**12**

votes

**1**answer

282 views

### Why are Witten-Reshetikhin-Turaev invariants expected to be integral?

A Witten-Reshetikhin-Turaev (WRT) Invariant $\tau_{M,L}^G(\xi)\in\mathbb{C}$ is an invariant of closed oriented 3-manifold $M$ containing a framed link $L$, where $G$ is a simple Lie group, and $\xi$ ...

**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 ...

**45**

votes

**8**answers

3k 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 ...

**23**

votes

**9**answers

5k views

### Why are polynomials so useful in mathematics?

This is perhaps unanswerable,
or perhaps I am too algebraically ignorant to phrase it cogently, but:
Is there some identifiable reason that polynomials over
$\mathbb{C}$,
$\mathbb{R}$, ...

**7**

votes

**5**answers

911 views

### Advice on choosing an area of specialization

I'm not sure if this is an appropriate question for MO, but I figured it couldn't hurt to ask. I'm a second year graduate student, currently gearing up to construct a committee and syllabus for my ...

**5**

votes

**0**answers

65 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 ...

**6**

votes

**1**answer

230 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 ...

**10**

votes

**0**answers

375 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 ...

**4**

votes

**1**answer

153 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

126 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 ...

**3**

votes

**0**answers

462 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

**1**answer

379 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 ...

**-2**

votes

**2**answers

608 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 ...

**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 ...

**0**

votes

**2**answers

168 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 ...

**33**

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. ...

**9**

votes

**1**answer

201 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 ...

**12**

votes

**2**answers

694 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 ...

**22**

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 ...

**1**

vote

**0**answers

237 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 ...

**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 ...

**7**

votes

**1**answer

342 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 ...

**10**

votes

**1**answer

878 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 ...

**17**

votes

**2**answers

846 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

**1**answer

226 views

### Generalization of join of simplicial complexes

The join of two abstract simplicial complexes $K$ and $L$, denoted $K\star L$ is defined as a simplicial complex on the base set $V(K)\dot{\cup} V(L)$ whose simplices are disjoint union of simplices ...

**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 ...

**6**

votes

**2**answers

674 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 ...

**9**

votes

**2**answers

861 views

### Status of Beilinson conjectures?

(I hesitate to post that question here, but I received on answer on FB:)
Does anyone know how the current status of work on them is? And how the possible generalizations etc. which one thinks ...

**7**

votes

**2**answers

467 views

### Is there a deep reason for the fecundity of involutions?

You might have come across the book of involutions in your travels. A colleague of mine asked whether there is a natural global reason (versus ad-hoc trickery) for considering involutions in ...

**10**

votes

**0**answers

316 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 ...

**15**

votes

**3**answers

956 views

### Research level applications of “row rank = column rank”?

No less an authority than Gilbert Strang frames "row rank equals column rank" (and a couple of other facts) as "The Fundamental Theorem of Linear Algebra."
I'd simply like to assemble (for teaching ...

**7**

votes

**3**answers

500 views

### spectacular applications of functional analysis in resolutions of apparently unrelated problems

What are some of the spectacular applications of functional analysis to apparently unrelated problems.One that
comes to my mind is Per Enflo's resolution of Hilbert's 5th problem.There are also ...

**13**

votes

**1**answer

405 views

### Skeleton category of the category of skeleton categories?

A category is a skeleton if, roughly speaking, no two distinct objects within the category are isomorphic. To every category is associated a skeleton, and two categories are categorically "equivalent" ...

**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 ...

**4**

votes

**1**answer

1k views

### Algebraic number theory: building and simplifying

This is a somewhat subjective question, about the past, present and especially future of algebraic number theory. I'm not at all in this area, but I'd be interested in an answer.
As we all know, ...

**20**

votes

**4**answers

1k views

### When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?

Every once in a blue moon it actually matters that some mathematical entity which might a priori only be a class is in fact a set. For clarification, here are some examples of what I do not ...

**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 ...

**19**

votes

**4**answers

2k views

### The Riemann Hypothesis and the Langlands program

On page 263 of this book review appears the following:
Given the centrality of L-functions to the Langlands program, nothing would seem more natural (than a presentation of elementary algebraic ...

**16**

votes

**1**answer

482 views

### Monte Carlo integration

As probably many other people here, I learned integration, as an undergrad, from Rudin's books. I recently realized, however, that I don't quite use Lebesgue integration in my work, or at least I use ...

**27**

votes

**3**answers

999 views

### “Softness” vs “rigidity” in Geometry

According to common wisdom, there are structures in Geometry that have a more "topological" flavor, others that are more "geometrical", and others that are halfway between. Usually, geometries${}^*$ ...

**46**

votes

**7**answers

4k views

### How does “modern” number theory contribute to further understanding of $\mathbb{N}$?

I hope this question is appropriate for MO. It comes from a genuine desire to understand the big picture and ground my own studies "morally".
I'm a graduate student with interest in number theory. I ...

**4**

votes

**1**answer

628 views

### Clean introduction to toric varieties for an undergraduate audience

I will be giving a talk to a (primarily) undergraduate audience on certain relatively concrete computations with toric varieties and their blowups. The talk is short, about 20 mins. As I result I need ...

**3**

votes

**1**answer

434 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 ...

**8**

votes

**3**answers

722 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 ...

**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 ...

**12**

votes

**6**answers

803 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 ...

**12**

votes

**0**answers

601 views

### What is the appropriate setting for Cauchy's Integral Formula?

For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula:
$$
f(\zeta) = ...

**9**

votes

**0**answers

878 views

### Status of the Leopoldt conjecture ? [closed]

In 2009 Mihailescu published a proof of the famous Leopoldt conjecture on the arxiv. Later on, in 2011, he published a 'lightweight' version that proves the conjecture for CM fields. Also compare
...

**20**

votes

**2**answers

1k views

### Elevator pitch for the Virtual Fibering Theorem

There has been a great deal of excitement among topologists about the proof of the Virtual Haken Theorem, and in fact of the Virtual Fibering Theorem (for closed hyperbolic 3-manifolds, but I'm ...

**7**

votes

**2**answers

777 views

### Explaining Mukai-Fourier transforms physically

A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality).
The basic algorithm is ...