**4**

votes

**2**answers

248 views

### Critical points in $ZF$ without Choice

Recall the definition of critical point for set theory:
A critical point of an elementary embedding of one transitive class into another transitive class is the smallest ordinal not mapped to ...

**5**

votes

**5**answers

552 views

### Important results with one or more than one proof [closed]

Can you give examples of deep, important results that have only one known proof, and not just because the first proof is fairly recent, or because not many people really cared to think about it? How ...

**12**

votes

**5**answers

513 views

### Applications of space filling curves

I am seeking articles where a space filling curve has been used as a theoretical application, such as in the study of general orthogonal polynomials.

**23**

votes

**1**answer

761 views

### Big list - Equivalent descriptions of Hodge conjecture?

I would like to know equivalent descriptions of the Hodge conjecture (with references).
Dan Freed's Version:
Consider a topological cycle (boundary less chains that are free to deform) on a ...

**4**

votes

**0**answers

95 views

### Examples of combinatorial bijections found by considering functors

Let us assume that I have two sets of combinatorial objects, $A$ and $B$,
and I am looking for a bijection (in particular a map) $\psi:A \to B$ between these sets, usually required to preserve some ...

**6**

votes

**3**answers

1k views

### Examples of high level math that can be motivated to laypeople

One of the difficulties of mathematics over other sciences is that our problems are harder to motivate to a general audience. A biologist studying a particular pathway in the body can say that he's ...

**70**

votes

**21**answers

17k views

### Examples of math hoaxes/interesting jokes published on April Fool's day?

What are examples of math hoaxes/interesting jokes published on April Fool's day?
For a start P=NP.

**22**

votes

**4**answers

1k views

### Expert, Intuitive, Organizing Analogies

In learning a new area it is very helpful to have high-level intuitive analogies that keep track of the various parts of an important argument or strategy. Experts have a store of such things, and ...

**36**

votes

**25**answers

7k views

### Where can square roots come from when they are not distances?

In a recent survey "Supergeometry in Mathematics and Physics", Kapranov points out cases in which observable quantities of immediate interest are represented as bilinear combinations of more ...

**5**

votes

**0**answers

136 views

### Current Main Areas of Research in Model Theory [closed]

Could someone gives a general picture of the present state of Model Theory as a field? What are the current main areas and directions of research? What are some examples of the current experts and the ...

**7**

votes

**23**answers

3k views

### A search for theorems which appear to have very few, if any hypotheses [closed]

I'm interested in theorems which appear to have very few, if any hypotheses. Essentially a search for unexpected regularity or pattern in a relatively unstructured situation.
By "few hypotheses" I ...

**35**

votes

**15**answers

3k views

### Free open-access peer-reviewed math journal

Is there any free (as in free beer, i.e., no publication fees or other fees whatsoever), open-access (free and open access to everyone) and peer-reviewed mathematics journal?
I am interested in a ...

**14**

votes

**3**answers

645 views

### Open problems in Hopf algebras

I couldn't find a list of open problems in Hopf algebras. So my question is the following:
In the theory of Hopf algebras, what are the (big) open problems?
Any kind of problem/question will be ...

**12**

votes

**4**answers

539 views

### List of counting proofs instead of linear algebra method in combinatorics

I've just come across this proof of the Graham-Pollak Theorem by Sundar Vishwanathan (thanks to Konrad Swanepoel's sporadic comments about it on this site), that must be called beautiful after its ...

**3**

votes

**0**answers

209 views

### Roadmap for the ideas expressed in Grothendieck's Esquisse d'un Programme

I would like to understand Grothendieck's Esquisse d'un Programme more. Are there any references that would help me, and are there modern works pursuing the same themes?
At this point I am still ...

**34**

votes

**1**answer

841 views

### A dictionary of Characteristic classes and obstructions

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians.
In an effort to ...

**9**

votes

**1**answer

522 views

### Problems which use S₄ → S₃

I need examples of problems which use, directly or indirectly, the homomorphism $S_4\to S_3$ in the solution (its kernel is $\mathbb{Z}_2\oplus\mathbb{Z}_2$).
Obvious candidates:
Lagrange resolvent ...

**31**

votes

**8**answers

6k views

### What are some important but still unsolved problems in mathematical logic?

In the past, First order logic and its completeness and whether arithmetic is complete was a major unsolved issues in logic . All of these problems were solved by Godel. Later on, independence of ...

**4**

votes

**3**answers

135 views

### Repository of graph classes that are tough to test non-isomorphic pairs from isomorphic pairs

(1) Which graph classes are extremely tough to test for graph non-isomorphic pairs from isomorphic pairs?
(2) Is there a repository of adjacencies from such classes?

**3**

votes

**2**answers

147 views

### Database of adjacency matrices on cospectral non-isomorphic graph pairs

Is there a repository of cospectral non-isomorphic graphs available somewhere?
I am looking for list of $0/1$ adjacency matrix pairs that can be input data in tools such as MATLAB.

**9**

votes

**0**answers

191 views

### Algebraic K-theory of a ring.

I started to learn some algebraic $K$-theory and its relation to geometric topology problems.
My question is : What is the list of rings such that all their algebraic $K$-theory groups are known ?
I ...

**16**

votes

**14**answers

1k views

### Applications of Representation Theory in Combinatorics

What are the examples of interesting combinatorial identities (e.g. bijection between two sets of combinatorial objects) that can be proved using representation theory, or has some representation ...

**9**

votes

**4**answers

992 views

### Which journals publish experimental results in pure maths?

All pure mathematicians know that the goal is to produce insight, rather than to simply obtain results. However, it might sometimes be of value to disseminate largely empirical work. In the same ...

**58**

votes

**15**answers

6k views

### Sophisticated treatments of topics in school mathematics

Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples ...

**5**

votes

**0**answers

147 views

### List of finitely presented groups with undecidable word problem

Is there any reasonably updated list of (representative) examples of finitely presented groups with undecidable word problem?
By "representative" I mean "avoiding obvious redundancy", i.e. examples ...

**15**

votes

**3**answers

499 views

### Alternate proofs of Hilberts Basis Theorem

I'm interested in proofs using ideas from outside commutative algebra of Hilbert's Basis Theorem.
If $R$ is a noetherian ring, then so is $R[X]$.
or its sister version
If $R$ is a noetherian ...

**71**

votes

**30**answers

11k views

### What are some very important papers published in non-top journals?

There has already been a question about important papers that were initially rejected. Many of the answers were very interesting. The question is here.
My concern in this question is slightly ...

**0**

votes

**1**answer

167 views

### Equivalence relations of topological spaces not comparable with homotopy [closed]

The question is pretty much contained in the title:
What are examples of equivalence relations of topological spaces which are neither stronger nor weaker than homotopy equivalence?
Something that ...

**9**

votes

**5**answers

519 views

### List of generic properties of Riemannian metrics

I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of ...

**4**

votes

**4**answers

586 views

### Which journals publish short notes in discrete mathematics?

The journal Discrete Mathematics contains a lot of short notes (i.e., less than 7 journal pages). What are some other journals that publish short notes in discrete mathematics? I've looked at other ...

**9**

votes

**3**answers

563 views

### Historical developement of analysis and partial differential equations (especially in the 20th century)

Q: Is there a set of some comprehensive surveys or monographs describing (in
technical detail) the historical development of the various
subareas of analysis and partial differential equations?
...

**46**

votes

**17**answers

5k views

### Essays and thoughts on mathematics

Many distinguished mathematicians, at some point of their career,
collected their thoughts on mathematics (its aesthetic, purposes,
methods, etc.) and on the work of a mathematician in written ...

**0**

votes

**1**answer

258 views

### Best way to find recent papers in a special field of mathematics?

My subjects of interest are Geometry of Banach spaces, renorming theory and fixed point theory. When I want to find recent papers in these fields of mathematics, mostly, I search name of paper, say, ...

**11**

votes

**1**answer

224 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 ...

**82**

votes

**15**answers

8k 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 ...

**36**

votes

**29**answers

8k views

### Most intriguing mathematical epigraphs

Good epigraphs may attract more readers. Sometimes it is necessary.
Usually epigraphs are interesting but not intriguing.
To pick up an epigraph is some kind of nearly mathematical problem: it ...

**16**

votes

**5**answers

992 views

### What are examples when the equality of some invariants is good enough in algebraic topology?

As far as my understanding goes, most of the tools of algebraic topology (homotopy groups, homology groups, cup product, cohomology operations, Hopf invariant, signature, characteristic classes, knot ...

**42**

votes

**3**answers

3k views

### Advice for PhD Supervisors

My first PhD student is having his viva tomorrow. Hence, I began contemplating a bit about the whole process of supervising. One thing I realized is that while there seems to be plenty of advice for ...

**71**

votes

**22**answers

7k views

### Special rational numbers that appear as answers to natural questions

Motivation:
Many interesting irrational numbers (or numbers believed to be irrational) appear as answers to natural questions in mathematics. Famous examples are $e$, $\pi$, $\log 2$, $\zeta(3)$ etc. ...

**2**

votes

**0**answers

120 views

### What other axioms for set theory can be written in the form: “If mathematical structures $X$ and $Y$ are equipotent, then they're isomorphic”?

The "injective continuum function hypothesis" (ICF) is the following statement.
ICF (Version 0). For all cardinal numbers $\kappa$ and $\nu$, we have $2^\kappa = 2^\nu \rightarrow \kappa = \nu.$
...

**75**

votes

**51**answers

12k views

### Important formulas in Combinatorics

Motivation:
The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post, for the poster, and cash prizes offered ...

**98**

votes

**17**answers

23k views

### Mathematical software wish list

Like many other mathematicians I use mathematical software like SAGE, GAP, Polymake, and of course $\LaTeX$ extensively. When I chat with colleagues about such software tools, very often someone has ...

**4**

votes

**2**answers

405 views

### $C^{*}$ algebras which do not admit nontrivial idempotent morphism

In this question which I flag it as a community wiki, I search for a big list of $C^{*}$ algebras(and a big list of criterions) which do not admit a non trivial idempotent $C^{*}-$morphism.
I ...

**6**

votes

**2**answers

807 views

### Survey papers on the role played by PDE in mathematics

There are already several questions on MathOverflow that inquire about the many diverse relationships between PDE and several other 'areas' of mathematics (e.g., algebraic and differential geometry ...

**1**

vote

**1**answer

176 views

### Noncommutative analogs of classical Banach geometric properties

The scale of Schatten-von Neumann classes is noncommutatitve analog of classical $\ell_p$-spaces. A lot of researchers devoted their lives to study Banach geometric structure of these spaces. ...

**0**

votes

**1**answer

450 views

### Collection of graduate research projects in Real Analysis [closed]

While there are many open problems in Real Analysis like Khabibullin's conjecture or Lehmer's conjecture, those are big enough to take an expert's life for several years, let alone some graduate ...

**58**

votes

**20**answers

6k views

### Solving algebraic problems with topology

Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem.
...

**8**

votes

**0**answers

135 views

### Nice applications of Liouville's theorem

I need a big list of nice-looking and simple applications of Liouville's theorem on geodesic flow in Riemannian geometry.
Please help.
Examples:
A Riemannian manifold with finite volume does not ...

**198**

votes

**9**answers

18k views

### John Nash's Mathematical Legacy

It would seem that John Nash and his wife Alicia died tragically in a car accident on May 23, 2015 (reference). My condolences to his family and friends.
Maybe this is an appropriate time to ask a ...

**3**

votes

**2**answers

182 views

### Examples of TVS with no non-trivial open convex subsets

I give here the classical example of the space $E = L^p([0,1])$ which has no open convex subsets apart from $\emptyset$ and $E$. Consequently, there is no non-trivial continuous linear form on $E$.
...