**6**

votes

**1**answer

126 views

### Under which constraints are there only finite numbers of irreducible eta product identities?

For the Dedekind eta function, defined as usual by $\eta(q) = q^{\frac1{24}} \prod\limits_{n=1}^{\infty} (1-q^{n})$, let for brevity $e_k:=\eta(q^k)$.
An eta product identity (or eta identity for ...

**46**

votes

**12**answers

8k 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 ...

**8**

votes

**4**answers

4k views

### Is the ABC conjecture known to imply the Riemann hypothesis?

I once heard from a graduate student that the ABC conjecture implies the Riemann hypothesis. I can't find a reference for this, but given the department the student is from I tend to believe he might ...

**66**

votes

**15**answers

13k 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 ...

**31**

votes

**33**answers

4k views

### Structures that turn out to exhibit a symmetry even though their definition doesn't

Sometimes (often?) a structure depending on several parameters turns out to be symmetric w.r.t. interchanging two of the parameters, even though the definition gives a priori no clue of that symmetry. ...

**56**

votes

**52**answers

17k views

### Theorems that are 'obvious' but hard to prove

There are several well-known mathematical statements that are 'obvious' but false (such as the negation of the Banach--Tarski theorem). There are plenty more that are 'obvious' and true. One would ...

**15**

votes

**12**answers

2k views

### What are some fundamental “sources” for the appearance of pi in mathematics?

I thought it might be fun to ask this question as a way of celebrating Pi Day. One way in which people popularize pi is that they say that even though it's defined in terms of properties of a circle, ...

**5**

votes

**0**answers

63 views

### Classifying two-faces of four-polytopes

Motivation: This question is related to my study of hyperbolic Coxeter polytopes. In general, if one put some restrictions on the type of their dihedral angles (say, all dihedral angles are equal to ...

**9**

votes

**0**answers

309 views

### How come Cartan did not notice the close relationship between symmetric spaces and isoparametric hypersurfaces?

Elie Cartan made fundamental contributions to the theory of Lie groups and their geometrical applications. Among those, we can list the introduction of the remarkable family of Riemannian symmetric ...

**31**

votes

**13**answers

2k views

### Equality vs. isomorphism vs. specific isomorphism

This question prompted a reformulation:
What is a really good example of a situation where keeping track of isomorphisms leads to tangible benefit?
I believe this to be a serious question because ...

**2**

votes

**1**answer

532 views

### Knowledge base about topology [closed]

We are studying topology. There are a lot of definitions and theorems. I wonder if there somewhere knowledge base about topology and reasoning system exists. So I expect some tool that systematizes ...

**186**

votes

**21**answers

21k 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 ...

**4**

votes

**1**answer

124 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, ...

**3**

votes

**1**answer

117 views

### Toric ideal of slice of a polytope?

Given a collection $A:=\{a_1, \ldots ,a_n \}$ of different integer points in $\mathbb{N}^d$, which span an affine hyperplane when viewed in $\mathbb{R}^d$, one can define a toric ideal $I_A$ from a ...

**18**

votes

**5**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 ...

**3**

votes

**0**answers

313 views

### Differential Galois number theory

Following http://math.stackexchange.com/questions/635659/can-the-error-term-involved-in-the-pnt-be-expressed-in-a-galois-theoretic-framew?noredirect=1#comment1341143_635659, I vainly tried to find ...

**6**

votes

**2**answers

663 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 ...

**6**

votes

**1**answer

358 views

### Getting the story of Dynkin and Satake diagrams straight

I've been trying to teach myself the theory of Lie groups. The sources I've been reading reference Lie algebras in the context of Dynkin and Satake diagrams, but not Lie groups (which I am more ...

**45**

votes

**18**answers

7k 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 ...

**-3**

votes

**2**answers

551 views

### Accidental, unplanned breakthroughs in Mathematics [closed]

In math/physics, or generally in science, there are many moments where the success and the triumph come from the accidental, unplanned attempts. Moreover, there are some cases that originally having ...

**12**

votes

**6**answers

797 views

### Proof by `universal receiver'

Anyone following the news knows about the major breakthoughs that have taken place recently in $3$-manifold topology. These have come via a route whose big-picture I find to be conceptually ...

**0**

votes

**2**answers

146 views

### Intuition about covariant derivative/connections on real and complex manifolds

I was hoping to gain more intuition about the similarities and differences between the covariant derivative (of any connection, not necessarily the Levi Civita one if it exists) on real and complex ...

**9**

votes

**1**answer

192 views

### Why do convex polytope options constrict with dimension, rather than expand?

There are an infinite number of regular polygons in the plane,
five regular polyhedra,
six regular polytopes in $\mathbb{R}^4$,
and then three regular polytopes in every dimension $d > 4$.
There ...

**7**

votes

**3**answers

674 views

### What are Penrose Tilings, and how do they relate to Quasicrystals?

The question is in the title, but let me elaborate a little.
Background
Penrose Tilings are really pretty and satisfy some remarkable properties. For instance, I believe the following is true: even ...

**100**

votes

**61**answers

18k 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 ...

**68**

votes

**36**answers

13k views

### The phenomena 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 ...

**11**

votes

**2**answers

640 views

### What is the status of (universal) algebra in type theory?

With the recent interest in homotopy type theory as a foundation for mathematics, it seems natural to develop algebra within the framework of type theory. So far, I can't find much literature ...

**48**

votes

**16**answers

7k views

### Why is it a good idea to study a ring by studying its modules?

This is related to another question of mine. Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module. You tell him that modules ...

**21**

votes

**4**answers

2k 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 ...

**49**

votes

**11**answers

4k views

### Why certain diophantine equations are interesting (and others are not) ?

It is quite clear why certain differential equations, among the jungle of possible diff equations that is possible to conceive, are studied: some come from physical problems, or from "spontaneous" ...

**36**

votes

**22**answers

5k views

### What's a groupoid? What's a good example of a groupoid?

Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?

**102**

votes

**3**answers

4k views

### Analytic tools in algebraic geometry

This is not a very precise question, but I hope it will get some good answers.
As someone with a background in smooth manifold theory, I have experienced algebraic geometry as a beautiful but foreign ...

**33**

votes

**2**answers

4k views

### current status of crystalline cohomology?

The great references given on Ilya's question make me wonder about the current status of the many conjectures and open questions in Illusie's survey from 1994 on crystalline cohomology. Obviously ...

**40**

votes

**14**answers

4k views

### What is the high-concept explanation on why real numbers are useful in number theory?

The utopian situation in mathematics would be that the statement and the proof of every result would live "in the same world", at the same level of mathematical complexity (in a broad sense), unless ...

**3**

votes

**1**answer

420 views

### What do orbital integrals have to do with reciprocity?

Hi, this is my first question (of many). I am blogging for the Fields Medal Symposium and would like to get into the mathematics involved with our program.
In an attempt to sort through the articles ...

**0**

votes

**4**answers

692 views

### What are other applications of difference equations in other branches of mathematics ?

What are some of interesting results that arise from using difference equations in number theory , Combinatorics or any other field ?

**54**

votes

**6**answers

7k views

### How to find ICM talks?

I am very interested in reading some and skimming through the list of invited talks at the International Congress of Mathematicians. Since the proceedings contain talks supposedly by top experts in ...

**62**

votes

**24**answers

26k 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 ...

**1**

vote

**0**answers

219 views

### Categories of sheaves and Kan Extensions

This is quite a broad question regarding constructions of categories of sheaves in geometry.
Let $\textbf{Sch}$ denote the category of schemes. Let $\textbf{SchAff}$ denote the full subcategory of ...

**32**

votes

**2**answers

1k views

### Recent fundamental new directions in PDEs

My main interests are in modern geometry/topology, algebra and mathematical physics. I observe that there is a raising communication, language and social barrier between this community and the ...

**23**

votes

**3**answers

2k views

### Contemporary mathematical themes

The presence of fruitful mathematical themes suggests the unity of mathematics. What I mean by a mathematical theme here is a basic idea or guiding principle that motivates or directs the central ...

**6**

votes

**1**answer

304 views

### Motives over the complex numbers versus mixed Hodge structures

Let $\mathsf{MM}(\mathbf C)$ be the hypothetical category of mixed motives over the complex numbers, and consider the realization functor $\Phi : \mathsf{MM}( \mathbf C) \to \mathsf{MHS}$ to integral ...

**32**

votes

**6**answers

4k views

### The 'real' use of Quantum Algebra, Non-commutative Geometry, Representation Theory, and Algebraic Geometry to Physics

In this question, Orbicular made the following comment to Feb7 and my own answers;
Please keep in mind that - even though it is stated very often - noncommutative geometry does not give "real" ...

**28**

votes

**3**answers

1k views

### What do theta functions have to do with quadratic reciprocity?

The theta function is the analytic function $\theta:U\to\mathbb{C}$ defined on the (open) right half-plane $U\subset\mathbb{C}$ by $\theta(\tau)=\sum_{n\in\mathbb{Z}}e^{-\pi n^2 \tau}$. It has the ...

**110**

votes

**130**answers

26k 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 ...

**8**

votes

**3**answers

321 views

### How does one identify properties of objects with good “inheritance”?

When you are dealing with a very general object like a topological space or a ring, usually you impose an additional condition (such as compact Hausdorff or Noetherian) with the property that the ...

**10**

votes

**1**answer

787 views

### Are overlaps among {algebraic geometry, arithmetic geometry, algebraic number theory} growing?

From a naive outsider's viewpoint, just watching the MO postings
in those three fields scroll by, and hearing of breakthroughs in the news,
it appears there might be increasing overlap among the ...

**35**

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 ...

**17**

votes

**2**answers

797 views

### At which level is it currently possible to write formal proofs?

I am wondering whether I should try to have some fun using proof systems. I have never used such a system, but I have some experiences in logic and programming. My question is: At which level of ...

**0**

votes

**3**answers

943 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 (?) ...