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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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13 votes
5 answers
5k views

Who says understanding physics helps mathematicians? (A reference request) [Take the word "who" literally.]

If I wanted to make a somewhat bold and rather vague claim in print that it is widely acknowledged among mathematicians that knowledge of mechanics (in the sense in which physicists understand that ...
46 votes
9 answers
2k views

Examples of theorems where numerical bounds on $\pi$ played a role

This is a whimsical question, motivated purely by curiosity rather than for any application. We are all familiar with countless mathematical results which use Archimedes' constant $\pi$ either in ...
Terry Tao's user avatar
  • 110k
24 votes
3 answers
991 views

Proofs of Poincaré duality

I know several proofs of Poincaré duality: The original proof using dual cell complexes. Probably the nicest version of this uses a handle decomposition. The argument (in Hatcher and many other ...
Misha's user avatar
  • 241
2 votes
2 answers
638 views

No starter "accessible" well known open problems

We all know very famous open problems. Usually the ones that become famous are well studied and lots of progress was achieved and conjectures, partial results, reductions and so on exist. This is one ...
7 votes
1 answer
287 views

High dimensional Lusin conjecture

Lusin, in 1913, while considering the properties of Hilbert's transform, conjectured that every function in $L^2[-\pi, \pi]$ has an a.e. convergent Fourier series. Kolmogorov, in 1923, gave an example ...
Tomas's user avatar
  • 869
19 votes
3 answers
2k views

Where do root systems arise in mathematics?

One often hears that root systems are ubiquitous in mathematics and physics. The most obvious occurrence of root systems is in the classification of complex simple Lie algebras. Where else do they ...
6 votes
0 answers
301 views

Theories of manifolds w/ extra structure and singularities

Many different objects in mathematics can be described as manifolds with extra structure. Among the most famous examples of these are smooth manifolds, Riemannian manifolds, complex manifolds, and ...
Will Sawin's user avatar
  • 138k
-6 votes
2 answers
511 views

Homotopy theory and algebraic topology last 10 years. Is it a dying field? [closed]

I'm under the impression that algebraic topology is a dying field in mathematics. That was my impression but I think I'm wrong. As every person I do need some evidence that my impression is not ...
6 votes
1 answer
455 views

Nice diophantine equations with large smallest solutions

Given a polynomial $P$ with integer coefficients in finitely many variables, we denote by $v(P)$ the product of the absolute values of the non-zero coefficients and the non-zero total degrees of the ...
-1 votes
1 answer
150 views

Categories that admit all finite products but not all finite coproducts

What are examples for categories that admit all finite products but not all finite coproducts? (See also this question: Categories that admit all products but not all coproducts .)
Yilmaz Caddesi's user avatar
3 votes
7 answers
1k views

Categories that admit all products but not all coproducts

What are examples for categories that admit all products but not all coproducts.
6 votes
1 answer
203 views

Results with a flavor “every automorphism of automorphisms is inner”

It seems that there are a number of results which take more or less the following form: let $X$ be some (specific) kind of structure, let $Y$ be the group of automorphisms of $X$ or perhaps ring of ...
3 votes
1 answer
239 views

Nonisomorphic central products on the same pair of groups?

A central product of two groups $G$ and $H$ is determined as follows. The groups $G$ and $H$ have respective central subgroups $A$ and $B$ which are isomorphic, let $\delta:A\rightarrow B$ be such ...
49 votes
1 answer
8k views

What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?

I am a research mathematician at a university in the United States. My training is in pure mathematics (geometry). However, for the past couple of months, I have been supervising some computer science ...
Ryan Hendricks's user avatar
37 votes
7 answers
17k views

Daunting papers/books and how to finally read them

Most people throughout their career encounter at least one paper that seems especially daunting to them. I'm interested in real stories of how you successfully overcame that to extract the knowledge ...
7 votes
1 answer
1k views

Which revolutions in topology and geometry can we expect in the next 20 years? [closed]

In my limited perspective on the history of mathematics, I can name at least two big revolutions in Topology and Geometry (broadly construed): the introduction of Schemes in Algebraic Geometry, and ...
9 votes
2 answers
512 views

What are applications of asymptotic freeness of random matrices?

In around 1990 Voiculescu showed asymptotic freeness of certain random matrices, i.e., free independence when the matrix size goes to infinity. Since then this link between free probability and random ...
Bart's user avatar
  • 191
87 votes
9 answers
12k views

Breakthroughs in mathematics in 2023

At the end of 2021, Johnny Cage asked about breakthroughs in 2021 in different mathematical disciplines. A similar question has been asked at the end of 2022, so it looks like Johnny Cage originated a ...
3 votes
0 answers
1k views

Most important results in 2023 [duplicate]

Last year I asked a question about the best results in the year 2022. This year I moved away from mathematics, but that does not eliminate my curiosity to know what great results were published, so ...
0 votes
1 answer
363 views

How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher number sets

How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher sets (Everything circled in red is what I'm interested in (+ the Cauchy integral to make it Dedekind ...
user1248224's user avatar
4 votes
0 answers
89 views

List of equivalent conditions for the invariant subalgebra to be polynomial

Let $k$ be a field, $P_n$ the polynomial algebra in $n$ indeterminates, and $G<\operatorname{GL}_n$ a finite group whose order is coprime to the characteristic of $k$, and that acts on $P_n$ by ...
jg1896's user avatar
  • 3,063
5 votes
0 answers
531 views

What sets are known to have cardinality equal to $\mathbb{N}$ or $\mathbb{R}$ but open as to which?

A long time ago a similar question was asked on math.stackexchange. There are many sets which we know to be either finite or infinitely countable but do not know which cardinality specifically. An ...
28 votes
1 answer
2k views

Useful ideas in category theory which violate the principle of equivalence

Or an alternate title: using evil for the greater good. In category theory, the principle of equivalence says that statements about things should be invariant under the appropriate notion of thing-...
8 votes
0 answers
462 views

What is the nicest bijection $\textbf{R}^p \to \textbf{R}^q$ that you know?

It is well-known that bijection between $\textbf{R}^p$ and $\textbf{R}$ exist (e.g. here, though many other examples exist). The problem with all these examples of bijections is that typically the ...
18 votes
2 answers
1k views

What notable theorems cannot be automatically proven without choice using Shoenfield absoluteness?

There have been a couple of recent questions, here and here, regarding the role of the axiom of choice in real-analytic results with applicability to general relativity. This lead me to look at some ...
James E Hanson's user avatar
24 votes
0 answers
780 views

Revising the proof of CFSG

This is an oft-quoted excerpt from John Thompson's article "Finite Non-Solvable Groups": “... the classification of finite simple groups is an exercise in taxonomy. This is obvious to the ...
semisimpleton's user avatar
20 votes
6 answers
3k views

What are some nice uses of ultraproducts/ultrapowers?

Motivated by a recent post (Non-definability of graph 3-colorability in first-order logic), I was wondering: what are some nice arguments based on ultraproducts? I don't mind definability results, but ...
6 votes
0 answers
230 views

What can lattices tell us about lattices?

A general group-theoretic lattice is usually defined as something like A discrete subgroup $\Gamma$ of a locally compact group $G$ is a lattice if the quotient $G/\Gamma$ carries a $G$-invariant ...
Mark Schultz-Wu's user avatar
1 vote
0 answers
214 views

Results that hold for the complex numbers but not for algebraically closed fields of characteristic zero

When a result is stated for the field of complex numbers it can usually be extended to a result for an algebraically closed field of characteristic zero. I would like to see a list of results that ...
Béla Fürdőház 's user avatar
22 votes
4 answers
3k views

Brute force open problems in graph theory

Usually, a graph theoretic problem asks whether some class of graphs $C$ possesses a quality $P$. For example, $C$ is the class of all graphs and $P$ is the reconstructability property in Kelly-Ulam ...
3 votes
0 answers
468 views

What are some of the big open problems in $4$-manifold theory?

I've recently been studying some Manifold Theory and got very interested in their topological as well as geometric properties. From my understanding of the current literature, most the big and ...
math-physicist's user avatar
14 votes
2 answers
1k views

Famous papers published in annotated form?

I very much enjoyed reading through The Annotated Turing which goes through Turing's "On Computable Numbers, with an Application to the Entscheidungsproblem" in careful detail. I saw this ...
Nobilis's user avatar
  • 323
10 votes
1 answer
382 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 ...
0 votes
1 answer
135 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"...
Arshak Aivazian's user avatar
5 votes
0 answers
195 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 $\mathbf{P}^1$...
Pulcinella's user avatar
  • 5,555
32 votes
3 answers
2k 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 ...
21 votes
1 answer
2k 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 ...
-4 votes
1 answer
207 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?
1 vote
0 answers
183 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 ...
12 votes
7 answers
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 ...
48 votes
2 answers
5k 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 ...
6 votes
0 answers
253 views

Usefulness of total algebras and exotic generating series

In his first Algebra volume, Bourbaki [1] 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 $f:\ M\to k$ ...
Duchamp Gérard H. E.'s user avatar
22 votes
2 answers
1k 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 ...
9 votes
1 answer
641 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, ...
69 votes
28 answers
8k 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 ...
35 votes
8 answers
3k 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 ...
35 votes
9 answers
4k 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 ...
17 votes
5 answers
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 ...
H.C Manu's user avatar
  • 743
2 votes
1 answer
269 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?
34 votes
2 answers
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 ...
Noah Schweber's user avatar

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