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 ...

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0
votes
0answers
93 views

Any result, example or conjecture about the computational complexity of transcendental number that can not be computed by linear time Turing Machine [on hold]

The Hartmanis-Stearns Conjecture is restated by Prof.Lipton as: Suppose that a linear time Turing Machine computes the first $n$ digits of the real number $r$ in base ten. Then, the number is either a ...
6
votes
1answer
222 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 ...
3
votes
1answer
233 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 ...
6
votes
4answers
751 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 ...
88
votes
19answers
20k 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 ...
12
votes
4answers
684 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. ...
7
votes
2answers
346 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 ...
23
votes
15answers
4k 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 ...
0
votes
0answers
43 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 ...
2
votes
1answer
131 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 ...
27
votes
26answers
4k 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. ...
67
votes
6answers
7k 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 ...
12
votes
9answers
1k 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 ...
19
votes
4answers
706 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 ...
10
votes
6answers
469 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
8answers
740 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
2answers
408 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
2answers
333 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 ...
2
votes
0answers
79 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 ...
5
votes
4answers
331 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
7answers
693 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
2answers
173 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 ...
4
votes
4answers
674 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). ...
66
votes
9answers
7k 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 ...
3
votes
0answers
435 views

Does Pure Mathematics glue Science together? [closed]

A little while ago, I was reading Cathy O'Neil's post Why is math research important (subtext: why does Pure Math deserve funding), where she discusses 3 possible answers. One of these is the usual ...
19
votes
0answers
637 views

The most important facts, modern surveys, and readable introductions to p-adic cohomology theories (crystalline cohomology and the mysterious functor)

I would like to organize a seminar on crystalline cohomology; I dream of understanding the Beilinson's recent paper on the mysterious functor ...
1
vote
1answer
330 views

Transfinite induction vs induction in mathematics

What are the nontrivial theorems one can prove about natural numbers which need transfinite induction? By "need transfinite induction" I mean one can show that the statement is not provable in ...
14
votes
9answers
1k views

What are your favorite concrete examples of limits or colimits that you would compute during lunch?

(The title was initially "What are your favorite concrete examples that you would compute on the table during lunch to convince a working mathematician that the notions of limits and colimits are not ...
29
votes
15answers
2k views

Counterexamples in universal algebra

Universal algebra - roughly - is the study, construed broadly, of classes of algebraic structures (in a given language) defined by equations. Of course, it is really much more than that, but that's ...
0
votes
0answers
161 views

Examples of 'bad' notations and definitions [duplicate]

I am trying to compile a list of notations and definitions that has become ingrained in mathematical folklore, yet are still on some objective scale unsatisfactory. I offer two starting examples. For ...
19
votes
32answers
1k views

Formalizations of category theory in proof assistants

What are the existing formalizations of category theory in proof assistants? I'm primarily interested in public-domain code implementing category theory in a proof assistant (Coq, Agda, ...
-2
votes
2answers
619 views

Accidental, unplanned breakthroughs in Mathematics [closed]

In math/physics, or generally in science, there are many moments where the success and the triumph come from the accidental, unplanned attempts. Moreover, there are some cases that originally having ...
45
votes
18answers
7k views

Examples of major theorems with very hard proofs that have NOT dramatically improved over time

This question complement a previous MO question: Examples of theorems with proofs that have dramatically improved over time. I am looking for a list of major theorems in mathematics whose proofs are ...
0
votes
2answers
175 views

Intuition about covariant derivative/connections on real and complex manifolds

I was hoping to gain more intuition about the similarities and differences between the covariant derivative (of any connection, not necessarily the Levi Civita one if it exists) on real and complex ...
33
votes
33answers
4k views

Structures that turn out to exhibit a symmetry even though their definition doesn't

Sometimes (often?) a structure depending on several parameters turns out to be symmetric w.r.t. interchanging two of the parameters, even though the definition gives a priori no clue of that symmetry. ...
6
votes
2answers
309 views

Adelic methods for classical modular forms

Many conjectures about properties of automorphic forms on $\mathrm{GL}(2)$ can be formulated in the basic language of classical modular forms (i.e. Hecke forms that are holomorphic on $\mathbb{H}$ or ...
18
votes
8answers
2k views

Fantastic properties of Z/2Z

Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...
6
votes
4answers
1k views

Interactions of number theoretic conjectures and other fields of mathematics

There are many interesting open conjectures in number theory. My question is not about partial results or possible ways to prove them. It is about their interactions with the other fields of ...
3
votes
1answer
198 views

Symmetries of the standard probability space

The standard probability space $(I, \mathcal B, \lambda)$ consists of the interval $I = [0,1]$, its Borel $\sigma$-algebra $\mathcal B := \mathcal B(I)$ and Lebesgue measure $\lambda$. In ...
11
votes
3answers
703 views

Writing Mathematics : Linking words

I'm trying to write mathematics in English and I'm clearly missing something : linking words. I'm writing "so, we get", "Observe that" too many times and I'm afraid to use some expressions : "it ...
77
votes
10answers
10k views

Work of plenary speakers at ICM 2014

The next International Congress of Mathematicians (ICM) will take place in 2014 in Seoul, Korea. The present question is meant to gather brief overviews of the work of the plenary speakers for the ICM ...
22
votes
4answers
961 views

Multiplying by irrational numbers in combinatorial problems

This is getting no attention on stackexchange. Everybody knows that the number of derangements of a set of size $n$ is the nearest integer to $n!/e$. It had escaped my attention until last week, ...
5
votes
1answer
896 views

Hodge Theory (Voisin)

I have a strong understanding of Representation Theory but am interested in learning from Voisin, Hodge Theory and Complex Algebraic Geometry. What are the prerequisites to learning from this ...
12
votes
2answers
512 views

Occurrences of D. H. Lehmer's 10-th degree polynomial

Salem numbers and Lehmer's minimum height problem are venerated not only in number theory and diophantine analysis, where they are considered naturally interesting for their own sake, but also in ...
35
votes
17answers
4k views

What are some deep theorems, and why are they considered deep?

All mathematicians are used to thinking that certain theorems are deep, and we would probably all point to examples such as Dirichlet's theorem on primes in arithmetic progressions, the prime number ...
18
votes
5answers
632 views

Online high quality colloquium talks

In my department we're thinking about showing online lectures one day per week at lunch, as sort of a virtual colloquium appropriate to mathematics undergraduates as well as faculty. To start with ...
12
votes
1answer
335 views

Applications of non-separable Hilbert spaces

In applications, Hilbert spaces of interest are often assumed to be separable. In addition to being extremely convenient mathematically, this assumption can often be justified on computational or ...
16
votes
3answers
2k views

Famous vacuously true statements

I am interested to know other examples vacuously true statements that are non-trivial. My starting example is Turan's result in regards to the Riemann hypothesis, which states Suppose that for each ...
2
votes
1answer
173 views

Properties of the weak-$*$ topology

Let $X$ be a topological affine space over a complete base field $\mathbb S := \mathbb C$, $\mathbb R$ or $\mathbb Q_p$. Let $X^*$ be the dual space of continuous affine functionals equipped with the ...
14
votes
10answers
2k views

An example of a proof that is explanatory but not beautiful? (or vice versa)

This question has a philosophical bent, but hopefully it will evoke straightforward, mathematical answers that would be appropriate for this list (like my earlier question about beautiful proofs ...