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.
39 questions from the last 365 days
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Relation between properties of functions/sets and Grzegorczyk's hierarchy
I know for example that the first level of the Grzegorczyk hierarchy contains the functions which enumerate the c.e sets and that it has an interesting relation to the provably total functions in ...
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47
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10
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Algebraic theorems with no known algebraic proofs
What are some good examples of algebraic theorems that have no known algebraic proofs?
A few I know concern classifications of (not necessarily associative) division algebras over $\mathbb{R}$: the ...
5
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2
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668
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Recent breakthroughs with applied origins
Historically, the boundary between pure mathematics and its applications was much less defined. However, with the increasing complexity of modern mathematics and the resulting need for specialization, ...
18
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13
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When is 4 qualitatively different than $n\leq 3$?
Inspired by When is 2 qualitatively different from 3?
Also similar to Are there mathematical concepts that exist in dimension 4, but not in dimension 3? (Math SE), but with the restriction of being ...
61
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71
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When is 2 qualitatively different from 3?
I'd like to get a list of instances in mathematics where a problem with two parameters (or some parameter set to $2$) is qualitatively different from the instance of that problem with the value set to ...
9
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3
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Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?
In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach ...
45
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10
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Has the mathematics research community ever been led astray by a dumb mistake?
This is a highly subjective question, but here goes.
Has anyone ever published a result that was "taken seriously" by the research community, but was then discovered to be incorrect because ...
7
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0
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166
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Examples of finitary problems/theorems of high logical complexity? [duplicate]
Generally, number theoretic conjectures which are well-known and easy to explain are either obviously $\Pi^0_1$ or $\Pi^0_2$, which is to say, their truth can be decided by a single membership query ...
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2
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0
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158
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What rational zeta series with non-integer arguments appear in mathematics?
Background
Rational zeta series are series of the form $$\sum_{n=2}^{\infty} q_{n} \zeta(n + p, m), \label{1} \tag{1} $$ where $\zeta(x,m)$ is the Hurwitz zeta function and $q_{n}, \ p \in \mathbb{Q} \...
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3
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0
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159
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Proofs of the loop-suspension adjunction in infinity-categories
$\DeclareMathOperator{\Map}{Map}$$\DeclareMathOperator{\Fun}{Fun}$$\DeclareMathOperator{\const}{const}$$\DeclareMathOperator{\colim}{colim}$$\DeclareMathOperator{\lim}{lim}$In Elements of $\infty$-...
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3
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0
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270
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Categorical General Relativity
What are some good references for GR from a categorical point of view?
This is essentially just a big-list reference request.
I'm aware that the subject exists and can do some basic sleuthing to find ...
2
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0
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136
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Elementary functions such that $\sum_{n=2}^{\infty} f(n) \left( \zeta(n)-1 \right)$ can be evaluated, but $\sum_{n=2}^{\infty} f(n)$ can't
Background
The general context for this question is the topic of rational zeta series. What I've found so far, is that it usually the case that sums of the form $$\zeta_{f} := \sum_{n=2}^{\infty} f(n) ...
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5
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1k
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Uniqueness results that follow from CH
Recently, Joel David Hamkins presented a historical thought experiment that shows that CH could have been adopted as an axiom if we had been using the hyperreal field $\mathbb{R}^*$ instead of $\...
2
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1
answer
526
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What are some (popular) references on variants of the classical gambler's ruin problem that exists in literature?
It is fascinating that the gambler's ruin problem which is so ubiquitous in modern probability theory (cf. the Levin-Peres text on Markov chain and Mixing Times) actually dates back to a letter from ...
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9
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2
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473
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Completing half of Hilbert's program: Foundations that are conservative over Peano Arithmetic
The goal of the Hilbert program was to find a complete and consistent formalization of mathematics. Gödel's first incompleteness theorem establishes that completeness is impossible with first-order ...
- 6,353
10
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0
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274
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Open problems in complete theories
It is well-known that every complete recursively enumerable first-order theory is decidable. Does that mean that such theories are "trivial", or are there still interesting open problems ...
user507115
19
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6
answers
2k
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Book recommendation introduction to model theory
Next semester I will be teaching model theory to master students. The course is designed to be "soft", with no ambition of getting to the very hardcore stuff. Currently, this is the syllabus....
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9
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1
answer
506
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Current state of the art in geometric complexity theory
I came across this interesting question from almost 7 years ago:
What are the current breakthroughs of Geometric Complexity Theory?
My question is quite simple: Have there been any breakthroughs in ...
11
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4
answers
950
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Is there a name for finite unions of intervals?
Finite unions of intervals are simple sets that are used quite often, e.g. in measure theory. (The construction of the Cantor set is a noble example). I realised that I do not have a name for them. Is ...
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11
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6
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2k
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Hard problems with an easy-to-understand answer
I am very interested by problem in mathematics which are difficult (go at least 10 years without a resolution, say) but which have a solution that is short and elementary.
In this video Launay gave an ...
14
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5
answers
5k
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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 ...
51
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9
answers
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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 ...
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28
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3
answers
1k
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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 ...
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3
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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
400
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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 ...
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3
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2k
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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 ...
7
votes
0
answers
480
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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 ...
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2
answers
859
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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 ...
5
votes
1
answer
505
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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
178
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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 .)
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7
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2k
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Categories that admit all products but not all coproducts
What are examples for categories that admit all products but not all coproducts.
6
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1
answer
242
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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
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1
answer
263
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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 ...
53
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1
answer
9k
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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 ...
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38
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7
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18k
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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
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1
answer
2k
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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 ...
10
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2
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615
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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 ...
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93
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9
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13k
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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
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0
answers
1k
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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 ...