0
votes
0answers
43 views

Models for events where position and time are correlated

Apologies in advance if this question is not sufficiently research-level: What are the standard models that are used to describe phenomena in which events that occur at the same time are likely to be ...
93
votes
19answers
21k 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
721 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. ...
24
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 ...
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. ...
72
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 ...
20
votes
4answers
720 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
8answers
757 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 ...
2
votes
2answers
340 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 ...
68
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
438 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 ...
0
votes
0answers
163 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 ...
11
votes
3answers
722 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
11k 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 ...
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
636 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 ...
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 ...
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 ...
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 ...
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 ...
2
votes
2answers
248 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
990 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 ...
7
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 ...
5
votes
0answers
901 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: ...
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, ...
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
917 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 ...
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
866 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 ...
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 ...
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 ...
7
votes
9answers
998 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 ...
19
votes
3answers
663 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 ...
6
votes
8answers
2k views

Beautiful theorems with short proof [closed]

I like to ask for beautiful mathematical theorems with short proof. A proof is short in my sense if it is at most one page assuming basic notations and very basic results a second year student will ...
5
votes
3answers
2k views

Battle of the brains; cultural mathematics

Firstly, I apologize if my question is long. Three years ago, I watched a video with the name Battle of the Brains. It was a wonderful video about challenging some famous peoples to solve some ...
18
votes
19answers
3k views

History Question: AUTObiography of Mathematicians

According to Wikipedia, an autobiography is an account of the life of a person, written by that person sometimes with a collaborator. An autobiography offers the author the ability to recreate ...
29
votes
11answers
5k views

“Must read” papers in numerical analysis

In 1993, Prof. L.N. Trefethen published a NA-net posting with a list of thirteen paper he used for teaching the seminar Classic Papers in Numerical Analysis. In Trefethen's words, ... this course ...
25
votes
65answers
7k views

Fiction books about mathematicians? [closed]

What are some fiction books about mathematicians? It seems to me rather difficult for writers to create good books on this subject. Some years ago I thought there were no such books at all. There ...
8
votes
10answers
3k views

Music: mathematical point of view (revised) [closed]

Mathematical analysis of music started when Pythagoras made his observations about consonant intervals and ratios of string lengths. ADDED: In the paper Mathematical Music Theory -- Status Quo 2000, ...
10
votes
1answer
365 views

List of Whitehead-like Problems

Whitehead problem is a rather well known problem: Suppose that $G$ is an abelian group and $\mathrm{Ext}^1(G,\Bbb Z)=0$, is $G$ free? It wasn't long before it was proved that if $G$ is ...
67
votes
60answers
7k views

Blackbox Theorems [closed]

By a blackbox theorem I mean a theorem that is often applied but whose proof is understood in detail by relatively few of those who use it. A prototypical example is the Classification of Finite ...
3
votes
2answers
1k views

Papers whose title defines a new terminology [duplicate]

To explain a new signal processing technique based on Fourier Transform, Bogert et al went on to define a new vocabulary. The new terminology was published in a paper with the title: The Quefrency ...
39
votes
35answers
6k 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 ...
14
votes
12answers
2k views

Excellent uses of induction and recursion

Can you make an example of a great proof by induction or construction by recursion? Given that you already have your own idea of what "great" means, here it can also be taken to mean that the chosen ...
5
votes
5answers
2k views

Easy and Hard problems in Mathematics [closed]

Modified question: I would like to know some examples of problems in Mathematics, for pedagogical purposes, which do not involve difficult techiques to solve the problem but with a change of context ...