**3**

votes

**0**answers

185 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 ...

**30**

votes

**1**answer

659 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

483 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 ...

**29**

votes

**8**answers

5k 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

114 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

86 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

175 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

935 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 ...

**59**

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 ...

**1**

vote

**0**answers

72 views

### On Different Ways of Proving Isoperimetric Inequalities [closed]

Update: Thanks to Douglas Zare's comment, My previous questions in this thread turned out to be equivalent to the Isoperimetric problem. Thus I edited my question to make it a bit different.
...

**5**

votes

**0**answers

142 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

**2**answers

370 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 ...

**70**

votes

**30**answers

10k 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

143 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

479 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

449 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

531 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?
...

**45**

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 ...

**1**

vote

**1**answer

234 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

212 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 ...

**78**

votes

**17**answers

6k 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 ...

**34**

votes

**29**answers

7k 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

923 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 ...

**70**

votes

**22**answers

6k 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 $2^\kappa = 2^\nu \rightarrow \kappa = \nu.$
...

**71**

votes

**48**answers

10k 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 ...

**97**

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

399 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

787 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

174 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

323 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.
...

**7**

votes

**0**answers

128 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 ...

**193**

votes

**9**answers

17k 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

180 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 $E$.
...

**-4**

votes

**3**answers

165 views

### Collecting sufficient conditions for Sorli's conjecture on odd perfect numbers [closed]

Sorli's conjecture predicts that, for an odd perfect number $N$ given in the Eulerian form $N = {q^k}{n^2}$ (where $q$ is prime with $\gcd(q, n) = 1$ and $q \equiv k \equiv 1 \pmod 4$), the condition ...

**0**

votes

**1**answer

345 views

### Physics that needs “new” math [closed]

Just curious: I can't think of a single example that a physicist did not had his mouth open in amazement when he learnt that all (OK, lets say the foundations) the math he needs for his brand-new ...

**11**

votes

**7**answers

675 views

### Finite-space dynamical systems

This question is quite open-ended, but I will formulate several sub-questions that I'll try to make precise. It is about finite-state dynamical system: start with a finite set $X$, with say $n$ ...

**1**

vote

**1**answer

159 views

### Maximality statements that cannot be proved using $\mathsf{ZL}$ [closed]

What are examples for maximality statements that cannot be proved using Zorn's Lemma?

**49**

votes

**4**answers

2k views

### When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.
For example, it gives some evidence that there are finitely many ...

**2**

votes

**4**answers

355 views

### Higher Moments, what are they good for? [closed]

Absolutely nothing?
And now seriously - When I studied the basics of probability theory, and even in more advanced topics (random walks, stochastic processes, etc.), I always felt that the mean and ...

**3**

votes

**3**answers

363 views

### When few simple conditions yield a unique intricate structure

If people were asked to do a brainstorming related to the title, everyone would probably come up with dozens of examples. Those could include things as different as
the Mandelbrot set, Julia sets ...

**10**

votes

**7**answers

964 views

### Open problems in continued fractions theory

I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.

**15**

votes

**2**answers

1k views

### What are the applications of operator algebras to other areas?

Question: What are the applications of operator algebras to other areas?
More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by ...

**2**

votes

**1**answer

2k views

### Famous examples of PhD advisors younger than their student [closed]

What are the most famous examples of PhD advisors in mathematics, younger than their student?
(if possible put the date of birth and/or the difference in age).

**-2**

votes

**1**answer

222 views

### Degree of a rational function [closed]

I would like to have a simple proof for the following result:
Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). ...

**4**

votes

**2**answers

314 views

### Proving results about complete Boolean algebras in ZFC using Boolean valued models

I want to know what non-trivial ZFC theorems (not consistency results) about complete Boolean algebras (or more generally of partially ordered sets) one can prove using forcing.
I am mainly ...

**2**

votes

**1**answer

427 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 ...