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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Adjunctions in the real world

What concepts in the real world can be described by adjunctions? For example, parents and children are adjoint to one another. Specifically, work in $ZFC$ plus a finite class of atoms $\mathscr{X}$ (...
Alec Rhea's user avatar
  • 8,977
8 votes
1 answer
353 views

Formalisation of intuitive concepts in the language leading to mathematical progress

In his work, Albert Lautman thinks the genesis of some mathematical works as a dialectic that takes place between opposite notions, like between global and local. He argues that while those notions, ...
Johan's user avatar
  • 501
15 votes
2 answers
1k views

Definitions of determinant by unique features

A well-known definition of the determinant is: The determinant is the only function of a vector space of dimension $n$ to its underlying field which is multilinear, alternating and normalized. See e....
6 votes
2 answers
305 views

Differences between $p$-groups and $q$-groups

First, let me include the same disclaimer that goes in the first line of any article I write: all groups considered herein are finite. Academically, I work with connecting the arithmetic structure of ...
4 votes
1 answer
542 views

Novel examples, proofs or results in mathematics from arithmetic billiards

The goal of the post is get a repository of mathematical results, proofs or examples by users of the site, arising from arithmetic billiards in number theory, analysis, geometry,…. Wikipedia has an ...
4 votes
1 answer
331 views

Sufficient conditions for a SDE to have a stationary probability measure

Apologies if this question is too basic for MathOverflow. For a smooth Wiener-driven SDE on a non-compact manifold $M$ taking the form $$ dX_t = b(X_t) dt + \sum_{i=1}^k \sigma_i(X_t) \ast dW_t^i $$ ...
Julian Newman's user avatar
1 vote
0 answers
375 views

Mathematical technicalities that few people know [closed]

I am looking for a list of mathematical technicalities that are not so well-known, even in the mathematical community. What I mean is, I am looking for examples of phenomenon where it is important to ...
3 votes
0 answers
233 views

Maths books or works by originators or pioneers of fields of mathematics [closed]

I am looking for a (hopefully eventually comprehensive) list of examples of books or works that are: written by an originator of a field of mathematics, and about that field written by a pioneer of a ...
1 vote
2 answers
402 views

Naturally occurring examples of categories where composition depends on objects

In the comments and answer to another recent question, it became apparent that category theorists who work with the ‘many hom-class’ definition of a category implicitly view composition as a function ...
2 votes
2 answers
654 views

What is the most "informative" Yes/No math question you know? [closed]

Imagine that alien civilization contacted you and offered to answer one math question. This should be a Yes/No question (so, you cannot ask for a million-digit binary string encoding the answers to a ...
24 votes
15 answers
4k views

When forgetting structure doesn't matter

What forgetful functors are equivalences? The motivation here is understanding when some part of a structure can be 'safely' forgotten, even if remembering it might make our lives easier. There is ...
46 votes
7 answers
8k views

Swimming against the tide in the past century: remarkable achievements that arose in contrast to the general view of mathematicians

I would like to ask a question inspired by the title of a book by Sir Roger Penrose ([1]). The germ of this is to ask about the role, if any, of the fashion in research of pure and applied mathematics....
6 votes
2 answers
416 views

Common/well-known results with natural and/or useful reformulations

$\DeclareMathOperator{\pp}{\mathbb{P}}$My aim here is to have a collection of "natural" not-so-common reformulations/extensions of common/well-known results such that the reformulation/...
19 votes
3 answers
2k views

Unnecessary uses of the Continuum Hypothesis

This question was inspired by the MathOverflow question "Unnecessary uses of the axiom of choice". I want to know of statements in ZFC that can be proven by assuming the Continuum Hypothesis,...
63 votes
14 answers
6k views

Unnecessary uses of the axiom of choice

What examples are there of habitual but unnecessary uses of the axiom of choice, in any area of mathematics except topology? I'm interested in standard proofs that use the axiom of choice, but where ...
4 votes
1 answer
347 views

Examples of rich theories that started out as an infinite-dimensional inquiry

It seems that when a mathematical theory was newly invented, or a particular phenomenon was discovered, it is often while tackling a specific hard problem, but as more of the theory was developed it ...
liuyao's user avatar
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53 votes
14 answers
9k views

Modern results that are widely known, yet which at the time were ignored, not accepted or criticized

What is your favorite example of a celebrated mathematical fact that had a hard time to become accepted by the community, but after overcoming some initial "resistance" quickly took on? It ...
8 votes
1 answer
365 views

Examples of non-adjoint equivalences

What are some examples of equivalences whose canonical unit/counit fail to satisfy the triangle identities? It is common knowledge that not all equivalences satisfy the triangle identities, but that ...
Alec Rhea's user avatar
  • 8,977
4 votes
0 answers
413 views

What are your common strategies/remedies when your new theory/idea stuck in most cases?

Sorry if this is not a suitable post for MO. Sometimes after reading the origin of a theory/idea in differential topology I put myself in the shoes of that mathematician and ask myself, Did you do the ...
C.F.G's user avatar
  • 4,165
38 votes
4 answers
4k views

Interesting and surprising applications of the Ising Model

One of the most famous models in physics is the Ising model, invented by Wilhelm Lenz as a PhD problem to his student Ernst Ising. The one-dimensional version of it was solved in Ising's thesis in ...
IamWill's user avatar
  • 3,151
3 votes
2 answers
404 views

Properties that only Lorentzian manifolds have

I wonder what are some statements that although they can be formulated for pseudoriemannian manifolds of arbitrary signature they turn out to be true only in the lorentzian case. I admit things like: &...
0 votes
0 answers
212 views

Stories where a different definition lead to an inaccurate conclusion/a misunderstanding/etc

The overall question: What are some good examples where a different understanding of terminology or notation caused you to misinterpret a result in a way that was inaccurate? The intent here is of ...
63 votes
7 answers
4k views

What well known results with countability assumptions can be naturally extended to uncountable settings?

In many of the common categories of spaces (or algebras) in mathematics, one often restricts attention to those spaces or algebras which are "countable" or "countably generated" in ...
1 vote
0 answers
143 views

Pairs of functions with $\sum_{n} (f \circ g)(n) = \sum_{n} (g \circ f)(n) $

I was wondering there there are any pairs of functions $(f,g)$ such that $$\sum_{n=1}^{\infty} (f \circ g)(n) = \sum_{n=1}^{\infty} (g \circ f)(n) $$ on condition that they're not commutative with ...
Max Muller's user avatar
  • 4,485
15 votes
4 answers
919 views

What are some examples of understanding a space by studying the functions on this space?

In Quantum theory, groups and representations, Peter Woit writes: A fundamental principle of modern mathematics is that the way to understand a space $M$, given as some set of points, is to look at $...
103 votes
17 answers
15k views

Theorems that are essentially impossible to guess by empirical observation

There are many mathematical statements that, despite being supported by a massive amount of data, are currently unproven. A well-known example is the Goldbach conjecture, which has been shown to hold ...
16 votes
2 answers
1k views

Major applications of the internal language of toposes

What are the major applications of the internal language of toposes? Here are a few applications I know: Mulvey's proof of the Serre–Swan theorem in which he interprets the intuitionistically valid ...
120 votes
9 answers
13k views

Breakthroughs in mathematics in 2021

This is somehow a general (and naive) question, but as specialized mathematicians we usually miss important results outside our area of research. So, generally speaking, which have been important ...
18 votes
4 answers
3k views

What are the "hot" topics in mathematical QFT at the time?

I am currently finishing my Master's studies in mathematical physics. One topic which always interested me a lot were modern mathematical approaches to Quantum Field Theory (QFT) as well as the ...
6 votes
0 answers
271 views

Mathematical questions or areas amenable to AI [duplicate]

This question regards the new paper "Advancing mathematics by guiding human intuition with AI" by Davies et al. (Nature, 2021) (DOI link in open access) in which researchers at Deepmind ...
80 votes
22 answers
15k views

How would you have answered Richard Feynman's challenge?

Reading the autobiography of Richard Feynman, I struck upon the following paragraphs, in which Feynman recall when, as a student of the Princeton physics department, he used to challenge the students ...
15 votes
1 answer
388 views

Surprising invertibility results

There are results in category theory that imply that some morphism is invertible when a priori one might not have expected it. For instance, Given a monoidal natural transformation $\tau$ between ...
37 votes
17 answers
10k views

Listing applications of the SVD

The SVD (singular value decomposition) is taught in many linear algebra courses. It's taken for granted that it's important. I have helped teach a linear algebra course before, and I feel like I need ...
40 votes
11 answers
4k views

Results with short, advanced proofs or long, elementary proofs

Recently I was preparing an undergrad-level proof of (a form of) the Jordan Curve Theorem, and I had forgotten just how much work is involved in it. The proof stored my head was just using Alexander ...
22 votes
1 answer
3k views

What is so special about Chern's way of teaching?

First of all sorry for this non-research post. I was watching Jeffrey Blitz Lucky documentary movie and it was interesting to me that a winner of Lottery was a math Ph.D. from Berkeley. In the movie ...
C.F.G's user avatar
  • 4,165
57 votes
5 answers
6k views

What about a mathematics journal for 'negative' results?

In the empirical sciences, there are a number of journals that publish 'negative' results. Negative or null results occur when researchers are unable to confirm the findings obtained from earlier ...
29 votes
15 answers
5k views

Lunch seminars for PhD students

The problem that I would like to ask about is metamathematical, but I hope the question is appropriate. I would like to know if there exist mathematical departments that run a regular seminar for all ...
17 votes
3 answers
1k views

Theoretical results on neural networks

With this question I'd like to have a recollection of theoretical rigorous results on neural networks. I'd like to have results that have been settled, as opposed to hypothesis. As an example, this ...
14 votes
2 answers
896 views

Open problems in symbolic dynamics

I would like to know which are some noticeable open problems in symbolic dynamics, including substitution dynamics. I'm especially interested in connections with topological chaos of various forms. ...
49 votes
30 answers
7k views

Taking a theorem as a definition and proving the original definition as a theorem

Gian-Carlo Rota's famous 1991 essay, "The pernicious influence of mathematics upon philosophy" contains the following passage: Perform the following thought experiment. Suppose that you are ...
16 votes
4 answers
1k views

Categories disguised as other structures

It is common to hear that category theory unifies many apparently disparate areas of mathematics. One way it does so is by allowing us to take other mathematical structures and organize them into ...
15 votes
3 answers
1k views

Why it is convenient to be cartesian closed for a category of spaces?

In 1967 Steenrod wrote what later became a quite celebrated paper, A convenient category of topological spaces (Michigan Math. J. 14 (1967) 133–152). The paper conveys the work of many (among the most ...
Ivan Di Liberti's user avatar
4 votes
2 answers
522 views

Theorems with finite sets of exceptions

Exceptions are interesting. Sometimes, they're also important. If a theorem with exceptions is important for a subject, there are liable to be many corollaries of the form "either this is true... ...
2 votes
1 answer
178 views

Limit of line bundles on smooth curves degenerating to double line

Consider a family of smooth plane conics $f_\lambda(x,y,z)=0$ as a family $T_\lambda = (C,L,v_1,v_2,v_3)_\lambda$ of genus zero curves with a degree 2 line bundle $L$ and an ordered basis $v_i$ for ...
Caliper's user avatar
  • 21
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.
5 votes
3 answers
1k views

Consequences of Goldbach's Conjecture

In a letter of 1742 to Euler, Goldbach expressed the belief that ‘Every integer $N>5$ is the sum of three primes’. Euler replied that this is easily seen to be equivalent to the following statement ...
41 votes
23 answers
8k views

Theorems with many distinct proofs

I was told that whenever one learns a new technique, it is a good idea to see if one can prove a well-known theorem using the new technique as an exercise. I am hoping to build a list of such theorems ...
-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 ...
65 votes
21 answers
9k views

Situations where “naturally occurring” mathematical objects behave very differently from “typical” ones

I am looking for examples of the following situation in mathematics: every object of type $X$ encountered in the mathematical literature, except when specifically attempting to construct ...
10 votes
6 answers
1k views

Homology software

What software is there to efficiently compute homology? Specifically: What software can take a simplicial complex (provided by a file listing maximal simplices, for example) and quickly compute its ...

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