**78**

votes

**12**answers

16k views

### What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...

**25**

votes

**6**answers

2k views

### A toolbox for algebraic topology

This question has a very general part and a rather concrete part.
General:
When one wants to prove something in algebraic topology (actually in all parts of mathematics) one obviously needs some ...

**61**

votes

**12**answers

5k 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 ...

**18**

votes

**2**answers

701 views

### how do you visualize characteristic class?

For cohomology, there are some equivalent definitions when the object we consider is sufficiently nice. Since I mainly work with algebraic variety, I will restrict the objects I am considering to be ...

**7**

votes

**2**answers

545 views

### Hilbert's 3rd problem,number theory, motives, cyclic homology,…

This talk by Jinhyun Park connects a lot of interesting themes, making me curious to read more about that. Do you know where?

**135**

votes

**72**answers

23k 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 ...

**139**

votes

**136**answers

30k 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 ...

**68**

votes

**16**answers

7k 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 ...

**230**

votes

**21**answers

28k views

### Thinking and Explaining

How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words ...

**1**

vote

**0**answers

93 views

### Programmatically computing dual Hopf algebras: state of the art

Given a graded Hopf algebra of finite type, we know the (graded) linear dual is also a graded Hopf algebra. For instance the dual Hopf algebra to the polynomial algebra on an even degree generator, ...

**79**

votes

**41**answers

10k views

### Experimental Mathematics

I would like to ask about examples where experimentation by computers have led to major mathematical advances.
A new look
Now as the question is five years old and there are certainly more examples ...

**21**

votes

**8**answers

4k views

### How should one think about sheafification and the difference between a sheaf and a presheaf

The first time I got in touch with the abstract notion of a sheaf on a topological space $X$, I thought of it as something which assigns to an open set $U$ of $X$ something like the ring of continuous ...

**57**

votes

**22**answers

8k 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 ...

**10**

votes

**1**answer

201 views

### On Bailey–Borwein–Plouffe formula for irrational numbers

A BBP-type formula for an irrational number $\alpha$ in the integer base $b\geq 2$ is a formula in the form $\alpha=\Sigma_{k=0}^{\infty}\frac{1}{b^k}\frac{p(k)}{q(k)}$ ($p, q$ are polynomials in ...

**95**

votes

**25**answers

42k views

### Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows-
An engineer, a physicist, and a mathematician are discussing how to visualise four ...

**21**

votes

**6**answers

2k views

### Residual finiteness: why do we care?

Residually finite groups have been studied for a long time. However, I am struggling to work out why we care, or perhaps, why they continue to be of interest. Let me explain.
Magnus, in his 1968 ...

**8**

votes

**1**answer

339 views

### Grothendieck - sheaves as meter sticks

I'm trying to read parts of McLarty's Grothendieck on Simplicity and Generality. In the article, I read Grothendieck thought of sheaves over some topological space as meter sticks measuring it.
...

**42**

votes

**24**answers

7k 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.
...

**60**

votes

**32**answers

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

**3**answers

2k views

### On what kind of objects do the Galois groups act?

I am neither number theorist nor algebraic geometer. I am wondering
whether Galois groups of number fields (say the absolute Galois
group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$) act on objects which
...

**35**

votes

**22**answers

3k views

### What are examples of good toy models in mathematics?

This post is community wiki.
A comment on another question reminded me of this old post of Terence Tao's about toy models. I really like the idea of using toy models of a difficult object to ...

**27**

votes

**3**answers

1k views

### Underlying idea for (automorphic) L-function?

Edit: So with a few more months of math under my belt, I recognize some of the issues with this question. I still hope for an answer, so let me say a few things.
Within the Langlands philosophy, ...

**32**

votes

**5**answers

5k views

### What is a cumulant really?

A cumulant is defined via the cumulant generating function
$$ g(t)\stackrel{\tiny def}{=} \sum_{n=1}^\infty \kappa_n \frac{t^n}{n},$$
where
$$
g(t)\stackrel{\tiny def}{=} \log E(e^{tX}).
$$
Cumulants ...

**26**

votes

**9**answers

6k views

### Intuition and/or visualisation of Ito integral/Ito's lemma

Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve.
See e.g. Wikipedia:Riemann sum
The Ito integral has due to the unbounded total variation ...

**92**

votes

**42**answers

18k views

### Examples of eventual counterexamples

Define an "eventual counterexample" to be
$P(a) = T $ for $a < n$
$P(n) = F$
$n$ is sufficiently large for $P(n) = T\ \ \forall n \in \mathbb{N}$ to be a 'reasonable' conjecture to make.
where ...

**54**

votes

**13**answers

11k views

### Logic in mathematics and philosophy

What are the relations between logic as an area of (modern) philosophy and mathematical logic.
The world "modern" refers to 20th century and later, and I am curious mainly about the second half of ...

**4**

votes

**1**answer

1k views

### Why we study Geometric invariant theory?

I am trying to learn Geometric invariant theory like it was introduced by Mumford. But I do not have a strong motivation and so I want to know the reason of studying Geometric invariant theory. I just ...

**5**

votes

**1**answer

331 views

### Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hierarchy

Philip Ehrlich's paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45. claims as a theorem that, in NBG, ...

**18**

votes

**4**answers

3k views

### Integrals from a non-analytic point of view

I've mentioned before that I'm using this forum to expand my knowledge on things I know very little about. I've learnt integrals like everyone else: there is the Riemann integral, then the Lebesgue ...

**5**

votes

**2**answers

618 views

### Mazur's torsion theorem on elliptic curves and its generalisations

I want to study Mazur's torsion theorem for elliptic curves over $Q$ and its generalizations for number fields, i.e., papers by Kamienny, Kenku & Momose, Filip Najman. So please suggest to me what ...

**16**

votes

**5**answers

2k views

### Category theory and model theory as “natural” counterparts

I am aware of the profound discussion of the relationship between category theory and model theory (e.g. at The n-Category Café) but do wonder why category theory (CT) is not opposed to model theory ...

**11**

votes

**2**answers

1k views

### What´s essential to learn about complex spaces and several complex variables for an algebraic geometer?

Hi, I don´t know if this question is suitable for this site. The field of several complex variables is too broad, so I would like to know what´s essential to learn about complex spaces and several ...

**23**

votes

**9**answers

2k views

### What are some interesting problems in the intersection of Algebraic Number Theory and Algebraic Topology?

I'm a beginning graduate student and while my background is primarily in algebraic number theory, I've found myself a bit smitten with the subject of algebraic topology recently after only having read ...

**4**

votes

**1**answer

288 views

### What is the applications of the dg-enhancements of derived categories of sheaves

Let $X$ be a scheme and let $D^b_{\text{coh}}(X)$ be the derived category of complexes of sheaves with bounded, coherent cohomologies.
We know that the category $D^b_{\text{coh}}(X)$ has some ...

**80**

votes

**11**answers

9k views

### Why are modular forms interesting?

Well, I'm aware that this question may seem very naive to the several experts on this topic that populate this site: feel free to add the "soft question" tag if you want... So, knowing nothing about ...

**9**

votes

**11**answers

3k views

### What advanced Area of Mathematics can be delved into with only basic Calculus and Linear Algebra

Hello Mathoverflow Community,
I would really appreciate some advice on this:
All I know is Basic Calculus and Basic Linear Algebra,
I want to start learning more advanced material on my own while ...

**39**

votes

**8**answers

11k views

### Modern algebraic geometry vs. classical algebraic geometry

Can anyone offer advice on roughly how much commutative algebra, homological algebra etc. one needs to know to do research in (or to learn) modern algebraic geometry. Would you need to be familiar ...

**13**

votes

**7**answers

2k views

### What is lambda calculus related to?

So I'm not much of a math guy but I've really enjoyed programming in Lisp and have become interested in the ideas of lambda calculus which it is based.
I was wondering if anyone had a suggestion ...

**0**

votes

**3**answers

1k views

### Intuitions/connections/examples for “eigen-*”

There are many concepts in mathematics that begin with the German word "eigen": eigenvector, eigenvalue, eigenspace, eigenstate, eigenfunction, eigensystem etc. (to name just the most important (?) ...

**4**

votes

**2**answers

693 views

### What's so special about $1$-categories?

I have been pretty thoroughly convinced for some time now that, when thinking about mathematics, one really should be thinking 'categorically', that is, one should always be thinking of the morphisms ...

**24**

votes

**1**answer

1k views

### more on “Transalgebraic Theories” (a 19th century yoga)?

Among the talks at occasion of the Galois Bicentennial, one is about "Transalgebraic Theories". Unfortunately I found only this article describing that fascinating idea as " an extremely powerful ...

**77**

votes

**18**answers

7k views

### How do you decide whether a question in abstract algebra is worth studying?

Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my ...

**35**

votes

**0**answers

4k views

### Grothendieck's manuscript on topology

Edit: Infos on the current state by Lieven Le Bruyn: http://www.neverendingbooks.org/grothendiecks-gribouillis
Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are ...

**23**

votes

**6**answers

4k views

### Why the triangle inequality?

[Maybe this is asking to be closed; but I thought I'd risk it.]
A metric satisfies the axioms:
$d(x,y)=0$ if and only if $x=y$.
$d(x,y) = d(y,x)$.
$d(x,y) \leq d(x,z) + d(z,y)$.
Similarly (and ...

**2**

votes

**1**answer

383 views

### Statistical distance between discrete and continuous distributions

Are there any statistical distance functions that are capable of comparing a continuous and a discrete distribution? From reading this list
http://en.wikipedia.org/wiki/Statistical_distance
the only ...

**-2**

votes

**1**answer

336 views

### How many of Ramanujan's discoveries have had a practical application? [closed]

I was reading about the Indian mathematician Srinivasa Ramanujan who, before dying at the age of 32, independently compiled nearly 3900 results (this is from Wikipedia). So based on this he seems to ...

**2**

votes

**1**answer

1k views

### Why Riemann Hypothesis so important [closed]

I am often hearing people emphasized how important the RH is,one of them said that it should lead to an efficient way of determining whether a given large number is prime,and the other said,RH would ...

**2**

votes

**0**answers

200 views

### Interpretation of Shannon Entropy Application

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Let entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$ be given
by ...

**4**

votes

**2**answers

313 views

### Can one make a category concrete by “enlarging the universe”?

This is more or less a followup of this question. There, it was established that (it is well known that) the homotopy category of topological spaces is not concrete, in other words, there is no ...

**16**

votes

**5**answers

1k views

### How and how much do the notations and diagrams influence our understanding of mathematical concepts?

How and how much do the notations and diagrams influence
our understanding of mathematical
concepts?
This question was stimulated by the MathOverflow questions Thinking and Explaining and ...