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 ...

**86**

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**25**answers

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### 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 ...

**-2**

votes

**2**answers

309 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

**0**answers

108 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

**1**answer

278 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)\...

**68**

votes

**18**answers

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

**6**answers

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

**6**answers

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

**1**answer

324 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

**2**answers

354 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

**1**answer

151 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

**0**answers

235 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

**0**answers

192 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 ...

**31**

votes

**7**answers

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

**0**answers

157 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

**0**answers

67 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

**0**answers

56 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

**0**answers

160 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 ...

**40**

votes

**13**answers

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

**0**answers

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

**1**answer

103 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

**3**answers

713 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 ...

**2**

votes

**2**answers

179 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

**10**answers

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

**1**answer

60 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

**0**answers

108 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

**9**answers

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

**6**answers

1k 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

**5**answers

725 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

**23**answers

16k 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

**1**answer

424 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 ...

**28**

votes

**7**answers

2k 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

**8**answers

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

**2**answers

658 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

**0**answers

177 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

**59**answers

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

**8**answers

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

**4**answers

514 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 ...

**5**

votes

**1**answer

880 views

### Is this equivalent to RH - Riemann hypothesis?

$\pi = 3\prod_{\substack{t = \Im(r)\\r = (1/2+it) \\\zeta(r) = 0}}\frac{9+4t^2}{1+4t^2}\iff\text{RH is true}$

**2**

votes

**0**answers

94 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

**11**answers

3k 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

**1**answer

279 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 ...

**11**

votes

**2**answers

446 views

### Examples of (Git) open math (texts) projects

I am an active part of a research project on the positive effects of open math projects on the community. With open math projects I have in mind a particular thing, namely a GIT project on mathematics ...

**33**

votes

**4**answers

1k views

### Important open exposition problems?

Timothy Chow, in his article A beginner's guide to forcing, defines an open exposition problem as a certain concept or topic in mathematics that has yet to be explained "in a way that renders it ...

**37**

votes

**5**answers

4k views

### How to improve writing mathematics?

My first language is not English. How can I improve my mathematical writing. I feel like the only things I can write down are numbers and equations. Is there any good suggestion for improving writing, ...

**7**

votes

**1**answer

471 views

### Steps in Geometric Complexity Theory

GCT purports to provide a program to show that $NP \not \subset P/poly$.
At the high level what are the steps involved in the program and what stage is each step in?
What difficulties ...

**2**

votes

**0**answers

123 views

### Right adjoint completions

Forgive me if this question is not well thought out. I don't know how else to ask it.
The nlab page on completion gives some examples of completions which are left adjoints. These completions are "...

**21**

votes

**2**answers

1k views

### Interpretations of permanent

The standard interpretation of permanent of a $0/1$ matrix if considered as a biadjacency matrix of a bipartite graph is number of perfect matchings of the graph or if considered as a adjacency matrix ...

**18**

votes

**2**answers

1k views

### Any important consequences with presupposition of $\mathbf{P} \neq \mathbf{NP}$

As we know, there are lots of consequences with the presupposition of the Riemann Hypothesis.
Similarly, are there any important consequences with the presupposition of $\mathbf{P} \neq \mathbf{NP}$ ?...

**48**

votes

**9**answers

22k views

### Mathematically interesting screensavers

A screensaver is a computer program that fills a computer screen with a moving pattern that eluminates each pixel for approximately the same proportion of time. Originally designed to prevent burn-in ...

**13**

votes

**2**answers

792 views

### Terminology: Lost in translation or multiple-meanings

I was reading Uniformization of Riemann Surfaces by Henri Paul de Saint Gervais (not a real person, but a group of French mathematicians), and the translator kindly points out that the name of "the ...