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# 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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332 views

### Big list: barycentric subdivision of simplicial sets

I'm preparing a seminar on the barycentric subdivision of simplicial sets and I'm looking for some examples of this construction appearing in the literature. Since it's a useful technique (at least in ...
112 views

### Examples of cartesian-closed model categories

One of the main settings of my research are Cartesian-closed model categories. I would like to know as many interesting and/or important examples of such categories as possible. "Interesting"...
156 views

### Relations between Whittaker functions/W algebras and Stokes data/resurgence

Skippable background: A Whittaker function is more or less a function on a flag manifold which is twisted-invariant for the action of a unipotent subgroup. E.g. consider functions $f$ on $\mathbf{P}^1$...
1k views

### The probabilistic method outside of discrete mathematics

The probabilitic method is a genius idea in combinatorics, graph theory etc, where instead of constructing something by hand, you construct the thing randomly and show that there is a positive ...
994 views

### Expected applications of condensed mathematics

As a student of algebraic geometry (in an advanced stage, but still far from an expert on anything), I am quite excited about learning some condensed mathematics. I have been told that the theory has ...
165 views

### What are the applications of spin geometry? [closed]

What are applications of spin geometry to physics? Does it have something to do with gravity?
151 views

### A zoo of derivations

Recall that given a $k$-algebra $A$, a derivation on $A$ is a $k$-linear morphism $d:A\to A$ such that $$d(ab)=d(a)b+ad(b).$$ The use of derivations is of paramount importance in mathematics. I think ...
3k views

### Books containing new results

In Endless controversy about the correctness of significant papers, Denis Serre writes: The research community is able to point out incorrect statements, at least among those which have some ...
4k views

### Well known theorems that have not been proved

I believe that there are numerous challenging theorems in mathematics for which only a sketch of a proof exists. To meet the standards of rigor, a complete proof of these theorems has yet to be ...
181 views

### Usefulness of total algebras and exotic generating series

In his first Algebra volume, Bourbaki  defines the structure of a “total algebra” i.e. the space of functions on a monoid $M$ (to a ring $k$) with the convolution product ( a function $f:\ M\to k$ ...
997 views

### Statements in differential geometry independent from ZFC

It is well known that some problems in functional analysis and in general topology are independent from ZFC: to name a few, Kaplansky's conjecture, the existence of outer automorphisms of the Calkin ...
486 views

### Popular mistakes in probability

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...
6k views

### Results from abstract algebra which look wrong (but are true)

There are many statements in abstract algebra, often asked by beginners, which are just too good to be true. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup ...
2k views

### Examples of errors in computational combinatorics results

I would like to collect examples of errors in published numerical results in computational combinatorics: where a result (typically a counting of some objects, or an extremal quantity within some ...
3k views

### Places where one can post open problems

(This must have been asked before and exist somewhere in Community Wiki, but I can't find it...) Where can you post open (math) problems? And what are the advantages and disadvantages? Example: This ...
2k views

### What are some interesting applications/corollaries of Kleene's Recursion theorem?

Lately I became very interested in the theory of computability and a fundamental early result you learn is the Recursion Theorem also known as the Fixed point theorem. At first sight you can see it's ...
219 views

### Examples of new results found via exams [closed]

I suspect that there have been many instances throughout history where a new proof of an existing result has been discovered by a student while taking an exam. Does anyone have an example of this?
2k views

### Ur-elemental surprises

For most of my (mathematical) life, I believed that there was really no essential difference between set theory without urelements and set theory with urelements. However, while that may be true in ...
1 vote
194 views

### Named sets of permutations

I am looking into interesting subsets of permutations, and there are several classes of permutations which are named. For example, there are Derangements, Alternating, Grassmann permutations (at most ...
420 views

### Landau's century-old problems: Anything comparable?

Landau's four problems are now over a century old (1912), and each still unsolved. This seems remarkable, even though he was not the originating author all four (maybe only the 4th?). Still, he ...
5k views

### Most important results in 2022

Undoubtedly one of the news that attracted the most attention this year was the result of Yitang Zhang on the Landau–Siegel zeros (see Consequences resulting from Yitang Zhang's latest claimed ...
760 views

### Compilation of strategies to show that some constant is irrational

I'm looking into expanding my knowledge in ways to show that some constant is irrational. I'm gonna give some examples of irrationality proofs, and I'm interested in learning what strategies you guys ...
413 views

### Funding programs for mathematical research [closed]

In the USA, as far as I know, the main grants available to mathematicians are collected on the NSF or the AMS websites [please, correct me if this perception is inaccurate]. On the other hand, for ...
1 vote
329 views

### 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 $\mathscr{X}$ (...
322 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, ...
777 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....
267 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 ...
418 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 ...
172 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 $$dX_t = b(X_t) dt + \sum_{i=1}^k \sigma_i(X_t) \ast dW_t^i$$ ...
1 vote
347 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 ...
204 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
346 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 ...
616 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 ...
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 ...
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 (). The germ of this is to ask about the role, if any, of the fashion in research of pure and applied mathematics....
390 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/...
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,...
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 ...
323 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 ...
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 ...
317 views

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 ...
401 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 ...
3k 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 ...
335 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: &...
204 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 ...
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
125 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 ...
842 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 \$...