# All Questions

**136**

votes

**31**answers

17k views

### What should be learned in a first serious schemes course?

I've just finished teaching a year-long "foundations of algebraic
geometry" class. It
was my third time teaching it, and my notes are gradually converging.
I've enjoyed it for a number of reasons ...

**6**

votes

**4**answers

445 views

### SO$(4)$ (& SO$(n)$) characterization?

I believe it is the case that
any finite subgroup of SO$(3)$
(the $3 \times 3$ orthogonal matrices of determinant $1$)
is either a cyclic group $C_n$,
or a dihedral group $D_n$, or one of the groups ...

**5**

votes

**1**answer

265 views

### Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved).
Then the Fourier transform of this function is given by
...

**25**

votes

**1**answer

1k views

### Is there an arbitrarily long arithmetic progression whose members are palindromes?

I suspect there isn't for some simple reason, but I could not find anything on it and if the opposite would hold in a more general sense, then that would solve this question.

**144**

votes

**14**answers

10k views

### What elementary problems can you solve with schemes?

I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary ...

**70**

votes

**16**answers

10k views

### Are there examples of non-orientable manifolds in nature?

Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence:
"The unorientable surfaces are never discussed ...

**56**

votes

**26**answers

7k views

### Sexy vacuity …

I included this footnote in a paper in which I mentioned that the number of partitions of the empty set is 1 (every member of any partition is a non-empty set, and of course every member of the empty ...

**47**

votes

**13**answers

9k views

### How has modern algebraic geometry affected other areas of math?

I have a friend who is very biased against algebraic geometry altogether. He says it's because it's about polynomials and he hates polynomials. I try to tell him about modern algebraic geometry, ...

**6**

votes

**0**answers

358 views

### Prerequisites for reading Gregory Perelman's work

What are the prerequisites for understanding the work of Perelman concerning the Poincaré conjecture?
I am referring to the last three papers here.

**12**

votes

**2**answers

331 views

### ODE properties true in finite dimension but not in Banach spaces of infinite dimension

Some properties of Ordinary Differential Equations - ODE are true in finite dimension spaces but not in Banach spaces of infinite dimension.
The first one I know is the Peano existence theorem. I ...

**86**

votes

**10**answers

10k views

### Are there any very hard unknots?

Some years ago I took a long piece of string, tied it into a loop, and tried to twist it up into a tangle that I would find hard to untangle. No matter what I did, I could never cause the later me any ...

**17**

votes

**7**answers

2k views

### Recreational mathematics: where to search?

I am not sure I can strictly define recreational mathematics. But we all feel what it is about: puzzles, problems you can ask your mathematical friends, problems that will bother them for a couple of ...

**20**

votes

**2**answers

911 views

### What do you do if you believe a problem is undecidable?

While the title of this question is subjective, I hope to make what I'm looking for quite concrete. The first, and main question is this: If you believe that a problem you are working on is formally ...

**84**

votes

**12**answers

17k views

### What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...

**32**

votes

**14**answers

7k views

### Examples of using physical intuition to solve math problems

For the purposes of this question let a "physical intuition" be an intuition
that is derived from your everyday experience of physical reality. Your
intuitions about how the spin of a ball affects ...

**5**

votes

**1**answer

107 views

### Submersion theorem for smooth tame Frechet manifolds

If $M$ and $N$ are Banach manifolds, $f:M\rightarrow N$ is a smooth map, and $q\in N$ is a regular value, so $f$ is a submersion on $f^{-1}(q)$, it is well known that the level set $f^{-1}(q)$ is a ...

**7**

votes

**0**answers

157 views

### A “universally non Hypercomplete” $\infty$-topos?

My question is :
Is there a classifying $\infty$-topos for $\infty$-connected objects ? Does this $\infty$-topos has a nice description (as an $\infty$-category ) ?
What I mean by $\infty$-connected ...

**25**

votes

**3**answers

2k views

### Intuition behind the ricci flow

I hope you don't shoot me for this question.
I try to understand among other things the Ricci flow. However I have no idea of the intuition behind the definition. So my questions is:
What is the ...

**59**

votes

**26**answers

5k views

### What would you want on a Lie theory cheat poster?

For some long time now I've thought about making a poster-sized "cheat sheet" with all the data about Lie groups and their representations that I occasionally need to reference. It's a moving target, ...

**26**

votes

**2**answers

710 views

### A more natural proof of Dold-Kan?

The Dold-Kan correspondence gives an equivalence of categories between $SAb$, the category of simplicial abelian groups, and $Ch_{\geq 0}$, the category of non-negatively graded chain complexes of ...

**7**

votes

**4**answers

1k views

### Is the Manickam-Miklós-Singhi Conjecture solved? [closed]

This arXiv paper is claimed to contain a proof for the MMS conjecture. But it seems that this manuscript is not yet peer reviewed by other mathematicians. I personally tried to follow the paper, but ...

**64**

votes

**12**answers

20k views

### What are the big problems in probability theory?

Most branches of mathematics have big, sexy famous open problems. Number theory has the Riemann hypothesis and the Langlands program, among many others. Geometry had the Poincaré conjecture for a long ...

**15**

votes

**2**answers

471 views

### Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?

Let $X$ be a Banach space. Consider the map
$$
\alpha\colon X\hat{\otimes} X^* \to B(X)^*,
$$
defined one simple tensors as
$$
\alpha(\xi\otimes\eta)(a) = \eta(a(\xi)).\quad (\xi\in X, \eta\in X^*, ...

**28**

votes

**4**answers

2k views

### Why the Dold-Thom theorem?

Dold-Thom Theorem: $$\pi_i(SP(X))\cong\tilde{H}_i(X)$$
It's pretty miraculous, no? I've seen its proof, where you show that the composition of the functors on the left-side satisfies the axioms of a ...

**39**

votes

**9**answers

11k views

### What is the exterior derivative intuitively?

Hi,
actually I have several related questions, not worth opening different threads:
What is the of the exterior derivative intuitively? What is its geometric meaning?
A possible answer I know is, ...

**6**

votes

**1**answer

267 views

### Non-negativity of invariant polynomials

Certifying the non-negativity of a symmetric polynomial is much easier than
certifying the non-negativity of an arbitrary polynomial function:
for instance, in (1) is proved that the complexity of ...

**15**

votes

**2**answers

1k views

### (co)homology of symmetric groups

Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...

**15**

votes

**1**answer

856 views

### Applications of $p$-adic Hodge theory

I am trying to learn $p$-adic Hodge theory. I found some materials explaining main theorems (or aspects) of the theory. However, I could not find references which explaining how to use the theory. ...

**35**

votes

**11**answers

8k views

### Standard model of particle physics for mathematicians

If a mathematician who doesn't know much about the physicist's jargon and conventions had the curiosity to learn how the so called Standard Model (of particle physics, including SUSY) works, where ...

**4**

votes

**2**answers

348 views

### Yang–Baxter explanation

What are the most simple examples which can explain the meaning of Yang–Baxter equation? Is there any way to explain this mysterious object to a person who is not a professional in quantum groups? ...

**11**

votes

**1**answer

237 views

### How many Fréchet manifolds are there?

Clearly the title needs clarifying. Allow me to let "how many" to mean a set larger than a skeleton of the category of Fréchet manifolds and smooth maps, if this category is indeed essentially small.
...

**40**

votes

**1**answer

3k views

### Are the primes normally distributed? Or is this the Riemann hypothesis?

Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes.
Let $\mathrm{Li}(x)$ be the offset logarithmic ...

**2**

votes

**2**answers

146 views

### Question on the number of equilibria

Let $f: C \to C$ be a smooth function and $C$ be a compact set, subset of $\mathbb{R}^n$.
We assume that all the fixed points are hyperbolic. Is it true that the number of fixed points is finite or ...

**53**

votes

**7**answers

17k views

### Is Mac Lane still the best place to learn category theory?

For a student embarking on a study of algebraic topology, requiring a knowledge of basic category theory, with a long-term view toward higher/stable/derived category theory, ...
Is Mac Lane still ...

**20**

votes

**3**answers

1k views

### What's a noncommutative set?

This issue is for logicians and operator algebraists (but also for anyone who is interested).
Let's start by short reminders on von Neumann algebra (for more details, see [J], [T], [W]):
Let $H$ ...

**41**

votes

**23**answers

23k views

### Taking lecture notes in lectures

Do you find it a good idea to take lecture notes (even detailed lecture notes) in mathematical lectures?
Related question: Digital Pen for Math: Your Experiences?

**1**

vote

**2**answers

211 views

### Mathematical statistical qm book-recommendation

I feel that there are quite a few good and rigorous books on the mathematical foundations of quantum mechanics, but I am currently looking for a book that covers mathematical statistical quantum ...

**20**

votes

**1**answer

543 views

### How to make the Capelli's identity less mysterious?

The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity
To ...

**0**

votes

**0**answers

86 views

### Random Cluster Model only for bond percolation?

Can someone please tell me which of the following statements I make are true of the current state of the art:
The Random Cluster Model is a generalization of bond percolation (with possibly ...

**27**

votes

**7**answers

1k views

### Concise model of modern fiat money and its non-conservation

A confession: I have never really understood the basic model of fiat money and central banking, by which a central bank controls the money supply. By the standards of someone trained in mathematics, ...

**4**

votes

**1**answer

417 views

### weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$.
$$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$
...

**7**

votes

**7**answers

906 views

### Almost-converses to the AM-GM inequality

Let us consider the Arithmetic Mean -- Geometric Mean inequality for nonnegative real numbers:
$$ GM := (a_1 a_2 \ldots a_n)^{1/n} \le \frac{1}{n} \left( a_1 + a_2 + \ldots + a_n \right) =: AM. $$
...

**5**

votes

**1**answer

908 views

### Riemann hypothesis and Kakeya needle problem

The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are ...

**59**

votes

**3**answers

11k views

### Slick proof?: A vector space has the same dimension as its dual if and only if it is finite dimensional

A very important theorem in linear algebra that is rarely taught is:
A vector space has the same dimension as its dual if and only if it is finite dimensional.
I have seen a total of one proof ...

**7**

votes

**3**answers

671 views

### Is it consistent with ZFC (or ZF) that every definable family of sets has at least one definable member?

I consider definability to mean one of either cases:
Definability without parameters (in the language of set theory), or
Definability from ordinals and a real (in the same language).
So my ...

**23**

votes

**7**answers

2k views

### Intuition for failure of Implicit Function theorem on Frechet Manifolds

When dealing with moduli spaces of, say connections or metrics, I am using the notions of Frechet spaces/manifolds/groups. I have become familiar with Banach manifolds (I think), but Frechet manifolds ...

**1**

vote

**1**answer

192 views

### Polynomial convex coefficients

Assume we have an arbitrary high order polynomial $$f(L)=1-L\theta_1-L^2\theta_2-L^3\theta_3-...-L^N\theta_N$$ and we know all roots of this polynomial site outside the unit circle. It is obvious that ...

**18**

votes

**5**answers

4k views

### What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

**22**

votes

**8**answers

2k views

### Between mu- and primitive recursion

It is well known that primitive recursion is not powerful enough
to express all functions, Ackermann function being probably the best
known example.
Now, in the logic courses (that I have had look ...

**378**

votes

**1**answer

28k views

### Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?

Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}\ $ such that $f:\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?