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 ...

learn more… | top users | synonyms

13
votes
8answers
3k views

Failures that lead eventually to new mathematics [duplicate]

Possible Duplicate: Most interesting mathematics mistake? In the 25-centuries old history of Mathematics, there have been landmark points when a famous mathematician claimed to have proven a ...
109
votes
39answers
32k views

Most interesting mathematics mistake?

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincaré's 3d sphere characterization or the search to prove ...
27
votes
35answers
2k 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, ...
3
votes
0answers
184 views

Roadmap for the ideas expressed in Grothendieck's Esquisse d'un Programme

I would like to understand Grothendieck's Esquisse d'un Programme more. Are there any references that would help me, and are there modern works pursuing the same themes? At this point I am still ...
58
votes
20answers
6k views

Solving algebraic problems with topology

Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem. ...
67
votes
77answers
12k views

Elementary+Short+Useful

Imagine your-self in front of a class with very good undergraduates who plan to do mathematics (professionally) in the future. You have 30 minutes after that you do not see these students again. You ...
144
votes
77answers
24k views

Your favorite surprising connections in Mathematics

There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the ...
4
votes
2answers
543 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 ...
532
votes
207answers
135k views

Examples of common false beliefs in mathematics

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...
59
votes
22answers
9k 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 ...
77
votes
17answers
6k views

Proposals for polymath projects

Background Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by ...
14
votes
10answers
3k views

nonstandard analysis book recommendation

I wish to learn nonstandard analysis. Are there any good book recommendations? I'm familiar with the ZFC system, and learnt analysis the classical way. I've found some undergraduate texts, but they ...
177
votes
30answers
18k views

Which journals publish expository work?

I wonder if anyone else has noticed that the market for expository papers in mathematics is very narrow (more so than it used to be, perhaps). Are there any journals which publish expository work, ...
30
votes
1answer
658 views

A dictionary of Characteristic classes and obstructions

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians. In an effort to ...
53
votes
20answers
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 ...
98
votes
26answers
31k views

What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
65
votes
8answers
4k views

Examples where physical heuristics led to incorrect answers?

I have always been impressed by the number of results conjectured by physicist, based on mathematically non-rigorous reasoning, then (much) later proved correct by mathematicians. A recent example is ...
113
votes
46answers
36k views

Ways to prove the fundamental theorem of algebra

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there? Please give a new way in each answer, and if possible give reference. I start by giving two: ...
9
votes
1answer
480 views

Problems which use S₄ → S₃

I need examples of problems which use, directly or indirectly, the homomorphism $S_4\to S_3$ in the solution (its kernel is $\mathbb{Z}_2\oplus\mathbb{Z}_2$). Obvious candidates: Lagrange resolvent ...
125
votes
27answers
30k 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 ...
57
votes
41answers
9k views

Examples of theorems with proofs that have dramatically improved over time

I am looking for examples of theorems that may have originally had a clunky, or rather technical, or in some way non-illuminating proof, but that eventually came to have a proof that people consider ...
20
votes
17answers
33k views

Suggestions for a good Measure Theory book

I have taken analysis and have looked at different measures, but I am currently looking at realizing a certain problem in a different light and feel that I need a better background in various measures ...
87
votes
17answers
8k views

Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$?

I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...
26
votes
5answers
1k 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, ...
104
votes
59answers
18k views

Jokes in the sense of Littlewood: examples? [closed]

First, let me make it clear that I do not mean jokes of the "abelian grape" variety. I take my cue from the following passage in A Mathematician's Miscellany by J.E. Littlewood (Methuen 1953, p. 79): ...
119
votes
37answers
15k views

Books you would like to read (if somebody would just write them…)

I think that the title is self-explanatory but I'm thinking about mathematical subjects that have not received a full treatment in book form or if they have, they could benefit from a different ...
72
votes
25answers
9k views

Papers that debunk common myths in the history of mathematics

What are some good papers that debunk common myths in the history of mathematics? To give you an idea of what I'm looking for, here are some examples. Tony Rothman, "Genius and biographers: The ...
29
votes
8answers
5k views

What are some important but still unsolved problems in mathematical logic?

In the past, First order logic and its completeness and whether arithmetic is complete was a major unsolved issues in logic . All of these problems were solved by Godel. Later on, independence of ...
11
votes
7answers
682 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 ...
8
votes
8answers
12k views

Pseudo-random number generation algorithms

What algorithms are used in modern and good-quality random number generators?
10
votes
10answers
941 views

Statements reliant on conjectures

There are lots of statements that have been conditionally proved on the assumption that the Riemann Hypothesis is true. What other conjectures have a large number of proven consequences?
4
votes
3answers
114 views

Repository of graph classes that are tough to test non-isomorphic pairs from isomorphic pairs

(1) Which graph classes are extremely tough to test for graph non-isomorphic pairs from isomorphic pairs? (2) Is there a repository of adjacencies from such classes?
9
votes
6answers
2k 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 ...
3
votes
2answers
86 views

Database of adjacency matrices on cospectral non-isomorphic graph pairs

Is there a repository of cospectral non-isomorphic graphs available somewhere? I am looking for list of $0/1$ adjacency matrix pairs that can be input data in tools such as MATLAB.
16
votes
14answers
1k views

Applications of Representation Theory in Combinatorics

What are the examples of interesting combinatorial identities (e.g. bijection between two sets of combinatorial objects) that can be proved using representation theory, or has some representation ...
60
votes
19answers
21k views

Why were matrix determinants once such a big deal?

I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. ...
168
votes
36answers
46k views

Widely accepted mathematical results that were later shown wrong?

I wonder if there are any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time ...
59
votes
15answers
6k views

Sophisticated treatments of topics in school mathematics

Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples ...
9
votes
0answers
175 views

Algebraic K-theory of a ring.

I started to learn some algebraic $K$-theory and its relation to geometric topology problems. My question is : What is the list of rings such that all their algebraic $K$-theory groups are known ? I ...
80
votes
17answers
21k views

What recent discoveries have amateur mathematicians made?

E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what ...
8
votes
4answers
934 views

Which journals publish experimental results in pure maths?

All pure mathematicians know that the goal is to produce insight, rather than to simply obtain results. However, it might sometimes be of value to disseminate largely empirical work. In the same ...
63
votes
34answers
11k views

What notions are used but not clearly defined in modern mathematics?

"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions." Felix Klein What notions are used but not ...
29
votes
18answers
10k views

Interesting and Accessible Topics in Graph Theory

This summer, I will be teaching an introductory course in graph theory to talented high school seniors. The intent of the course is not to establish proficiency in graph theory, per se. Rather, I hope ...
70
votes
30answers
10k views

What are some very important papers published in non-top journals?

There has already been a question about important papers that were initially rejected. Many of the answers were very interesting. The question is here. My concern in this question is slightly ...
1
vote
0answers
72 views

On Different Ways of Proving Isoperimetric Inequalities [closed]

Update: Thanks to Douglas Zare's comment, My previous questions in this thread turned out to be equivalent to the Isoperimetric problem. Thus I edited my question to make it a bit different. ...
58
votes
38answers
12k views

nontrivial theorems with trivial proofs

A while back I saw posted on someone's office door a statement attributed to some famous person, saying that it is an instance of the callousness of youth to think that a theorem is trivial because ...
82
votes
18answers
8k views

Occurrences of (co)homology in other disciplines and/or nature

I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive ...
17
votes
21answers
24k views

Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?
61
votes
16answers
6k views

Facts from algebraic geometry that are useful to non-algebraic geometers

A professor of mine (a geometric topologist, I believe) once criticized the core graduate curriculum at my institution because it teaches all sorts of esoteric algebra, but does not include basic ...
21
votes
29answers
14k views

Alternative Undergraduate Analysis Texts

Other than the standard baby Rudin, Spivak, and Stein-Shakarchi, are there other alternative and comprehensive analysis texts at the undergraduate level? For example something that has general results ...