Questions designed to get an overview of a specific subject or body of results or to understand the relations among similar definitions, techniques or concepts appearing in different sub-fields of mathematics. While such questions by their very nature sometimes cannot be made very narrow and ...

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64
votes
16answers
6k 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 ...
128
votes
72answers
22k 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 ...
92
votes
25answers
37k 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
6answers
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 ...
6
votes
1answer
290 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. ...
40
votes
24answers
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
32answers
10k 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
3answers
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
22answers
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
3answers
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, ...
30
votes
5answers
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
9answers
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 ...
0
votes
0answers
65 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, ...
89
votes
42answers
17k 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 ...
55
votes
13answers
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
1answer
908 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
1answer
299 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
4answers
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
2answers
592 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 ...
131
votes
132answers
29k 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 ...
16
votes
5answers
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
2answers
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 ...
22
votes
9answers
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
1answer
238 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 ...
79
votes
11answers
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
11answers
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 ...
38
votes
8answers
10k 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
7answers
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
3answers
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
2answers
686 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 ...
60
votes
34answers
9k 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 ...
24
votes
1answer
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 ...
76
votes
18answers
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 ...
33
votes
0answers
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
6answers
3k 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
1answer
301 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 ...
-3
votes
1answer
291 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
1answer
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
0answers
170 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
2answers
298 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
5answers
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 ...
7
votes
3answers
2k views

The resolution of which conjecture/problem would advance Mathematics the most? [closed]

This is an impossibly broad question, and makes the unwarranted assumption that Mathematics is a uniform field. It might make more sense to ask the same question restricted to, say, Mathematical ...
-3
votes
1answer
132 views

Decidable theorem or result that is not weaker than Tarski's theorem

I am wondering what other decidable theorem or results that is not weaker or stronger than Tarski's theorem. Could any one give reference or a simple introduction about such result known in their ...
9
votes
4answers
974 views

What is about J. v. Neumann's “Continuous geometry”?

I am curious about von Neumann's "Continuous geometry", but found no recent text or survey on it. Does anyone know the book and would be so nice to share their impression, and if/how the concept of ...
38
votes
7answers
5k views

“Algebraic” topologies like the Zariski topology?

The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact. ...
55
votes
9answers
5k views

How does one find out what's happening in contemporary mathematics research?

How does one find out what's happening in contemporary mathematics research? EDIT: I should have mentioned that I am looking for open access online sources. It so happens that I have been outside ...
15
votes
1answer
582 views

3D generalizations of permutations, RSK correspondence, contingency tables, etc.

I want to gather facts and questions related to 3D generalizations of permutations, RSK correspondence, contingency tables, etc. One reason I am interested in this is because it is potentially related ...
11
votes
5answers
3k views

Brownian motion, martingales, Markov Chains - Rosetta Stone

What are the most fundamental/useful/interesting ways in which the concepts of Brownian motion, martingales and markov chains are related? I'm a graduate student doing a crash course in ...
25
votes
9answers
1k views

Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous? In topology, category theory or in analysis? Is it possible, canonical, useful? I have a vague idea of the possibility of ...
21
votes
8answers
3k views

Examples of intuition from fields other than Physics to solve math problems

This is a chaser for the examples of using physical intuition to solve math problems question. Physical intuition seems to be used relatively frequently for solving math problems as well as stating ...