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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32
votes
9answers
2k views

Homotopy as a general organizing principle

One of the realizations that led to the development of Homotopy Type Theory (HoTT) is that the ideas of homotopy theory have very broad applicability in mathematics. Indeed, Quillen model categories ...
39
votes
1answer
2k views

What if the Riemann Hypothesis were false?

There are lots of known and interesting consequences of the Riemann Hypothesis being true. Are there any known and interesting consequences of the Riemann Hypothesis being false?
9
votes
6answers
945 views

Math research in the app store [closed]

What apps can be found in the Apple app store, Google Play, Blackberry World etc. showcasing specific mathematics research? (Edit - Since the first version of the question got closed, examples should ...
23
votes
3answers
1k views

Nearly all math classes are lecture+problem set based; this seems particularly true at the graduate level. What are some concrete examples of techniques other than the “standard math class” used at the *Graduate* level?

In the fall, I am teaching one undergraduate and one graduate course, and in planning these courses I have been thinking about alternatives to the "standard math class". I have found it much easier ...
18
votes
21answers
2k views

What are some examples of mathematicians who had an unconventional education? [duplicate]

Possible Duplicate: Famous mathematicians with background in arts/humanities/law etc What are some examples of mathematicians who had an unconventional education and yet, went on to make an ...
7
votes
5answers
842 views

Mathematics for ebook readers

Project Gutenberg has a mathematics section, and they prepare their more recent publications in a format that works very well on an ebook reader of moderate size: they generate PDFs in a size of ...
37
votes
19answers
5k views

Are there proofs that you feel you did not “understand” for a long time?

Perhaps the "proofs" of ABC conjecture or newly released weak version of twin prime conjecture or alike readily come to your mind. These are not the proofs I am looking for. Indeed my question was ...
60
votes
24answers
6k views

Modern Mathematical Achievements Accessible to Undergraduates

While there is tremendous progress happening in mathematics, most of it is just accessible to specialists. In many cases, the proofs of great results are both long and use difficult techniques. Even ...
12
votes
11answers
1k views

Classic applications of Baire category theorem

I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...
12
votes
0answers
428 views

What are some examples of weak ω-categories?

As is usual, let's say an (n, k)-category is something with objects, morphisms, 2-morphisms, ..., n-morphisms, such that all j-morphisms for j > k are invertible, everything meant in the weak sense. ...
2
votes
2answers
247 views

Stronger theorem not resulting from proof analysis

Suppose that we proved $\varphi$ from a theory $T$. Often we ask whether or not we could have proved $\varphi$ with a weaker theory, to find out we usually analyze the proof and try to figure out ...
28
votes
13answers
2k views

Great mathematics books by pre-modern authors

Last summer, I read Euclid's Elements, and it was an eye-opening experience; I had assumed that three thousand years' difference would make the notation incomprehensible and the reasoning alien, but ...
7
votes
7answers
975 views

Gelfand representation and functional calculus applications beyond Functional Analysis

I think it is fair to say that the fields of Operator Algebras, Operator Theory, and Banach Algebras rely on Gelfand representation and functional calculus in a crucial way. I am curious about ...
28
votes
33answers
4k views

Fixed point theorems

It is surprising that fixed point theorems (FPTs) appear in so many different contexts throughout Mathematics: Applying Kakutani's FPT earned Nash a Nobel prize; I am aware of some uses in logic; and ...
4
votes
2answers
1k views

List of Charlatans in Mathematics [closed]

Recently, while looking for articles and documents to learn about the Riemann Hyopthesis, I came across a strange funny document of a chinese "mathematician" called Jiang Chun-Xuang who claimed to ...
6
votes
2answers
2k views

Two questions about combinatorics journals

Hello, I have two questions regarding combinatorics journals. I hope that this is the right place for such questions. Which combinatorics/DM journals would you consider as the "top tier"? I tried ...
9
votes
4answers
1k views

Role of applications in modern mathematics [closed]

Older days scientists were universalists and philosophy, physics and mathematics were a part the same question - understanding the world. Nowadays one may get feeling that the role of applications ...
9
votes
4answers
716 views

Connections between topos theory and topology

What are some "applications to" / "connections with" topology that one could hope to reasonably cover in a first course on topos theory (for master students)? I have an idea of what parts of the ...
0
votes
1answer
262 views

Applications of the natural bilinear forms on the direct sum between a vector space and its dual

As it is know, the vector space $V\oplus V^\ast$ admits the natural symmetric and skew symmetric bilinear forms $$\langle X+\xi,Y+\eta\rangle|_\pm:=\frac 1 2 \(\xi(Y) \pm \eta(X)\)$$ I am interested ...
15
votes
3answers
965 views

Research level applications of “row rank = column rank”?

No less an authority than Gilbert Strang frames "row rank equals column rank" (and a couple of other facts) as "The Fundamental Theorem of Linear Algebra." I'd simply like to assemble (for teaching ...
5
votes
0answers
772 views

“Must read ”papers on analytic number theory

Question: What would be some must-read papers for an aspiring analytic number theorist? In other words, what are the papers that any analytic number theorist would have read? (Background: ...
3
votes
1answer
231 views

Repertory of the different sorts of operads

Many different types of operads have emerged in recent years (symmetric, shuffle, cyclic, anticyclic, coloured, etc.). I would like, for any of these, list the following data: Description of the ...
49
votes
5answers
2k views

Math Annotate Platform?

Suppose most mathematical research papers were freely accessible online. Suppose a well-organized platform existed where responsible users could write comments on any paper (linking to its doi, ...
6
votes
3answers
775 views

problems from the scottish book

Which of the problems from the Scottish Book (pdf of English version) by Stefan Banach are still open? I know that one of the problems was solved by Per Enflo for which he got a live goose from ...
11
votes
2answers
864 views

Using schemes to prove things about rings

I apologize for asking a big list question, I've tried to avoid doing so for a while. I'll give my justification in a moment. The question is as follows: What are examples of strict applications ...
38
votes
19answers
6k views

Mathematicians whose works were criticized by contemporaries but became widely accepted later

Gauss famously discarded Abel's proof that an algebraic equation of degree five or more cannot have a general solution (Abel himself had rejected divergent series as the work of the devil). Cantor's ...
5
votes
3answers
873 views

group of diffeomorphisms of a manifold

How much has been the group of diffeomorphisms of a manifold " been studied. I got this information from wiki. " Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra ...
18
votes
8answers
2k views

How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there?

Offhand I can think of two ways in classical homotopy theory: Show that $\pi_k(S^n)=0$ for $k\lt n$ by deforming a map $S^k\to S^n$ to be non-surjective, then contracting it away from a point not in ...
1
vote
0answers
265 views

applications of gauss-bonnet theorem

In wikipedia,I was pretty amazed to find a proof of fundamental theorem of algebra using gauss bonnet theorem.I think given how central it is to mathematics with its far reaching generalizations like ...
32
votes
31answers
6k views

Trichotomies in mathematics

Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the ...
6
votes
4answers
863 views

fourier analytic proofs

While searching through Mathoverflow, I found out a fourier analytic proof of the Isoperimetric Inequality.Also, by google search I found a fourier analytic proof of Quadratic Reciprocity theorem.I ...
19
votes
10answers
2k views

Learning through guided discovery

I have been working through Kenneth P. Bogart's "Combinatorics Through Guided Discovery". You can download it from this page: http://www.math.dartmouth.edu/news-resources/electronic/kpbogart/ I've ...
5
votes
4answers
1k views

What are conjectures that are true for primes but then turned out to be false for some composite number?

Note: This is an update formulation since many people misunderstood the question before. Of course it is easy to make a statement like "Every n is a prime or at most 1000", which is true for every ...
9
votes
2answers
1k views

Math Zeitgeist 2012 [closed]

Dear MO-fellows, Happy New Year ! Taking google as an example: https://www.google.com/zeitgeist/2012/ Why not to make a list of noted math events in 2012 ? E.g. 15 December 2012 Museum of Math was ...
52
votes
61answers
8k views

Old books still used

It's a commonplace to state that while other sciences (like biology) may always need the newest books, we mathematicians also use to use older books. While this is a qualitative remark, I would like ...
23
votes
7answers
2k views

Excellent mathematical explanations

In the Stanford Encyclopedia of Philosophy there is an entry on mathematical explanation. The basic philosophical question is: What makes a proof explanatory? Two main "models" of mathematical ...
12
votes
4answers
1k views

Statements which were given as axioms, which later turned out to be false.

I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness ...
6
votes
4answers
1k views

On similar concepts in mathematics whose similarity is a non-trivial fact.

Recently, while undertaking a study of commutative algebra, I learned three concepts: (i) a local ring, (ii) a regular local ring and (iii) a regular ring. At the end, I found myself asking this ...
6
votes
12answers
5k views

Useless math that became useful

I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless. My idea is to amend my article with some theories that seemed useless when they are created but ...
27
votes
9answers
2k views

Covering maps in real life that can be demonstrated to students

Edit: I've really enjoyed everyone's examples (especially the pictures!), but I was mostly looking for a general theorem. For instance, a similar statement to mine is, Can the mapping cylinder of ...
20
votes
15answers
2k views

Non-rigorous reasoning in rigorous mathematics

I was wondering what role non-rigorous, heuristic type arguments play in rigorous math. Are there examples of rigorous, formal proofs in which a non-rigorous reasoning still plays a central part? ...
11
votes
7answers
2k views

What are some Applications of Teichmüller Theory?

I'm trying to collect some specific examples of applications of Teichmüller Theory. Here are some things I have collected thus far: No-wandering-domain Theorem (Sullivan) Theorems of Thurston ...
5
votes
2answers
1k views

Consequences of Legendre's conjecture

I am looking for a list/reference which explores the consequences of Legendre's conjecture, which states that one can always find a prime number between $n^2$ and $(n+1)^2$.
7
votes
9answers
994 views

Examples where adding complexity made a problem simpler

I can think of a few situations in math where a problem becomes easier or an object becomes simpler when some complexity is added. Examples: $S^n$ is never contractible, but $S^{\infty}$ is. The ...
1
vote
0answers
253 views

Applications of the class number formula, etc.

This is a big list of applications of the class number formula and its generalizations. I'll start: The solution to Gauss's class number problem for imaginary quadratic fields, and more generally ...
19
votes
3answers
660 views

What classification theorems have been improved by re-categorizing?

Many classification theorems (e.g. of the finite subgroups of $SO(3)$, or the finite-dimensional complex simple Lie algebras, or the finite simple groups) have some infinite lists, plus some ...
4
votes
3answers
576 views

Quotations about the power of simple ideas [closed]

I'm looking for quotations about how very simple mathematical ideas can be very powerful. I know of a few, but they're not quite what I'm looking for insofar as they contain criticism of other ...
4
votes
0answers
190 views

Quotations about the class number formula, etc.

I'm looking for interesting and/or expressive quotations from mathematicians about the class number formula. I'm interested both in quotations from historical mathematicians and from modern ...
37
votes
11answers
2k views

Important open problems that have already been reduced to a finite but infeasible amount of computation

Most open problems, when formalized, naturally involve quantification over infinite sets, thereby obviating the possibility, even in principle, of "just putting it on a computer." Some questions ...
30
votes
10answers
3k views

Believing the Conjectures

In Believing the axioms (I and II), Penelope Maddy proposes five "rules of thumb" that she then uses to justify large cardinal axioms in set theory. These extrinsic rules are modeled after the ...