Questions tagged [big-list]

Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.

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5 votes
0 answers
597 views

Rings such that torsion-free/flat/projective modules are flat/projective/free

While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, ...
16 votes
3 answers
3k views

Current Research in Numeric Mathematics

To me, as an non-expert in the field, it seems as if numeric mathematics should have lost its importance because nowadays symbolic calculations or calculations with unlimited precision are generally ...
12 votes
2 answers
1k views

Negative sectional curvature and constant curvature

Good morning everyone, I was wondering about the difference between manifolds carrying a Riemannian metric with negative sectional curvature and hyperbolic manifolds. I was told once "there are ...
21 votes
6 answers
2k views

Combinatorial sequences whose ratios $a_{n+1}/a_{n}$ are integers

I have a proof technique in search of examples. I'm looking for combinatorially meaningful sequences $\{a_n\}$ so that $a_{n+1}/a_n$ is known or conjectured to be an integer, such that there is a ...
9 votes
3 answers
718 views

Judging whether a finitely presented group is a 3-manifold group?

Given a finitely presented group $G$, how many necessary conditions do people know for $G$ to be isomorphic to the fundamental group of some closed connected 3-manifold? (e.g. residually finite)
10 votes
5 answers
4k views

Examples of inductive proofs that can be generalized by transfinite induction

Hello. I am currently searching for some nice examples of proofs by induction in the finite case, that can be generalized to the infinite case using transfinite induction (and dont become trivial ...
2 votes
2 answers
517 views

Characterizations of amenable groups which use the space $\ell_1(G)$ and convolution

Let $G$ be a discrete group. Do you know characterizations of amenable groups which use the space $\ell_1(G)$ and convolution? I only know Johnson's theorem: A group is amenable if and only if the ...
18 votes
22 answers
4k views

LaTeX based document editors

I'm afraid my first question isn't a math puzzle per se, but rather question of math "presentation" . Basically I've been out of school for a year or two - so I'm a bit out of practice in writing up ...
16 votes
3 answers
1k views

What are the main open problems in the theory of amenability of groups?

I have read the Paterson and Runde books about amenability of groups, but I do not know what are the most intriguing questions in this area today. A survey or a list of questions would be welcome.
80 votes
23 answers
18k views

Algebraic geometry examples

What are some surprising or memorable examples in algebraic geometry, suitable for a course I'll be teaching on chapters 1-2 of Hartshorne (varieties, introductory schemes)? I'd prefer examples that ...
35 votes
12 answers
8k views

How to write math well?

Let's learn about writing good mathematical texts. For some people it could be especially interesting to answer about writing texts on Math Overflow, though I personally feel like I've already ...
22 votes
1 answer
2k views

A list of proofs of the Hasse–Minkowski theorem

I am currently doing a project in which I intend to include the most insightful possible proof of the Hasse–Minkowski theorem (also known as the Hasse principle for quadratic forms, among other names) ...
39 votes
4 answers
6k views

Linear algebra in terms of abstract nonsense?

The categories of vector spaces and finite dimensional vector spaces are pretty much as nice as can be, I think. I was wondering what portions of basic linear algebra (first couple of courses) fall ...
127 votes
19 answers
69k views

Periods and commas in mathematical writing

I just realized that I am a barbarian when it comes to writing. But I am not entirely sure, so this might be the right place to ask. When typing display-mode formulae do you guys add a period after ...
-1 votes
1 answer
280 views

List of obscure summation identities [closed]

I am trying to evaluate a fairly simple summation: $\sum_{k=1}^n ka^kb^{n-k}$ Which is related to the common identity for $\sum_{k=1}^n ka^k$ available on Wikipedia. I've previously seen lengthy lists ...
78 votes
12 answers
94k views

What practical applications does set theory have?

I am a non-mathematician. I'm reading up on set theory. It's fascinating, but I wonder if it's found any 'real-world' applications yet. For instance, in high school when we were learning the ...
74 votes
29 answers
13k views

(Preferably rare) Audio/Video recordings of famous mathematicians?

Terence Tao's homepage has a link to a collection of quotes, and one among them was Hilbert's famous "We must know, we will know" quote. This quote also had an audio link to it. Now although I'm not ...
57 votes
14 answers
18k views

Open problems in Euclidean geometry?

What are some (research level) open problems in Euclidean geometry ? (Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet) I should clarify a bit ...
53 votes
6 answers
5k views

Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false

What are some examples of serious mathematical theory-building around hypotheses that are believed or known to be false? One interesting example, and the impetus for this question, is work in number ...
33 votes
9 answers
4k views

Theorems first published in textbooks?

According to Wikipedia, the Bohr-Mollerup Theorem (discussed previously on MO here) was first published in a textbook. It says the authors did that instead of writing a paper because they didn't think ...
119 votes
33 answers
14k views

Examples of theorems misapplied to non-mathematical contexts

For something I'm writing -- I'm interested in examples of bad arguments which involve the application of mathematical theorems in non-mathematical contexts. E.G. folks who make theological arguments ...
5 votes
2 answers
484 views

Critical points in $ZF$ without Choice

Recall the definition of critical point for set theory: A critical point of an elementary embedding of one transitive class into another transitive class is the smallest ordinal not mapped to ...
6 votes
0 answers
544 views

What are the topics in noncommutative algebraic geometry?

Preface: I know very little about noncommutative algebra and noncommutative geometry, so please feel free to make improvement suggestions for my question. Also, to my knowledge there are several ...
13 votes
2 answers
1k views

A comprehensive list of random walk inequalities?

I am interested in finding a comprehensive list of all noticeable random walk inequalities. ie. $S_n = \sum_{k\leq n} X_i$ for i.i.d symmetric $X_i$ I can only seem to find books/papers that list ...
5 votes
5 answers
1k views

Interesting topics for (very) short talks [closed]

Part of the requirements for my Honours is that I record a short 4-7 minute digital talk, which is then distributed to all the other students and staff at my university’s mathematics department. The ...
0 votes
4 answers
2k views

Math journals which publish/reject quickly [closed]

I would like to publish a math paper quickly. The level of journal is not that important (except that it should not send out spam with its own ads). I am looking for a math journal which decides ...
75 votes
13 answers
12k views

What precisely Is "Categorification"?

(And what's it good for.) Related MO questions (with some very nice answers): examples-of-categorification; can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$?; categorification-request.
53 votes
10 answers
7k views

Changes forced by the pandemic

The Covid-19 pandemic has changed our work-lives in ways few of us could have anticipated. These exceptional circumstances have forced each one of us and each one of our institutions to adapt, ...
108 votes
15 answers
10k views

Are there any good websites for hosting discussions of mathematical papers?

I was wondering if there are any websites out there which systematically provide space for the discussion of mathematics articles (particularly those on the arXiv, though not necessarily just those),...
6 votes
3 answers
548 views

Anomalous phenomena [closed]

What are examples of strikingly anomalous phenomena in mathematics, where just one or a rather small number of cases stand out because they don't fit a general pattern? This is most interesting when ...
11 votes
11 answers
1k views

Lattices on classical combinatorial families

I am asking for examples of lattices defined on classical combinatorial families, such as Permutations, Catalan objects, set partitions or integer partitions, graphs. I am mosty interested in lattices ...
86 votes
14 answers
9k views

LaTeX tricks that save time in typesetting

In ${\rm\LaTeX}$ typesetting, when we repeat a long and complex formula in long documents, it is appropriate to create a new command that just by calling this new command we get the desired output. ...
144 votes
21 answers
20k views

How does one justify funding for mathematics research?

G. H. Hardy's A Mathematician's Apology provides an answer as to why one would do mathematics, but I'm unable to find an answer as to why mathematics deserves public funding. Mathematics can be ...
34 votes
8 answers
4k views

Uncountable counterexamples in algebra

In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
68 votes
30 answers
36k views

A book you would like to write

Writing a book from the beginning to the end is (so I heard) a very hard process. Planning a book is easier. This question is dual in a sense to the question "Books you would like to read (if somebody ...
86 votes
27 answers
19k views

Which popular games have been studied mathematically?

I'm planning out some research projects I could do with undergraduates, and it struck me that problems analyzing games might be appropriate. As an abstract homotopy theorist, I have no experience with ...
36 votes
14 answers
5k views

What are interesting families of subsets of a given set?

Motivation The usual starting point of both Topology and Measure Theory is the definition of a family of subsets of a set $S$. Indeed, one defines a topology on $S$ to be a family of subsets ...
45 votes
39 answers
8k views

What out-of-print books would you like to see re-printed?

It's excellent news that the LMS are to re-publish Cassels & Fröhlich. There are many other excellent mathematics books which are just about impossible (or at least very expensive) to get hold of,...
35 votes
12 answers
3k views

No canonical isomorphism [duplicate]

I thought that it would be interesting to collect into a big list various instances of isomorphic structures with no preferred isomorphism between them. I expect the examples to be interesting since ...
21 votes
9 answers
2k views

Naturally occurring examples of badly behaved categories

What are some examples of naturally occurring badly behaved (possibly higher) categories? When working with a specific category like ${\bf Set}$ or ${\bf Cat}$, we usually understand/explain them by ...
114 votes
96 answers
16k views

What would you want to see at the Museum of Mathematics? [closed]

EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...
92 votes
74 answers
26k views

Pseudonyms of famous mathematicians

Many mathematicians know that Lewis Carroll was quite a good mathematician, who wrote about logic (paradoxes) and determinants. He found an expansion formula, which bears his real name (Charles ...
127 votes
13 answers
29k views

Why are modular forms interesting?

Well, I'm aware that this question may seem very naive to the several experts on this topic that populate this site: feel free to add the "soft question" tag if you want... So, knowing nothing about ...
10 votes
2 answers
1k views

How professional mathematicians deal with discouragement? [closed]

All professional mathematicians feel discouraged occasionally due to some issue. My question is: How do professional mathematicians deal with discouragement? In this link , Andrew Wiles say ...
66 votes
17 answers
12k views

Shortest/Most elegant proof for $L(1,\chi)\neq 0$

Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$? Background: The non-vanishing ...
47 votes
23 answers
76k views

Good programs for drawing (weighted directed) graphs

Does anyone know of a good program for drawing directed weighted graphs?
40 votes
15 answers
28k views

Mathematical podcasts/audio

Just to ask if anyone is aware of any interesting math podcasts? I am particularly interested in podcasts describing mathematics in the wider world; but interesting academic podcasts would also be ...
6 votes
0 answers
310 views

Which journals publish mathematics book reviews?

Which mathematics journals publish book reviews? So far I have the following: Notices of the American Mathematical Society Bulletin of the American Mathematical Society (From looking at its website ...
13 votes
5 answers
1k views

Connectedness in the plane

There are several open problems in topology which concern connectedness and subsets of the plane. The biggest of these is undoubtedly: Question. Does every non-separating plane continuum have the ...
6 votes
0 answers
200 views

What is known about "dimension two" vertex algebras?

In the paper Chiral Koszul duality, Gaitsgory and Francis develop a notion of a chiral algebra living on an arbitrary variety $X$. When $X=\mathbf{A}^1$ and the chiral algebra is translation invariant,...

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