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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57
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28answers
5k views

Proofs where higher dimension or cardinality actually enabled much simpler proof?

I am very interested in proofs that become shorter and simpler by going to higher dimension in $\mathbb R^n$, or higher cardinality. By "higher" I mean that the proof is using higher dimension or ...
17
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3answers
718 views

Examples of improved notation that impacted your research?

The intention of this question is to find practical examples of improved mathematical notation that enabled actual progress in someone's research work. I am aware that there is a related post ...
13
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5answers
1k views

Striking existence theorems with mild conditions, and simple to state: more recent examples?

I would like to write an article about powerful existence theorems that assert, under mild and simple conditions, that some basic pattern or regularity exist. See some examples below. By mild ...
5
votes
0answers
192 views

Interesting things you learned while grading/marking? [closed]

What are some interesting mathematical things you have learned while grading (or marking, if you prefer) student work? For example, clever proofs that students came up with; nice counterexamples or ...
11
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3answers
601 views

Series and sequences in physical systems & closed form expressions

I gave a colloquium a while ago about physics inspiring recent developments in mathematics and as is almost borderline cliche in such talks, I mentioned the Fibonacci sequence with closed form ...
0
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1answer
823 views

Do mathematicians ignore mathematical works from non-mathematicians? [closed]

Is it true that mathematicians ignore and do not like to take a look at or comment on any mathematical work or manuscript from a person outside the field of mathematics (meaning is not a professional ...
42
votes
10answers
4k views

List of long open, elementary problems which are computational in nature

I would like to ask a question of a similar vein to this question. Question: I'm asking for a list of long open problems which are computational in nature which a beginning graduate student can ...
3
votes
1answer
161 views

Abstract mathematical concepts/tools appeared in machine learning research

I am interested in knowing about abstract mathematical concepts, tools or methods that have come up in theoretical machine learning. By "abstract" I mean something that is not immediately related to ...
60
votes
7answers
15k views

Results that are widely accepted but no proof has appeared

The background of this question is the talk given by Kevin Buzzard. I could not find the slides of that talk. The slides of another talk given by Kevin Buzzard along the same theme are available here....
13
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2answers
798 views

Contrasting theorems in classical logic and constructivism

Is it possible there are examples of where classical logic proves a theorem that provably is false within constructivism? Is so what are some examples? What are some examples of most contrasting ...
62
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30answers
4k views

Atlas-like websites on specific areas of mathematics

In this post, we look for the existing atlas-like websites providing well-presented classifications or database about some specific areas of mathematics. Here are some examples: GroupNames: https://...
7
votes
0answers
125 views

Importance of textbooks in health of a sub-discipline

I am interested in published articles, and also more informal writing (blog posts, talk slides etc.) which discuss the importance of textbooks (where this word encompasses research monographs etc.) in ...
5
votes
3answers
596 views

Update on “Hopf algebras: their status and pervasiveness” by Hazewinkel

Hazewinkel wrote this article in 2005. Perhaps it's time for an update. For example, updating item 34: Ordinary differential equations much work has been done on the underlying Hopf algebra (HA) of ...
53
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60answers
10k views

Mathematicians with both “very abstract” and “very applied” achievements

Gödel had a cosmological model. Hamel, primarily a mechanician, gave any vector space a basis. Plücker, best known for line geometry, spent years on magnetism. What other mathematicians had so distant ...
7
votes
7answers
531 views

Important combinatorial and algebraic interpretations of the coefficients in the polynomial $[n]!_q = (1+q)(1+q+q^2) \ldots (1+q+\cdots + q^{n-1})$

What are some important combinatorial and algebraic interpretations of the coefficients in the polynomial $$[n]!_q = (1+q)(1+q+q^2) \ldots (1+q+\cdots + q^{n-1})?$$ As motivation, I will give ...
1
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0answers
107 views

Mathematical problems reducing to the traveling salesman problem

The superpermutation problem is: what is the shortest word that contains every permutation of $k$ letters as a substring. This can phrased as a Travelling Salesman Problem, where the nodes of your ...
1
vote
1answer
238 views

Examples of “irregularities” in mathematics, other than prime numbers [closed]

Prime numbers are the prime example (no pun intended) for something that arises apparently without describable patters; we know that infinitely many exist, that gaps between them can be arbitrarily ...
5
votes
1answer
222 views

The Idea of Kroneckerian geometry

Let $X$ be a complex, projective algebraic variety and assume that $X$ has a model $X_0$ over $\mathbb Z$ i.e. $X\cong X_0\times_{\operatorname{Spec }\mathbb Z}\operatorname{Spec }\mathbb C$. Let's ...
27
votes
18answers
6k views

Modeling in pure math

We all know that models play a major role in scientific practice. (By "model" here I mean any of various kinds of entities that function as representations or descriptions of real-world phenomena. ...
15
votes
8answers
956 views

Conceptual insights and inspirations from experimental and computational mathematics [duplicate]

I am interested in whether experiments on computers can help identifying new ideas or concepts in Mathematics. I am not talking about confirming particular conjectures up to certain numbers (for ...
4
votes
1answer
211 views

A list of locally finitely presentable topoi that are not coherent

Coherent topoi play an important role in topos theory, especially in the interaction with logic. Their most handy characterization is provided by Johnstone. Sketches, D3.3.1. Every coherent topos is ...
27
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7answers
4k views

Decision problems for which it is unknown whether they are decidable

In computability theory, what are examples of decision problems of which it is not known whether they are decidable?
5
votes
0answers
192 views

Theorems which are not numerically verified

Perhaps one of the best forms of justification for pure mathematics, in my experience, is the ability to demonstrate the truth of some statements despite the lack of numerical evidence. A rather ...
1
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0answers
100 views

Open problems in fiber bundles theory

As the title says, what are some problems in fiber bundles theory (especially principal bundles) that are still open?
2
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2answers
203 views

Measuring failure of a setup to preserve some structure giving interesting notions

I am looking for some examples of failure of some structures giving interesting notions. For example, we have the following situation: Let $P(M,G)$ be a principal bundle. Let $\Gamma\subseteq TP$ be ...
7
votes
3answers
623 views

Lesser known examples of perseverance with a successful ending [closed]

The stories of Wiles, of Perelman, and of Zhang, are very well-known to illustrate that sometimes great results are achieved through particularly long perseverance. What are lesser known-examples ...
29
votes
1answer
971 views

Which of the proofs of the fundamental theorem of algebra can actually produce bounds on where the roots are?

One of the old classic MO questions is a big-list of proofs of the fundamental theorem of algebra. Here is a second big-list question about this big list: Which of the FTA proofs can, even in ...
36
votes
15answers
7k views

Examples of “unsuccessful” theories with afterlives

I am looking for examples of mathematical theories which were introduced with a certain goal in mind, and which failed to achieved this goal, but which nevertheless developed on their own and ...
4
votes
0answers
91 views

Fredholm theory of non elliptic operators

In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in ...
15
votes
7answers
1k views

Examples of proofs by making reduction to a finite set [closed]

This is a very abstract question, I hope this is appropriate. Suppose $T$ is some claim over some infinite set $A$, for example, let $A$ be the set of all loopless planar graphs, and $T$ be the claim "...
20
votes
2answers
2k views

How to accelerate progress in mathematical research? [closed]

After completing a Ph.D. in pure mathematics, 10 years ago I left academia for working in industry. There, a typical question is "What can we do to accelerate $x$?" when a project is slowed down, and ...
56
votes
5answers
6k views

Examples where “thin + thin = nice and thick”

I'm interested in examples where the sum of a set with itself is a substantially bigger set with nice structure. Here are two examples: Cantor set: Let $C$ denote the ternary Cantor set on the ...
14
votes
2answers
414 views

Surprising appearances of Painlevé transcendents

What are some of your favorite examples of enumerative problems whose answer ended up being (related to) a solution to one of the Painlevé equations? I have seen examples from enumeration of classes ...
8
votes
1answer
206 views

Prominent examples of $q$-analogs without known cyclic sieving

The cyclic sieving phenomenon is nicely summarized in the following AMS Notices "What is...?" article: https://www.ams.org/notices/201402/rnoti-p169.pdf. In that article, Reiner, Stanton, and White ...
1
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0answers
160 views

Research-level blogs on complex networks:

I'm an applied mathematician that has a research interest in complex networks for modelling biological systems and I wondered whether the MathOverflow community might know of research-level blogs that ...
21
votes
1answer
820 views

When simple cohomological computations predict ingenious algebro-geometric constructions?

Classical algebraic geometry is full of ingenious constructions and miraculous coincidences: 27 lines on a cubic surface are related to Weyl lattice of type $E_6,$ lines on an intersection of four-...
34
votes
27answers
5k views

Examples of simultaneous independent breakthroughs

I'm looking for examples where, after a long time with little progress, a simultaneous mathematical discovery, solution, or breakthrough was made independently by at least two different people/groups. ...
14
votes
1answer
470 views

Legendary extra parameters to simplify a counting problem

I am reading Proofs and Confirmations, the history behind the alternating sign matrix conjecture, regarding counting $n \times n$ alternating sign matrices. In the introduction, it is written that ...
5
votes
0answers
120 views

Where have you encountered “arrangement spaces”?

I am compiling a paper in which I advertise (and use) the following notion of arrangement spaces (I made up the name, as I found no standard name in the literature). Let $v_i\in\Bbb R^d,i\in N:=\{1,.....
9
votes
0answers
357 views

True on stalks, false on affine opens [closed]

In scheme theory, there are some properties that can be specified purely on the stalks of the structure sheaf but they "lift" to the properties of the values of structure sheaf on affine opens, e.g. ...
53
votes
5answers
5k views

Arriving at the same result with the opposite hypotheses

I am pretty distant from anything analytic, including analytic number theory but I decided to read the Wikipedia page on the Riemann hypothesis (current revision) and there is some pretty interesting ...
10
votes
3answers
1k views

Errata for Bott and Tu's book “Differential Forms in Algebraic Topology”

My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Tu is a prequel. Is there a good list of errata for Bott and Tu available? ...
4
votes
1answer
331 views

Functors on the category of abelian groups which satisfy $F(G\times H) \cong F(G)\otimes_{\mathbb{Z}} F(H)$

Edit: According to the comment of Todd Trimble, I revise the question. What are some examples of functors $F$ on the category of Abelian groups or category of rings which satisfy $$F(G\times H)\cong ...
4
votes
1answer
285 views

Very canonical constructions

You have two categories $C_1$ and $C_2$. We call a map of the classes $\mathrm{Ob}(C_1)\rightarrow \mathrm{Ob}(C_2)$ a construction. Sometimes you can find a functor $C_1\rightarrow C_2$ inducing this ...
9
votes
2answers
555 views

A comprehensive list of random walk inequalities?

I am interested in finding a comprehensive list of all noticeable random walk inequalities. ie. $S_n = \sum_{k\leq n} X_i$ for i.i.d symmetric $X_i$ I can only seem to find books/papers that list ...
9
votes
0answers
151 views

Examples of automorphic representations to keep in mind

I have recently started studying the automorphic science and find it somewhat hard to form intuition. Can we have a list of examples of automorphic representations that you usually use to test a new ...
15
votes
1answer
1k views

Telling right from left

I know a lot of people, some of them mathematicians, who have trouble telling right from left. This can lead to problems when you are composing functions, for example. When did this seemingly ...
4
votes
0answers
258 views

What arithmetic would you do in parallel?

This is a post asking for references, and soliciting problems and people interested in accelerated computing. I will add the big-list tag and make it community-wiki. If this interests you strongly, ...
11
votes
5answers
840 views

Equivalent forms of the P vs. NP problem

Many things in math can be formulated quite differently; see the list of statements equivalent to RH here, for example, with RH formulated as a bound on lcm of consecutive integers, as an integral ...
1
vote
1answer
238 views

Topological Invariants for Group

Let $\mathbf{Grp}$ be the category of groups and $\mathbf{Top}$ be the category of topological spaces. To each group $(G, \circ_G)$, we can associate a topological space $(G,\tau_G)$ the basis for ...

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