**14**

votes

**9**answers

2k 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 ...

**7**

votes

**3**answers

2k views

### How to find/guess a polynomial sequence?

My question is motivated by the recent question and more recent
appearance of its author Bruce Westbury.
Most of you know that the best way to find a sequence of integers is looking for it on
The ...

**48**

votes

**13**answers

4k views

### What makes four dimensions special?

Do you know properties which distinguish four-dimensional spaces among the others?
What makes four-dimensional topological manifolds special?
What makes four-dimensional differentiable manifolds ...

**0**

votes

**1**answer

238 views

### a book comparable to Development of mathematics in the 19th century by F.Klein? [on hold]

This book is apparently very interesting according to Vladimir Arnold. I couldn't get my hand on a copy yet, therefore I would to ask you for any reference similar to it, and also can you post ...

**27**

votes

**6**answers

1k views

### Negative impact of wrong or non-rigorous proofs

The recent talks of Voevodsky (for example, http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf), which describe subtle errors in proofs by him as well as others, as well ...

**2**

votes

**4**answers

314 views

### Should all equations which appear in a thesis be numbered?

I was just wondering if there is any sort of consensus on the topic of when to number math expressions.
For example different lines in a proof, these should be tagged or not tagged?

**24**

votes

**5**answers

4k views

### What is the etymology of the term “perverse sheaf”?

Grothendieck famously objected to the term "perverse sheaf" in Récoltes et Semailles, writing "What an idea to give such a name to a mathematical thing! Or to any other thing or living being, except ...

**12**

votes

**19**answers

7k views

### Textbook recommendations for undergraduate proof-writing class

I am teaching the proof-writing class (for the 3rd time) in the Fall and plan to buck the party line and use a different text than the default Bond and Keane. My parameters are as follows:
Logic, ...

**39**

votes

**18**answers

5k views

### How can an extremely mathematically talented young person be helped to fulfill his/her potential?

Obviously, this question is not a research level mathematics question at all. But, I've just met an extremely mathematically talented 11 years old student and I don't know how I can help him. For ...

**72**

votes

**16**answers

6k views

### How does one justify funding for mathematics research?

G. H. Hardy's A Mathematician's Apology provides an answer as to why one would do mathematics, but I'm unable to find an answer as to why mathematics deserves public funding. Mathematics can be ...

**3**

votes

**4**answers

233 views

### Which fields could be applied to neurosciences?

I have a friend who wants to study something applied to neurosciences. He is going to begin his grad studies in mathematics.
He asked me which areas of mathematics could be applied to neurosciences.
...

**39**

votes

**11**answers

5k views

### What areas of pure mathematics research are best for a post-PhD transition to industry?

I have a student who is looking to start a PhD in pure mathematics. She is talented and motivated, and will do quite well. She is still in a phase of her development where she is still open to the ...

**37**

votes

**20**answers

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 ...

**9**

votes

**3**answers

714 views

### Proof correctness problem

I was watching this talk by Vladimir Voevodsky which was given at the Institute of Advanced Study in 2006.
In his talk the first slide he shows has the following written on it:
...

**5**

votes

**0**answers

278 views

### Refereeing a paper containing strong statements about other papers

The title says it all. Suppose you are refereeing a paper where the author A makes strong statements about other papers by a different author B, like: the proof of Theorem 1 in paper [B] is wrong and ...

**38**

votes

**10**answers

2k views

### Why is Set, and not Rel, so ubiquitous in mathematics?

The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations.
Why was there the necessity of singling ...

**27**

votes

**11**answers

2k views

### Are there elementary-school curricula that capture the joy of mathematics?

UPDATE: Wow, thank you everyone for the great insights!
A couple of months ago I stumbled across Paul Lockhart's essay A Mathematician's Lament and it made perfect sense to me. I'm not meaning to ...

**92**

votes

**59**answers

15k 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):
...

**81**

votes

**61**answers

18k views

### Which math paper maximizes the ratio (importance)/(length)?

My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7-manifolds which are ...

**72**

votes

**45**answers

29k views

### Interesting mathematical documentaries

I am looking for mathematical documentaries, both technical and non-technical. They should be "interesting" in that they present either actual mathematics, mathematicians or history of mathematics. I ...

**2**

votes

**0**answers

66 views

### Countable model theory for $\omega$-stable theories?

This is a bit of a fishing expedition, because I'm not sure what I'm looking for. Very vaguely stated, here's the driving question:
What conditions on an $\omega$-stable theory make the class of ...

**9**

votes

**3**answers

2k views

### Reading Papers in a Language you don't Speak

First, I apologize if I'm posting this to the wrong place, but it seems correct.
My adviser sent me the SGA text of Grothendieck which is in French. Though I can piece together parts of the text, I'm ...

**3**

votes

**0**answers

76 views

### Literature that helps explain what the theory of numerosities contributes with

Since 2003 a group of Italian mathematicians (Benci, Di Nasso and Forti) has developed a new measure for infinite sets that satisfies the Euclidian principle: The whole is greater than the part. The ...

**31**

votes

**5**answers

3k views

### What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?

I think we all occasionally come across terminology that we'd like to see supplanted (e.g. by something more systematic). What I'd like to know is, under what circumstances is it reasonable to believe ...

**32**

votes

**10**answers

13k views

### Mathematics as a hobby [closed]

I would like to know if practicing mathematics, constituting a hobby for some of you who are neither academics nor (advanced) mathematics, is an important part of your career. How do you go and learn ...

**2**

votes

**0**answers

39 views

### Almost everywhere in a rectangle [duplicate]

I would like to ask a question about the product (Lebesgue) measure on rectangle. I tried to solve the problem but I couldn't.
Let $S$ be a subset of a region, say $R$ which is enclosed by a ...

**56**

votes

**23**answers

21k views

### What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...

**6**

votes

**1**answer

250 views

### Differences in philosophy between Lie Groups and Differential Galois Theory

As far as I have heard,Sophus Lie's aim was to construct an analogue of galois theory for differential galois theory. I am familiar with lie group but not with differential galois theory. What is the ...

**21**

votes

**2**answers

1k views

### Do's and don'ts of writing survey papers

I am not sure if this is the appropriate forum to ask as it is not directly related to a research level (math) problem, but I figured it was worth a try. I recently attended a conference and felt that ...

**4**

votes

**0**answers

77 views

### Why is the density function of a sum of many iid random variables (i.e., the Gaussian) self-dual?

In the usual proof of the central limit theorem via characteristic functions, we note that if $X_1, \dots, X_n$ are independent and identically distributed, with probability density function $f(x)$, ...

**18**

votes

**5**answers

3k views

### Why differential forms are important?

Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated ...

**5**

votes

**1**answer

212 views

### Are there any organized websites for seminar/conference videos?

These days, there are many conference centers and universities recording seminars and conference talks and make them available on the web. Some examples:
http://www.fields.utoronto.ca/video-archive
...

**27**

votes

**26**answers

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. ...

**17**

votes

**0**answers

397 views

### Good ways to organize old personal mathematical resources

I am wondering how the other Mathematicians organize their old mathematical resources, like calculation drafts, class and seminar notes etc.
These old resources may be related to a wide range of ...

**94**

votes

**37**answers

62k views

### Too old for advanced mathematics? [closed]

Kind of an odd question, perhaps, so I apologize in advance if it is inappropriate for this forum. I've never taken a mathematics course since high school, and didn't complete college. However, ...

**4**

votes

**1**answer

218 views

### Where can I find resources for creating a mathematics “bridge course”?

My department is in the very early stages of developing a "bridge course" or "introduction to proofs" course, motivated by our lower-level courses not currently doing a good job of preparing our ...

**6**

votes

**5**answers

1k views

### Optical methods for number theory?

I found a paper: 'A New Method of Finding the Distribution of Prime Number', saying
We stack discs and annuluses with certain rules then turn on the light to illuminate. The projection of ...

**37**

votes

**24**answers

6k views

### The concept of Duality

I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come.
...

**0**

votes

**1**answer

209 views

### What do Hilbert and Bernays mean when they say “finitist number theory”? [closed]

Perhaps it is not a fair question to be addressed here. Anyway, when I read that Hilbert and Bernays develop finitist number theory. What does "finitist number theory" mean here?

**10**

votes

**8**answers

715 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 ...

**45**

votes

**27**answers

4k views

### books well-motivated with explicit examples

It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is ...

**93**

votes

**11**answers

13k views

### Have any long-suspected irrational numbers turned out to be rational?

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...

**10**

votes

**4**answers

1k views

### Does formalizing math require search and creativity, or is it near-mechanical?

I remember reading somewhere that it takes about a week to convert a page of math into something a proof-assistant like Isabelle or HOL Light would accept.
Is this type of conversion something that ...

**83**

votes

**9**answers

4k views

### What non-categorical applications are there of homotopical algebra?

(To be honest, I actually mean something more general than 'homotopical algebra' - topos theory, $\infty$-categories, operads, anything that sounds like its natural home would be on the nLab.)
More ...

**6**

votes

**10**answers

821 views

### Examples of “Unusual” Classifications

When one says "classification" in math, usually one of a handful of examples springs to mind:
-Classification of Finite Simple Groups with 18 infinite families and 26 sporadic examples (assuming one ...

**5**

votes

**4**answers

1k views

### Advice for number theory library

Hi I just got a faculty position and it comes with a generous start up funds for "office supplies", which I must use or lose. What does a pure mathematician need? I have good computers already. I ...

**16**

votes

**12**answers

5k views

### What is your favorite proof of Tychonoff's Theorem?

Here is mine. It's taken from page 11 of "An Introduction To Abstract Harmonic Analysis", 1953, by Loomis:
http://www.archive.org/details/introductiontoab031610mbp
...

**66**

votes

**6**answers

6k 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 ...

**69**

votes

**16**answers

15k 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 ...

**16**

votes

**8**answers

2k views

### Do there exist modern expositions of Klein's Icosahedron?

Reading Serre's letter to Gray
, I wonder if now modern expositions of the themes in Klein's book
exist. Do you know any?