfrak{gl}_n$, $X/\mathfrak{g}$,
$\mathfrak{g}\oplus\mathfrak{g}$, $\mathfrak{su}$, ...
- 151k
35
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7
answers
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Textbook for Etale Cohomology
What is the best textbook (or book) for studying Etale cohomology?
user16974
35
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4
answers
2k
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Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?
A $4\times 4$ symmetric matrix
$$
\left(
\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{12} & a_{22} & a_{23} & a_{24} \\
a_{13} & a_{23} & a_{33} & ...
- 2,527
35
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7
answers
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Paradoxical Mathematical Objects Pending for Construction [duplicate]
The possible properties and applications of some mathematical objects have been described far before their rigorous mathematical definition. Some of them even had a seemingly paradoxical description ...
35
votes
4
answers
6k
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How are infinite-dimensional manifolds most commonly treated?
I originally posted this question on StackExchange, where it was suggested I post here. It was also suggested I read about Hilbert manifolds and Fréchet manifolds. Nevertheless, I am still looking for ...
- 655
35
votes
5
answers
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A reference for geometric class field theory?
The classic reference of this topic is Serre's Algebraic Groups and Class Fields. However, many parts of this book use Weil's language, which I find quite hard to follow. Is there another reference ...
- 805
35
votes
6
answers
10k
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Roadmap for Mirror Symmetry
I am interested in learning Mirror Symmetry, both from the SYZ and Homological point of view. I am taking a reading course in Mirror Symmetry, which will focus on the SYZ side.
I know basic Complex ...
- 3,218