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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 question to be made CW.

14 votes
16 answers
1k views

Generalized notions of solutions in various areas of mathematics

In many areas of mathematics (PDE, Algebra, combinatorics, geometry) when we have difficulty in coming with a solution to a problem we consider various notions of "generalized solutions". (There are a …
147 votes
66 answers
39k views

Important formulas in combinatorics

Motivation: The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post, for the poster, and cash prizes offered …
110 votes
89 answers
29k views

Tweetable Mathematics

Update: Please restrict your answers to "tweets" that give more than just the statement of the result, and give also the essence (or a useful hint) of the argument/novelty. I am looking for examp …
217 votes
28 answers
52k views

The most outrageous (or ridiculous) conjectures in mathematics

The purpose of this question is to collect the most outrageous (or ridiculous) conjectures in mathematics. An outrageous conjecture is qualified ONLY if: 1) It is most likely false (Being hopeless is …
68 votes
30 answers
37k views

A book you would like to write

Writing a book from the beginning to the end is (so I heard) a very hard process. Planning a book is easier. This question is dual in a sense to the question "Books you would like to read (if somebody …
172 votes
36 answers
35k 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 …
91 votes
24 answers
22k 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 v …
75 votes
13 answers
13k views

What precisely Is "Categorification"?

(And what's it good for.) Related MO questions (with some very nice answers): examples-of-categorification; can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$?; categorification-request.
Gil Kalai's user avatar
  • 24.7k
63 votes
19 answers
12k views

Generalizations of the four-color theorem

The four color theorem asserts that every planar graph can be properly colored by four colors. The purpose of this question is to collect generalizations, variations, and strengthenings of the four c …
143 votes
12 answers
30k views

Solutions to the Continuum Hypothesis

Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? ; Completion of ZFC ; Complete resolutions of GCH How far wrong could the Continuum Hypothesis be? When was t …
Gil Kalai's user avatar
  • 24.7k
36 votes
21 answers
6k views

Generalizations of Planar Graphs

This is a follow up to Harrison's question: why planar graphs are so exceptional. I would like to ask about (and collect answers to) various notions, in graph theory and beyond graph theory (topology; …
Gil Kalai's user avatar
  • 24.7k
72 votes
13 answers
11k views

The use of computers leading to major mathematical advances II

I would like to ask about recent examples, mainly after 2015, where experimentation by computers or other use of computers has led to major mathematical advances. This is a continuation of a question …
156 votes
52 answers
23k views

Experimental mathematics leading to major advances

I would like to ask about examples where experimentation by computers has led to major mathematical advances. A new look Now as the question is five years old and there are certainly more examples of …
220 votes
140 answers
49k views

Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please) I'd love to learn about …