**11**

votes

**7**answers

712 views

### Finite-space dynamical systems

This question is quite open-ended, but I will formulate several sub-questions that I'll try to make precise. It is about finite-state dynamical system: start with a finite set $X$, with say $n$ ...

**1**

vote

**1**answer

159 views

### Maximality statements that cannot be proved using $\mathsf{ZL}$ [closed]

What are examples for maximality statements that cannot be proved using Zorn's Lemma?

**49**

votes

**4**answers

2k views

### When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.
For example, it gives some evidence that there are finitely many ...

**3**

votes

**4**answers

367 views

### Higher Moments, what are they good for? [closed]

Absolutely nothing?
And now seriously - When I studied the basics of probability theory, and even in more advanced topics (random walks, stochastic processes, etc.), I always felt that the mean and ...

**3**

votes

**3**answers

369 views

### When few simple conditions yield a unique intricate structure

If people were asked to do a brainstorming related to the title, everyone would probably come up with dozens of examples. Those could include things as different as
the Mandelbrot set, Julia sets ...

**11**

votes

**7**answers

1k views

### Open problems in continued fractions theory

I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.

**16**

votes

**2**answers

1k views

### What are the applications of operator algebras to other areas?

Question: What are the applications of operator algebras to other areas?
More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by ...

**2**

votes

**1**answer

2k views

### Famous examples of PhD advisors younger than their student [closed]

What are the most famous examples of PhD advisors in mathematics, younger than their student?
(if possible put the date of birth and/or the difference in age).

**-2**

votes

**1**answer

242 views

### Degree of a rational function [closed]

I would like to have a simple proof for the following result:
Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). ...

**4**

votes

**2**answers

334 views

### Proving results about complete Boolean algebras in ZFC using Boolean valued models

I want to know what non-trivial ZFC theorems (not consistency results) about complete Boolean algebras (or more generally of partially ordered sets) one can prove using forcing.
I am mainly ...

**2**

votes

**1**answer

516 views

### Statistical distance between discrete and continuous distributions

Are there any statistical distance functions that are capable of comparing a continuous and a discrete distribution? From reading this list
http://en.wikipedia.org/wiki/Statistical_distance
the only ...

**5**

votes

**0**answers

253 views

### What will be the consequences if second Hardy-Littlewood conjecture turns out to be true?

It is generally believed that the Second Hardy-Littlewood Conjecture is false. But it has not been proved (or disproved) yet. My question is,
What would be the consequences if Second ...

**13**

votes

**2**answers

1k views

### New research and re-discovering classic results in “basic” real analysis

Sometimes, it happens that researchers publish a new proof of an old well-known result in "basic real analysis" (I'm referring to what some American people may call "honors calculus"). For instance, ...

**6**

votes

**1**answer

147 views

### Which known theorems of Lie algebras are still valid for Leibniz algebras?

Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. Thus, it is common to see a lot of papers which topic is about a generalization of a classic theorem of Lie algebras ...

**19**

votes

**5**answers

2k views

### Collection of conjectures and open problems in graph theory

Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?

**25**

votes

**3**answers

1k views

### Graduate program applications that require questionnaires and other non-letter material

In the December 2014 AMS Notices, a letter to the editor (http://www.ams.org/notices/201411/rnoti-p1311.pdf) by Deconinck and Medlock addresses the problem of (math) graduate programs requiring letter ...

**7**

votes

**3**answers

2k views

### The resolution of which conjecture/problem would advance Mathematics the most? [closed]

This is an impossibly broad question, and makes the unwarranted assumption that Mathematics is a uniform field. It might make more sense to ask the same question restricted to, say, Mathematical ...

**-3**

votes

**1**answer

138 views

### Decidable theorem or result that is not weaker than Tarski's theorem

I am wondering what other decidable theorem or results that is not weaker or stronger than Tarski's theorem.
Could any one give reference or a simple introduction about such result known in their ...

**15**

votes

**10**answers

1k views

### Combinatorial Databases

At one point, I remember being excited by seeing the website Encyclopedia of Combinatorial Structures as an extension of Sloane's Online Integer Sequence Database site. Unfortunately, the site (ECS) ...

**25**

votes

**15**answers

2k views

### Big list of repositories of mathematical preprints and postprints

I'm looking for a extensive list of online repositories of mathematical preprints and postprints. I'm interested in every type of repository, including small informal and semi-formal collections, like ...

**18**

votes

**19**answers

7k views

### Math books for advanced high school students

I'm working in a program for teaching a group of students selected in a Olympiad competition. The program is aimed to acquaint the students with the diverse aspects of higher mathematics in a way ...

**4**

votes

**1**answer

87 views

### The link and equivalence between variant definition of computation model and computational complexity over reals

To unify the numerical computation and classic computability theory, or to pave a foundation for the numerical computation, mathematicians present variant computation model and computational ...

**2**

votes

**0**answers

95 views

### Non-negative, monotone polynomial sequences without combinatorial interpretation

I am wondering what sequences of integers there are, that are known to grow polynomially, are non-negative, monotone but lacks a combinatorial interpretation.
By combinatorial interpretation, they ...

**2**

votes

**0**answers

95 views

### Examples of noncommutative Bezout domains

I would like to see some (or many!) examples of noncommutative Bezout domains (one-sided principal ideals sum to one-sided principal ideals). I've read somewhere that it's not easy to find an example ...

**47**

votes

**37**answers

5k views

### Results true in a dimension and false for higher dimensions

Some theorems are true in vector spaces or in manifolds for a given dimension $n$ but become false in higher dimensions.
Here are two examples:
A positive polynomial not reaching its minimum. ...

**21**

votes

**5**answers

2k views

### Mathematical research papers in general science journals

I am interested in collecting a list of research papers with a mainly mathematical focus that appeared in high-reputation general science journals without a dedicated mathematics section. This would ...

**6**

votes

**2**answers

436 views

### Rational points techniques on curves not using their Jacobian

Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...

**5**

votes

**1**answer

415 views

### Does every mathematics article have a DOI (Digital Object Identifier)?

Most articles nowadays have DOI's. I am looking for a list of mathematics journals in which some (or all) articles lack this piece of metadata.
I don't have access to MathSciNet, but even if I had, a ...

**14**

votes

**6**answers

2k views

### Proofs of the Chevalley-Warning Theorem

A well known proof of the Chevally-Warning Theorem is the one listed on wikipedia: http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem
Are there any other proofs of this, or ...

**139**

votes

**27**answers

32k views

### Examples of unexpected mathematical images

I try to generate a lot of examples in my research to get a better feel for what I am doing. Sometimes, I generate a plot, or a figure, that really surprises me, and makes my research take an ...

**15**

votes

**5**answers

1k views

### Examples of research on how people perceive mathematical objects

What examples are there on research related to human perception and mathematical objects?
For example, the shape of a beer glass influences drinking habits,
since people are bad at integrating.
...

**8**

votes

**2**answers

561 views

### Undecidable puzzles

There are plenty of popular NP-hard puzzles,
for example, generalized Sudoku ($n^2 \times n^2$-board), Flow (I cannot give a source for this), Minesweeper, etc.
Recently, I read a bit about aperiodic ...

**28**

votes

**15**answers

5k views

### Examples of famous 'workhorse' theorems

I use the term 'workhorse' to describe a theorem which is technically challenging to prove, perhaps very deep, but the statement is either uninteresting at first glance or too imposing to be ...

**53**

votes

**20**answers

7k views

### Rediscovery of lost mathematics

Archimedes (ca. 287-212BC) described what are now known as the 13
Archimedean solids
in a lost work, later mentioned by Pappus.
But it awaited Kepler (1619) for the 13 semiregular polyhedra to be
...

**0**

votes

**0**answers

51 views

### The set of (property) elements of a locally compact group is closed

For which properties $(P)$ is the following statement known to be true?
In any locally compact group $G$, the elements of $G$ that satisfy $(P)$ form a closed subset of $G$. In other words, the ...

**3**

votes

**1**answer

253 views

### Distance matrices

We say that a matrix $M\in\mathbb{R}^{n\times n}$ is a distance matrix on a metric space $(X,d)$, if there exist $x_1,\cdots,x_n \in X$ such that $M=[d(x_i,x_j)]_{n\times n}$.
Question. For which ...

**28**

votes

**25**answers

5k views

### Mathematicians who made important contributions outside their own field? [closed]

It is often said that scientists who cross disciplinary borders can make unexpected discoveries thanks to their fresh view of the problems at hand.
I am looking for mathematicians who did just that. ...

**81**

votes

**7**answers

8k views

### Mistakes in mathematics, false illusions about conjectures

Since long time ago I have been thinking in two problems that I have not been able to solve. It seems that one of them was recently solved. I have been thinking a lot about the motivation and its ...

**10**

votes

**9**answers

2k views

### Probabilistic method used to prove existence theorems

I am aiming for a "big list" of theorems using probability techniques to prove existence of some objects. And in each case, there is an interesting question -- can we find an explicit example? Was the ...

**23**

votes

**4**answers

990 views

### Which journals allow authors to retain copyright…?

I became motivated to ask this question after seeing the inspiring "© The Author(s) 2013 " in the header of this very interesting article, published in Compositio Mathematica.
Apart from open access ...

**11**

votes

**7**answers

742 views

### Unconventional types of induction

Induction is one of the most common tools is mathematics, and everybody knows the ordinary induction and the strong induction. However, in some proofs induction is applied in an unexpected and elegant ...

**11**

votes

**8**answers

832 views

### Interesting examples of generic behavior of mathematical objects being either unreasonably structured or simply unreasonable

My experience seems to be that quite often "generic" mathematical objects tend to be either extremely well behaved or structured, or at the opposite extreme are as unstructured as possible.
For ...

**11**

votes

**2**answers

508 views

### References for particular topics related to Langlands

I have never really concentrated on Langlands, which explains my poor level of understanding of it. But I have read quite a few introductory papers related to Langlands, and to the circle of ideas ...

**2**

votes

**2**answers

379 views

### Beautiful constructions in algebraic topology that facilitate one's understanding of homotopy theory [closed]

There is an army of interesting constructions in AT, and Understanding them are usually very helpful for appreciate the theory underneath. So I would like to invite you to share those examples that ...

**4**

votes

**0**answers

158 views

### Isoperimetric inequality, isodiametric inequality, hyperplane conjecture… what are the inequalities of this kind known or conjectured?

I duplicate here a question I asked on math.stackexchange.
Question: Which inequalities similar to the famous isoperimetric inequality is known?
conjectured?
I recently learned about some ...

**7**

votes

**4**answers

858 views

### NP-hard problems in linear algebra and real analysis [closed]

I am curious about NP-hard problems in linear algebra and real analysis. An example in linear algebra would be the calculation of the permanent.
I would thus like to collect in this thread a list of ...

**11**

votes

**7**answers

863 views

### Properties of rings that have an elegant description in terms of the associated category of modules

Suppose $A$ is a ring. Then $A$ happens to be a division ring iff every left $A$-module is free. (See here for proofs). I think this is very beautiful; what other properties of rings have an elegant ...

**7**

votes

**2**answers

198 views

### Local-to-global inequalities for measures: Brunn-Minkowski, Ahlswede-Daykin, what else?

This question is motivated by an obvious formal analogy between two well-known inequalities:
Log-concavity and Brunn-Minkowski inequality
Let $\mu(dx) := m(x) dx$ be an absolutely continuous ...

**5**

votes

**3**answers

843 views

### The most unexpected and/or the least natural category theory theorem?

Theory of categories is all natural and abstract nonsense. Or is it? What would be the most unexpected and/or the least natural theorem of the theory of category? (It does not really have to be THE).
...

**77**

votes

**9**answers

9k views

### Analogues of P vs. NP in the history of mathematics

Recently I wrote a blog post entitled "The Scientific Case for P≠NP". The argument I tried to articulate there is that there seems to be an "invisible electric fence" separating the problems in P ...