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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3 votes
0 answers
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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
266 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
537 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 ...
21 votes
14 answers
3k 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 ...
45 votes
7 answers
7k 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....
5 votes
0 answers
132 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/...
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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,...
54 votes
13 answers
4k 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
290 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 ...
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51 votes
14 answers
8k 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 ...
7 votes
1 answer
255 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 ...
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4 votes
0 answers
375 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 ...
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38 votes
4 answers
2k 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 ...
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3 votes
2 answers
260 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
192 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 ...
56 votes
7 answers
3k 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
108 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 ...
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12 votes
4 answers
663 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 $...
99 votes
16 answers
14k 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 ...
0 votes
0 answers
46 views

Problems concerning the existence of matrices with specific conditions

I'd like to collect in this thread questions about the existence of matrices in general size fulfilling specific criteria. I have been thinking whether it would be a good idea to create a specific tag ...
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14 votes
2 answers
688 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 ...
112 votes
8 answers
10k 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
2k 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
214 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 ...
77 votes
21 answers
13k 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 ...
14 votes
0 answers
242 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 ...
35 votes
17 answers
4k 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 ...
33 votes
6 answers
3k 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 ...
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49 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 ...
27 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 ...
15 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
738 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. ...
46 votes
30 answers
6k 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 ...
14 votes
4 answers
930 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 ...
14 votes
3 answers
960 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 ...
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4 votes
2 answers
395 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
147 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 ...
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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.
4 votes
3 answers
665 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 ...
36 votes
22 answers
6k 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
152 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 ...
62 votes
20 answers
8k 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 ...
6 votes
5 answers
558 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 ...
50 votes
6 answers
4k 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 ...
67 votes
11 answers
8k views

The use of computers leading to major mathematical advances II

I would like to ask about recent examples, mainly after 2015, where experimentation by computers or other use of computers has led to major mathematical advances. This is a continuation of a question ...
31 votes
17 answers
2k views

Equivalent definitions of Gromov hyperbolicity

Let $X$ be a metric space. I'd like to collect as many definitions of Gromov hyperbolicity or $\delta$-hyperbolicity of $X$ as possible. I'm happy for the definitions to require some niceness ...
6 votes
0 answers
487 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 ...
5 votes
5 answers
995 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
776 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 ...

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