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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 ...

3
votes
1answer
250 views

Improvements to one's own theorems

What are some notable (famous?) instances where the following has occurred. A particular author proves: Every P which satisfies Q has property Z. A few years later (roughly speaking) the same ...
7
votes
0answers
190 views

List of modern points of view simplifying or clarifying classical topics

There are many modern mathematical achievements which greatly clarify or (and) simplify classical important topics. I believe a list of such achievements, among other benefits, would be a big help for ...
46
votes
11answers
5k views

What definitions were crucial to further understanding?

Often the most difficult part of venturing into a field as a researcher is to come up with an appropriate definition. Sometimes definitions suggest themselves very naturally, as when you solve a ...
0
votes
0answers
99 views

Reference request for bounds of $n$-th composite

Motivation I will briefly elaborate here my motivations for asking the question. If you are not interested in it then please go to the questions. Recently during trying to understand and prove the ...
3
votes
2answers
156 views

Free ergodic probability measure-preserving actions of the free group

Let $(X,\mathcal{B},\mu)$ be a standard Borel probability space. Let $\Gamma$ be a countable group. An action of $\Gamma$ on $X$ is: essentially free if for all $g \in \Gamma \setminus \{e \}$,...
0
votes
5answers
442 views

Mathematical phantoms, specifically but not exclusively in applied mathematics [duplicate]

A while ago over at our sister site, there was an interesting question [not by me] with next to no answers which I feel is, fleshed out in a more precise fashion, appropriate for MathOverflow. The ...
-5
votes
1answer
267 views

Big list of “outstanding paper awards” [closed]

What are the prizes awarded for significant papers published in mathematical journals? One of them is the SIAM Outstanding Paper Prize. I'd be particularly interested in hearing about prizes ...
5
votes
0answers
268 views

Theorems conditional on false conjectures

What is an example of a theorem that was conditional on a conjecture that later turned out to be false?
26
votes
8answers
5k views

How to explain to an engineer what algebraic geometry is?

This question is similar to this one in that I'm asking about how to introduce a mathematical research topic or activity to a non-mathematician: in this case algebraic geometry, intended as the most ...
90
votes
25answers
9k views

Which mathematical definitions should be formalised in Lean?

The question. Which mathematical objects would you like to see formally defined in the Lean Theorem Prover? Examples. In the current stable version of the Lean Theorem Prover, topological groups ...
-1
votes
2answers
324 views

What are some of the unsolved mathematical problems that we generally agree were posed before the beginning of the XX.-th century?

Because mathematics has been extremely well-developed in XX.-th and continues to do so in XXI.-th century and because there is an enormous number of open problems and conjectures and hypotheses posed ...
3
votes
0answers
129 views

Applications of Ambrose-Singer theorem on holonomy

I am planning to introduce to a group of Graduate students the notion of connections on Principal bundle, curvature of connection, Holonomy. I want to conclude with the statement of Ambrose-Singer ...
3
votes
1answer
308 views

What does reduction of structure group of principal bundle say?

Let $G$ be a Lie group and $\pi:P\rightarrow M$ be a principal $G$ bundle. The notion of reduction of structure group is standard but I will recall here in case some one needs it. Let $f:P(M,G)\...
69
votes
18answers
19k views

What programming language should a professional mathematician know? [closed]

More and more I am becoming convinced that one should know at least one programming language very well as a mathematician of this century. Is my conviction justified, or not applicable? If I am right,...
45
votes
6answers
4k views

Are there examples of conjectures supported by heuristic arguments that have been finally disproved?

The idea for this comes from the twin prime conjecture, where the heuristic evidence seems just so overwhelming, especially in the light of Zhang's famous result from 2014 about Bounded gaps between ...
18
votes
6answers
1k views

Lebesgue measure theory applications

I'm looking for reasonably simple examples of applications of Lebesgue measure theory outside the measure theory setting. I give an example. Theorem: Let $X$ be a differentiable submanifold of $\...
7
votes
1answer
345 views

What are some open problems in moduli spaces and moduli stacks?

I would like to know what are the open big and interesting problems related to moduli spaces and moduli stacks ? Thanks in advance for your help.
3
votes
2answers
359 views

Finite groups with small God's numbers

Let $G$ be a finite group and $S$ be generating set it. Now given all words with alphabet $S$, then there exists a minimum word length $N(S,G)$ such that all group elements are represented by a word ...
2
votes
1answer
158 views

Alternate descriptions of finite fields

The finite field of order $p^n$ is isomorphic to $(\mathbb Z/p \mathbb Z)[X]/(P)$, where $P$ is an irreducible polynomial in $(\mathbb Z/p \mathbb Z)[X]$ of degree $n$. This describes every finite ...
1
vote
0answers
238 views

Mathematical expressions involving weird constants [closed]

I hope this is a question that fits here and is not duplicated. Also that is clear since it can be a little ambiguous. I was wondering if you know deep expressions, theorems, isomorphisms or ...
1
vote
0answers
195 views

Do there exist similar programs which connect different field of Mathematics like Langlands program? [closed]

$2018$ Abel prize is awarded to Robert P. Langlands for his visionary program connecting representation theory to number theory. In particular, his program predicts the existence of a tight web of ...
32
votes
7answers
3k views

Papers in which the questions were more interesting than the results

I am looking for examples of recently (last 20 years, say) published math papers such that: the results/examples were fairly trivial (by this I mean anyone with the definitions and standard ...
9
votes
0answers
159 views

$p$-groups and the arithmetic of $p$

I cannot think of an example of a property of a $p$-group or a pro-$p$ group that depends on the arithmetic of the prime number $p$. To clarify l do not mean the size of $p$. Clearly lots of stuff ...
0
votes
0answers
70 views

Discrete spectrum of Schrodinger operator

Assume $\Omega$ is a non-compact region or manifold with dimension $\geq4$. Let $H=-\Delta+V$ be Schrodinger operator. Here $V$ is a (smooth)function. I know that if $V\geq c>0$ or $V\to c>0$,...
4
votes
0answers
59 views

Numerical and computational approaches to limit cycle theory

I am searching for a big list of papers or researches devoted to limit cycles of planar polynomial vector fields based on a computational and numerical approach. I would like to ask ...
1
vote
0answers
169 views

Textbooks on solidifying graduate knowledge

I am finishing my undergraduate program soon and start getting ready for graduate school. What I have realized is that although I have passed many subjects and with good grades I feel that ...
42
votes
13answers
7k views

PhD dissertations that solve an established open problem

I search for a big list of open problems which have been solved in a PhD thesis by the Author of the thesis (or with collaboration of her/his supervisor). In my question I search for every possible ...
0
votes
0answers
178 views

Important papers that were rejected several times before 1950 [duplicate]

There is no reasson to assume that an important paper has to be well written. For example, when Galois submited his paper on his theory of solvability by radicals it was rejected several times, and ...
2
votes
1answer
106 views

Exponential Decay on nolinear Schrodinger type equation with negative potential

Given a unbounded domain $\Omega$(a region or an open manifold) with $\dim \geq3$, consider the equation $$A u+Vu+|u|^ku=0,$$ here $A=\sum_{ij}\partial_ia^{ij}(x)\partial_j$ is an elliptic operator ...
5
votes
3answers
724 views

The “Spaces of Schwartz distributions are finite dimensional” challenge

The more I study Schwartz distributions and the corresponding spaces, the more the latter look "finite dimensional" to me. Of course they are not finite dimensional in the technical sense but they are ...
3
votes
2answers
183 views

Viewing parts of $\mathbb{V}$ 'from the top down' or 'from the bottom up'

I am curious about instances where we can glean nontrivial information about a certain piece of the universe by viewing it as being 'built over' a smaller part of the universe, or alternatively ...
105
votes
10answers
14k views

Do you know important theorems that remain unknown?

Do you know of any very important theorems that remain unknown? I mean results that could easily make into textbooks or research monographs, but almost nobody knows about them. If you provide an ...
0
votes
1answer
61 views

What are some relatively unknown solution concepts in cooperative game theory that are useful in a specific context?

In cooperative game theory, the payoffs for the grand coalition can be distributed in a number of ways. Each of those ways is a solution concept. Well-known examples of solution concepts include the ...
4
votes
0answers
109 views

What are some interesting examples of cooperative games that can be naturally generalised to a stochastic version of it?

In classical, deterministic cooperative game theory, there are $N$ players that can form $2^{N}$ coalitions. Each of these coalitions is assigned a value by means of the characteristic function $v ( \...
65
votes
9answers
7k views

Mathematical conjectures on which applications depend

What are some examples of mathematical conjectures that applied mathematicians assume to be true in applications, despite it being unknown whether or not they are true?
32
votes
6answers
2k views

What motivations for automorphic forms?

Automorphic forms are ubiquitous in modern number theory and stands as a mysterious Graal lying at the intersection of many fields, if not building valuable bridges between them. However, since this ...
13
votes
5answers
745 views

Examples of residually-finite groups

One of the main reasons I only supervised one PhD student is that I find it hard to find an appropriate topic for a PhD project. A good approach, in my view, is to have on the one hand a list of ...
66
votes
23answers
17k views

Which popular games have been studied mathematically?

I'm planning out some research projects I could do with undergraduates, and it struck me that problems analyzing games might be appropriate. As an abstract homotopy theorist, I have no experience with ...
7
votes
1answer
427 views

Example of a smooth family of projective surfaces with non-vanishing integrals of Todd classes

Motivation: Let $\pi\colon S \rightarrow B$ be smooth projective morphism of relative dimension 2 over a smooth projective scheme $B$. If the stucture sheaves of the fibres do not have higher ...
29
votes
7answers
3k views

Examples of integer sequences coincidences

For the time being, the OEIS website contains almost $300000$ sequences. Each of these sequences is the mark of a specific mathematical concept. Sometimes two (or more) distinct concepts have the ...
5
votes
8answers
2k views

Mathematical objects whose name is a single letter

(Not research-level, but perhaps not easily answered elsewhere — you decide if MO can afford the innocent fun. If so, it should likely be “community-wiki” i.e. one object per answer.) I am seeking ...
5
votes
2answers
697 views

Examples of analytic functions to motivate a first course in complex variables

[Changed title as a plea to re-open the question.] If one is to motivate a course in complex variables, what specific analytic (holomorphic/meromorphic) function of one variable would you cite as an ...
5
votes
0answers
179 views

On a Robin Forman's remark on combinatorial simplicial complexes

In a very captivating introduction to discrete Morse theory, Robin Forman makes the following remark: ...However, that does not explain why so many simplicial complexes that arise in combinatorics ...
106
votes
59answers
13k views

Nonequivalent definitions in Mathematics

I would like to ask if anyone could share any specific experiences of discovering nonequivalent definitions in their field of mathematical research. By that I mean discovering that in different ...
17
votes
8answers
2k views

Concepts in topology successfully transferred to graph theory and combinatorics with non-trivial applications?

What are some of the difficult concepts in topology that have been transferred to graph theory and combinatorics where a certain new application has been found. A good example is Lovász's proof of ...
8
votes
4answers
516 views

When is it easier to work projectively?

There are many instances in which theory over $\mathbb{C}$ is cleaner than theory over $\mathbb{R}$. For example, continuously differentiable functions over $\mathbb{R}$ are not necessarily twice ...
12
votes
1answer
1k views

Is this equivalent to RH - Riemann hypothesis?

$$\pi = 3\prod_{\zeta(1/2+it) = 0}\frac{9+4t^2}{1+4t^2}\iff\text{RH is true}.$$
2
votes
0answers
95 views

Conjectures that can be tested with large numbers of Hecke eigenvalues of GSp(4) automorphic forms

As part of my thesis work I have proved Ibukiyama's conjecture implies something about $\mathrm{SO}(5)$ forms associated to certain lattices lifting to $\mathrm{GSp}(4)$ (This was originally a ...
33
votes
11answers
4k views

Why is the definition of the higher homotopy groups the “right one”?

If someone asked me the question for the fundamental group, I would answer as follows: The connection to classification of covering spaces. The fundamental group of many spaces is an object of ...
4
votes
1answer
281 views

Maximality without Zorn

When confronted with finding an object that is maximal with regard to some ordering relation, most of us have the reflex to use Zorn's Lemma. I am interested in instances of proving the existence of ...