Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

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Jet bundles and partial differential operators

A geometric way of looking at differential equations In the literature for the h-principle (for example Gromov's Partial differential relations or Eliashberg and Mishachev's Introduction to the h-...
Willie Wong's user avatar
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32 votes
1 answer
2k views

Godel on recursion-theoretic hierarchies

At the end of his excellent article, "The Emergence of Descriptive Set Theory" (http://math.bu.edu/people/aki/2.pdf), Kanamori writes: "Another mathematical eternal return: Toward the end of his ...
Noah Schweber's user avatar
32 votes
0 answers
958 views

Is there a Mathieu groupoid M_31?

I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...
DavidLHarden's user avatar
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31 votes
19 answers
22k views

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
11 answers
23k views

A book for problems in Functional Analysis

I want to know if there's any book that categorizes problems by subjects of Functional Analysis. I'm studying Functional Analysis now a days and I really need to solve some problems in order to ...
31 votes
8 answers
8k views

"Modern" proof for the Baker-Campbell-Hausdorff formula

Does someone has a reference to a modern proof of the Baker-Campbell-Hausdorff formula? All proofs I have ever seen are related only to matrix Lie groups / Lie algebras and are not at all geometric (...
Mark.Neuhaus's user avatar
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31 votes
3 answers
12k views

What is the Katz-Sarnak philosophy?

It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some ...
Henry.L's user avatar
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31 votes
2 answers
3k views

Current status of Grothendieck's homotopy hypothesis and Whitehead's algebraic homotopy programme

(Disclaimer: I'm no expert in homotopy theory nor in higher categories!) If I understand it correctly, Grothendieck's homotopy hypothesis states that there should be an equivalence (of $(n+1)$-...
Qfwfq's user avatar
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31 votes
2 answers
1k views

Original source for Littlewood’s three precepts of refereeing in mathematics

I have a question regarding Littlewood’s three precepts of refereeing a mathematical paper, namely whether it is (1) new, (2) correct, and (3) interesting. I have found these mentioned in the ...
Christian Greiffenhagen's user avatar
31 votes
4 answers
2k views

Emergence of English as the dominant mathematical language

My impression is that most math papers (and almost all of the most important ones) are now published in English. Not long ago (historically) publishing in French, German, Russian, etc. were more ...
Noah Stein's user avatar
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31 votes
2 answers
3k views

Motivation behind Analytic Number Theory

I am an undergraduate student of mathematics and recently took an introductory course in analytic number theory, where the instructor roughly followed Apostol's first text on the subject. I have now ...
user135845's user avatar
31 votes
5 answers
8k views

How many binary operations are associative?

Let $X$ be a finite set of $n$ elements, and consider a binary operation $\odot: X \times X \rightarrow X$. There are $n^{n^2}$ such binary operations, as the $n \times n$ table entries can each be ...
Joseph O'Rourke's user avatar
31 votes
3 answers
3k views

References for Riemann surfaces

I know this question has been asked before on MO and MSE (here, here, here, here) but the answers that were given were only partially helpful to me, and I suspect that I am not the only one. I am ...
31 votes
2 answers
3k views

The Erdős-Turán conjecture or the Erdős' conjecture?

This has been bothering me for a while, and I can't seem to find any definitive answer. The following conjecture is well known in additive combinatorics: Conjecture: If $A\subset \mathbb{N}$ and $$\...
Eric Naslund's user avatar
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31 votes
1 answer
5k views

Modern proof of Serre's open image theorem?

Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic Curves'...
Adam Harris's user avatar
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31 votes
2 answers
3k views

On Grothendieck's idea on his Standard Conjecture B

Let me recall the Standard Conjecture B (see [1,2] below): The $\Lambda$-operation of Hodge theory is algebraic. It more or less says that the partial inverse to “cupping with the class of a ...
jmc's user avatar
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31 votes
2 answers
3k views

Mathematical Evidence Backing $|\mathbb{R}|=\aleph_2$

The "true" size of the real line, $\mathbb{R}$, has been the subject of Hilbert's first problem. Due to the Goedel and Cohen's work on the inner and outer models of $\text{ZFC}$, it turned ...
Morteza Azad's user avatar
31 votes
1 answer
2k views

Is this formal noncommutative power series identity known?

I recently discovered the following cute formal noncommutative power series identity: if $(x_i)_{i \in I}$ is some finite collection of noncommuting variables, then the formal power series $$ 1 + \...
Terry Tao's user avatar
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31 votes
1 answer
2k views

solving linear equations made difficult

(Note: This is a what's-in-the-literature question, not a what's-mathematically-true question, but I believe both are considered valid kinds of MathOverflow question.) I saw this amusing derivation ...
James Propp's user avatar
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31 votes
0 answers
1k views

"Three great cocycles" in Complex Analysis as cohomology generators

In his lecture notes, C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$ and the Schwarzian ...
Kostya_I's user avatar
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30 votes
7 answers
6k views

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 ...
Gray Taylor's user avatar
30 votes
11 answers
10k views

Introduction to deformation theory (of algebras)?

So I know that the idea of deformation theory underlies the concept of quantum groups; I haven't found any single introduction to quantum groups that makes me fully satisfied that I have some kind of ...
Harrison Brown's user avatar
30 votes
5 answers
5k views

Gossip about Grothendieck and distributive lattices

In Gian-Carlo Rota's Indiscrete Thoughts, there a list of mathematical gossip among which one reads: [...] What would have happened [...] if Grothendieck had known the theory of distributive ...
30 votes
8 answers
3k views

On independence and large cardinal strength of physical statements

The present post is intended to tackle the possible interactions of two bizarre realms of extremely large and extremely small creatures, namely large cardinals and quantum physics. Maybe after all ...
Morteza Azad's user avatar
30 votes
5 answers
8k views

Fermat's proof for $x^3-y^2=2$

Fermat proved that $x^3-y^2=2$ has only one solution $(x,y)=(3,5)$. After some search, I only found proofs using factorization over the ring $Z[\sqrt{-2}]$. My question is: Is this Fermat's original ...
Konstantinos Gaitanas's user avatar
30 votes
6 answers
9k views

learning crystalline cohomology

From which sources would you learn about crystalline cohomology and the de-Rham-Witt complex?
user avatar
30 votes
11 answers
13k views

Uniformization theorem for Riemann surfaces

How does one prove that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to ...
30 votes
8 answers
20k views

Reference book for commutative algebra

I'm looking for a good book in commutative algebra, so I ask here for some advice. My ideal book should be: More comprehensive than Atiyah–Macdonald More readable than Matsumura (maybe better ...
30 votes
4 answers
3k views

Algebraic P vs. NP

I recently attended a lecture where the speaker mentioned that what he was talking about was connected to the algebraic version of the $P$ vs. $NP$ problem. Could someone explain what that means in a ...
Sándor Kovács's user avatar
30 votes
6 answers
4k views

Book recommendation for cobordism theory

I am planning to organize a seminar on cobordism theory and I'm looking for a reference. Such a reference is preferably a book, but I'm open to other ideas. The audience is familiar with ...
Thomas Rot's user avatar
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30 votes
6 answers
5k views

Euclid with Birkhoff

I'm looking for a short and elementary book which does Euclidean geometry with Birkhoff's axioms. It would be best if it would also include some topics in projective (and/or) hyperbolic geometry. ...
Anton Petrunin's user avatar
30 votes
7 answers
14k views

What's the notation for a function restricted to a subset of the codomain?

Suppose I have a function f : A → B between two sets A and B. (The same question applies to group homomorphisms, continuous maps between topological spaces, etc. But for simpicity let's restrict ...
30 votes
3 answers
4k views

DG categories in algebraic geometry - guide to the literature?

Although my experience with DG categories is pretty basic I find them to be a very neat tool for organizing (co-)homological techniques in algebraic geometry. For someone who has algebro-geometric ...
Saal Hardali's user avatar
  • 7,549
30 votes
5 answers
6k views

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?) &...
André Henriques's user avatar
30 votes
2 answers
2k views

Unstable homotopy groups of spheres beyond Toda's range

In 1962 Toda published his book "Composition methods in homotopy groups of spheres", which contains computations of $\pi_{n+k}(S^n)$ for $k\le 19$ and $n\le 20$. The values of these groups are ...
Mark Grant's user avatar
30 votes
3 answers
5k views

When is an integral transform trace class?

Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator $$ K f(\xi) =\int_{X} f(x) k(x,\xi) d \mu(x),$$ the operator $K$ is Hilbert ...
Marc Palm's user avatar
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30 votes
6 answers
7k views

Algebraic stacks from scratch [closed]

I have a pretty good understanding of stacks, sheaves, descent, Grothendieck topologies, and I have a decent understanding of commutative algebra (I know enough about smooth, unramified, étale, and ...
30 votes
2 answers
15k views

A road to inter-universal Teichmuller theory

What would be a study path for someone in the level of Hartshorne's Algebraic Geometry to understand and study inter-universal Teichmuller (IUT) theory? I know that it heavily relies on anabelian ...
terett's user avatar
  • 1,069
30 votes
2 answers
3k views

An unfair marriage lemma

I am looking for a citeable reference to the following generalization of Hall's Marriage Theorem: Given a bipartite graph of boys and girls. In addition to gender difference, they are divided into ...
Sergei Ivanov's user avatar
30 votes
1 answer
588 views

Guess that group via product queries

Suppose someone (person B) knows a finite group $G$ of order $n$. You (person A) know only the order $n$, and that $1$ is the name of the identity element. The group elements are named $1,2,\ldots,n$ ...
Joseph O'Rourke's user avatar
30 votes
0 answers
989 views

Follow-up to Steinberg's problem (12) in his 1966 ICM talk?

Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (...
Jim Humphreys's user avatar
29 votes
5 answers
6k views

"The complex version of Nash's theorem is not true"

The title quote is from p.221 of the 2010 book, The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions by Shing-Tung Yau and Steve Nadis. "Nash's theorem" here ...
Joseph O'Rourke's user avatar
29 votes
19 answers
39k views

Good books on problem solving / math olympiad [closed]

I would want all book tips you could think of regarding problem solving and books in general, in elementary mathematics, with a certain flavour for "advanced problem solving". An example ...
29 votes
9 answers
10k views

Diophantine equation with no integer solutions, but with solutions modulo every integer

It's probably common knowledge that there are Diophantine equations which do not admit any solutions in the integers, but which admit solutions modulo $n$ for every $n$. This fact is stated, for ...
Faisal's user avatar
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29 votes
2 answers
4k views

Closed formula for a certain infinite series

I came across this problem while doing some simplifications. So, I like to ask QUESTION. Is there a closed formula for the evaluation of this series? $$\sum_{(a,b)=1}\frac{\cos\left(\frac{a}b\right)}{...
T. Amdeberhan's user avatar
29 votes
5 answers
3k views

Most manifolds are hyperbolic?

I heard the claim as in the title for a long time, but can not find the precise reference for this claim, what's the reference with proof for this claim? Thanks for the help. To be more precise, is ...
mmaatthh's user avatar
  • 789
29 votes
3 answers
2k views

All polynomials are the sum of three others, each of which has only real roots

It was asked at the Bulletin of the American Mathematical Society Volume 64, Number 2, 1958, as a Research Problem, if a Hurwitz polynomial with real coefficients (i.e. all of its zeros have negative ...
jack's user avatar
  • 2,929
29 votes
4 answers
5k views

What is Ricardo Pérez-Marco's eñe product? Does it explain his statistical results on differences of zeta zeros?

The number theory community here at University of Michigan is abuzz with talk of this paper recently posted to the arxiv. If you haven't seen it already, the punch line is that the global differences ...
Gene S. Kopp's user avatar
  • 2,190
29 votes
6 answers
1k views

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 ...
29 votes
6 answers
37k views

Reading materials for mathematical logic [closed]

Hi everyone, the summer break is coming and I am thinking of reading something about mathematical logic. Could anyone please give me some reading materials on this subject?

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