All Questions
23,892 questions
48
votes
6
answers
7k
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Is there an "elegant" non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?
Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway...
Consider this problem. I've been trying to find a formula to expand the "regular iteration" of "...
48
votes
2
answers
14k
views
Research situation in the field of Information Geometry
I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field.
I have read (4) and parts of (3). ...
- 8,071
47
votes
11
answers
5k
views
Reference request: Examples of research on a set with interesting properties which turned out to be the empty set
I've seen internet jokes (at least more than 1) between mathematicians like this one here about someone studying a set with interesting properties. And then, after a lot of research (presumably after ...
47
votes
1
answer
13k
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Lecture notes by Thurston on tiling
I am looking for a copy of the following
W. Thurston, Groups, tilings, and finite state automata, AMS Colloquium Lecture Notes.
I see that a lot of papers in the tiling literature refer to it but I ...
- 1,795
47
votes
3
answers
3k
views
Why is the Vandermonde determinant harmonic?
It can be checked that the Vandermonde determinant defined as
$$V(\alpha_1, \cdots, \alpha_n) = \prod_{1 \le i < j \le n}(\alpha_i-\alpha_j) $$
is a harmonic function, that is $\Delta V = 0$ where ...
- 763
47
votes
11
answers
15k
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Standard model of particle physics for mathematicians
If a mathematician who doesn't know much about the physicist's jargon and conventions had the curiosity to learn how the so called Standard Model (of particle physics, including SUSY) works, where ...
47
votes
6
answers
6k
views
Can we actually find any fixed points with Brouwer's theorem?
Background
At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is ...
- 15.5k
47
votes
4
answers
5k
views
What is the source of this famous Grothendieck quote?
I've seen the following quote many times on the internet, and have used it myself. It is usually attributed to Grothendieck.
It is better to have a good category with bad objects than a bad category ...
- 22.1k
47
votes
1
answer
3k
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Which small finite simple groups are not yet known to be Galois groups over Q?
The subject line pretty much says it all. To expand just a little bit:
1) What is the smallest simple group that is not yet known to occur as a Galois group over $\mathbb{Q}$? (Variants: not known ...
- 65.4k
47
votes
1
answer
1k
views
Summing infinitely many infinitesimally small variables makes sense in algebra
There is an identity $e^x=\lim_{n\to \infty} (1+x/n)^n$, and I always thought it is a purely analytic statement. But then I discovered its curious interpretation in pure algebra:
Consider the ring of ...
- 3,772
46
votes
0
answers
2k
views
Set-theoretic reformulation of the invariant subspace problem
The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and $l^2$...
- 42.8k
45
votes
7
answers
9k
views
What's an example of a space that needs the Hahn-Banach Theorem?
The Hahn-Banach theorem is rightly seen as one of the Big Theorems in functional analysis. Indeed, it can be said to be where functional analysis really starts. But as it's one of those "there ...
- 26.8k
45
votes
7
answers
16k
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What is an intuitive view of adjoints? (version 2: functional analysis)
After realising that I don't have an intuitive understanding of adjoint functors, I then realised that I don't have an intuitive understanding of adjoint linear transformations!
Again, I can use 'em, ...
- 26.8k
45
votes
8
answers
10k
views
A down-to-earth introduction to the uses of derived categories
When I was learning about spectral sequences, one of the most helpful sources I found was Ravi Vakil's notes here. These notes are very down-to-earth and give a kind of minimum knowledge needed about ...
45
votes
1
answer
3k
views
Hilbert's alleged proof of the Continuum Hypothesis in "On the Infinite"
As is known, Hilbert attempted a proof sketch of the Continuum Hypothesis in the latter part of his paper, "On the Infinite". It is also known that it is false.
Has there ever been a published ...
- 6,132
45
votes
10
answers
11k
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The functional equation $f(f(x))=x+f(x)^2$
I'd like to gather information and references on the following functional equation for power series $$f(f(x))=x+f(x)^2,$$$$f(x)=\sum_{k=1}^\infty c_k x^k$$
(so $c_0=0$ is imposed).
First things that ...
- 60.5k
45
votes
1
answer
2k
views
Existence and uniqueness of Haar measure on compacta; a cohomological approach
I am trying to use a modification of group cohomology to prove the existence and uniqueness of Haar measure on a compact Hausdorff group.
I think the best way of introducing the idea I am pursuing is ...
user30211
45
votes
1
answer
2k
views
Pach's "Animals": What if the genus is positive?
Janos Pach asked a deep question 23 years ago (1988) that remains unsolved today:
Can every animal—a topological ball in $\mathbb{R^3}$ composed of unit cubes glued face-to-face—be ...
- 151k
44
votes
10
answers
11k
views
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
44
votes
10
answers
47k
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Is square of Delta function defined somewhere?
I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere.
In the beginning, this question might look strange. But by restricting the space of the test functions, ...
44
votes
2
answers
6k
views
Clausen's modified Hodge Conjecture
In a recent talk at the University of Geneve, Dustin Clausen presented a "modified Hodge Conjecture". I found the abstract intriguing but couldn't find videos or notes available online.
If I'...
user497064
44
votes
2
answers
2k
views
Fermat's Last Theorem for integer matrices
Some years ago I was asked by a friend if Fermat's Last Theorem was true for matrices. It is pretty easy to convince oneself that it is not the case, and in fact the following statement occurs ...
- 1,889
44
votes
6
answers
12k
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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
1
answer
4k
views
Example of a compact set that isn't the spectrum of an operator
This question is somewhat ill-posed (due to the word easy) and is triggered by idle curiosity:
Is there an easy example of a (separable, infinite-dimensional) Banach space $X$ and a nonempty ...
- 5,743
43
votes
10
answers
7k
views
What is the shortest program for which halting is unknown?
In short, my question is:
What is the shortest computer program for which it is not known whether or not the program halts?
Of course, this depends on the description language; I also have the ...
- 23k
43
votes
16
answers
9k
views
Essential reads in the philosophy of mathematics and set theory
I am graduate student and have a decent understanding of logic and set theory.
Recently I have got interested in the philosophy of mathematics and set theory. I have read a number of papers by ...
43
votes
2
answers
4k
views
About a letter by Richard Palais of 1965.
In Cushman and Bates, Global Aspects of Classical Integrable Systems, 1997, I have read
In a widely circulated but unpublished letter in 1965, Palais explained the symplectic formulation of ...
- 4,306
43
votes
3
answers
7k
views
Could the Riemann zeta function be a solution for a known differential equation?
Riemann zeta function is a function of complex variable $s$ that analytically continous the sum of Dirichlet series .defined as :$$\zeta(s)=\sum_{n=1}^{\infty}\displaystyle \frac{1}{n^s} $$ for when ...
43
votes
4
answers
5k
views
Lists as a foundation of mathematics
I am wondering if there is a foundation of mathematics where not sets or "set-like objects" (such as objects of a suitable topos as in ETCS) are the primitive notion, but rather lists. These ...
- 63.1k
43
votes
8
answers
3k
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How to quantify noncommutativity?
If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is computable easily for ...
- 1,890
43
votes
3
answers
9k
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Why the name 'separable' space?
It is well known that a separable space is a topological space that has a countable dense subset. I am wondering how is this related to the name 'separable'? Any intuition where the name come from?
- 1,157
43
votes
1
answer
5k
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Can $L^p(\mathbb{R})$ and $ L^q(\mathbb{R})$ be isomorphic?
Let $p,q \in (1,\infty)$ with $p\neq q$. Are the Banach spaces $L^p(\mathbb{R})$, $L^q(\mathbb{R})$ isomorphic?
- 559
43
votes
3
answers
3k
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Is this integral representation of $\zeta(2n+1)$ known?
Background: I'm an undergraduate at an institution with no researchers in analytic number theory, and no ties to the analytic number theory community. I believe I have found what is, as far as I can ...
- 383
42
votes
2
answers
5k
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Homotopy groups of $S^2$
in the paper
Foundations of the theory of bounded cohomology,
by N.V. Ivanov, the author considers the complex of bounded singular cochains on a simply connected CW-complex $X$, and constructs a ...
- 3,749
42
votes
5
answers
14k
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The unproved formulas of Ramanujan
Are there any formulas due to Ramanujan that have still not been proved—or disproved?
If so, what are they?
I believe this conjecture is due to Ramanujan and still open: if $x$ is a real number and $2^...
- 22.3k
42
votes
1
answer
5k
views
Did Hilbert laugh?
Prof. D. C. McCarty recently gave an interesting
interview (published in January 2015, and easily
found on a large video hosting site), entitled
What are the limits of mathematical explanation?
I ...
- 6,051
42
votes
6
answers
4k
views
What are good articles/books on the psychology of mathematical research?
I am thinking about advanced texts similar to Polya's 'How to solve it?'. Quite a few good articles of such a kind are published under Philosophy of Mathematics, but that dwells on a very different ...
42
votes
4
answers
4k
views
Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?
Now that some of the previously MSE formulae that I left here have been applied Dec.2023 to compute high precision record values ($10^{12}$ decimal digits) of trascendental constants $\Gamma(1/3)$ (Eq....
- 2,836
41
votes
6
answers
4k
views
Measures of non-abelian-ness
Let $G$ be a finite non-abelian group of $n$ elements.
I would like a measure that intuitively captures the
extent to which $G$ is non-commutative.
One easy measure is a count of the non-commutative ...
- 151k
41
votes
11
answers
5k
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Topology in non-mathematical literature
A great piece of knowledge that I heard from a talk of Robert Ghrist, is that one of the earliest instances of non-trivial manifolds (i.e. of dimension higher than 2) appears in Dante's Paradise, ...
41
votes
2
answers
17k
views
Introductory text on Galois representations
Could someone please recommend a good introductory text on Galois representations? In particular, something that might help with reading Serre's "Abelian l-Adic Representations and Elliptic Curves" ...
41
votes
8
answers
16k
views
What are some good group theory references?
I'm curious about what books people use for a group theory reference. I don't currently own a dedicated group theory book, and I think I'd find such a book very helpful in my research. What is your ...
41
votes
3
answers
3k
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Did Emmy Noether ever publish under a man's name?
A recent article in the New York Times, http://www.nytimes.com/2012/03/27/science/emmy-noether-the-most-significant-mathematician-youve-never-heard-of.html?pagewanted=all says, among other things, "...
- 39.9k
41
votes
4
answers
16k
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Product of Borel sigma algebras
If $X$ and $Y$ are separable metric spaces, then the Borel $\sigma$-algebra $B(X \times Y)$ of the product is the $\sigma$-algebra generated by $B(X)\times B(Y)$. I am embarrassed to admit that I ...
- 31.5k
41
votes
2
answers
9k
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What should I read before reading about Arakelov theory?
I tried reading about Arakelov theory before, but I could never get very far. It seems that this theory draws its motivation from geometric ideas that I'm not very familiar with.
What should I read ...
41
votes
2
answers
2k
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Topple height of randomly stacked bricks
What is the expected height of a stack of unit-length bricks, each one
stacked on the previous with a uniformly random shift within $\pm \delta$?
The stack topples if the center of gravity of the top $...
- 151k
41
votes
0
answers
2k
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What does the theta divisor of a number field know about its arithmetic?
This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link).
Let ...
- 2,029
40
votes
29
answers
8k
views
Autobiographies of mathematicians
According to Wikipedia, an autobiography is an account of the life of a person, written by that person sometimes with a collaborator.
An autobiography offers the author the ability to recreate history....
40
votes
11
answers
12k
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Contemporary philosophy of mathematics
Starting to write an introduction to the philosophy of mathematics, I find tons of positions that are of historical interest. Which philosophical positions are explicitly considered these days, say in ...
40
votes
8
answers
12k
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What is the proper initiation to the theory of motives for a new student of algebraic geometry?
A preliminary apology is in order: I realize that most of my contributions to this site are in the form of reference requests. I understand that this makes it seem as though I do nothing more than sit ...
- 1,539