**11**

votes

**3**answers

677 views

### How much “Morse theory” can be accomplished given only a continuous transformation of a space?

If $M$ is a Riemannian manifold and $f:M\to \mathbb{R}$ a Morse-Smale function (which is just a rigorous way to say "generic smooth function"), then Morse theory essentially recovers the manifold ...

**9**

votes

**3**answers

1k views

### Which sequences can be extended to analytic functions? (e. g., Ackermann's function)

Let {an} be a sequence of complex numbers indexed by the positive integers. Does there always exist an analytic function f such that f(n) = an for n=1,2,...? If not, are there any simple necessary or ...

**8**

votes

**5**answers

994 views

### Analogues of the Weierstrass p function for higher genus compact Riemann surfaces

There was a previous post on the correspondence between Riemann surfaces and algebraic geometry. I want to ask a related but more detailed question.
BACKGROUND:
Engelbrekt gave an overview of how ...

**12**

votes

**3**answers

2k views

### cohomology and Eilenberg-MacLane spaces

This question is related to this question from Dinakar, which I found interesting but don't yet have the background to understand at that level.
Unless I'm mistaken, the rough statement is that ...

**8**

votes

**4**answers

2k views

### When does the sheaf direct image functor f_* have a right adjoint?

Say f: X → Y is a morphism of schemes. The sheaf direct image functor f★ always has a left adjoint, namely the sheaf inverse image functor f★ (with tensoring).
Under what ...

**11**

votes

**3**answers

1k views

### How should I approximate real numbers by algebraic ones?

Given a high precision real number, how should I go about guessing an algebraic integer that it's close to?
Of course, this is extremely poorly defined -- every real number is close to a rational ...

**6**

votes

**2**answers

462 views

### Can I finitely color Z^2 such that (x,a) and (a,y) are different for every x,y,a?

I ran into this obstacle in a harmonic analysis problem; I know epsilon about coloring problems.
Is it possible to finitely color Z^2 so that the points (x,a) and (a,y) are differently colored for ...

**8**

votes

**6**answers

1k views

### What is an example of a topological space that is not homotopy equivalent to a CW-complex?

It would also be nice if someone can explain this comment appearing on the Wikipedia page on CW-complexes:
"The homotopy category of CW complexes is, in the opinion of some experts, the best if not ...

**41**

votes

**9**answers

5k views

### understanding Steenrod squares

There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from ...

**5**

votes

**2**answers

326 views

### Characterizing the Radon transforms of log-concave functions

f:Rd→R≥0 is log-concave if log(f) is concave (and the domain of log(f) is convex).
Theorem: For all σ on the sphere Sd-1 and r∈R, gσ(r) := ∫σ.x=rf(x)dS(x) is a ...

**17**

votes

**4**answers

4k views

### Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?

An anonymous question from the 20-questions seminar:
Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?

**7**

votes

**5**answers

1k views

### When does the sequence of iterates of a rational function converge?

Darsh asks at the 20-questions seminar:
Let $f:P^1 \rightarrow P^1$ be rational function.
Can you say when the sequence $\{ f^n(x)\}_n=\{ x,f(x),f(f(x)),\cdots\} $ converges? What about the sequence ...

**10**

votes

**4**answers

657 views

### Can you describe the image of the exponential map B(H)->B(H).

James Tener asks at the 20-questions seminar:
The exponential map exp:B(H)->B(H) is just defined by its Taylor series. Can you describe its image?

**4**

votes

**2**answers

1k views

### Is there a category in which finite limits and directed colimits *don't* commute

Andrew Critch asks at the 20-questions seminar:
In Set and AbGrp (the categories of sets and abelian groups, respectively), finite limits commute with directed colimits. As an example, if you're ...

**21**

votes

**7**answers

2k views

### How do you see the genus of a curve, just looking at its function field?

Yuhao asked in the 20-questions seminar:
The genus of a curve is a birational invariant; the function field of a curve determines it up to birational equivelance.
How do you see the genus directly ...

**6**

votes

**2**answers

1k views

### Critical points on a fiber bundle

Consider a (smooth) bundle E→_B_, and a (smooth) function f: E → R on the total space. Then it makes sense to talk about the derivatives of f along the fibers. Let C be the subspace of E consisting ...

**6**

votes

**2**answers

432 views

### How can you find small denominators inside triangles?

Darsh asked over at the 20 questions seminar:
Take a triangle in R^2 with coordinates at rational points. Can we find the smallest denominator point in the interior? (Consider denominator of an ...

**38**

votes

**5**answers

4k views

### Can $N^2$ have only digits 0 and 1, other than $N=10^k$?

Pablo Solis asked this at a recent 20 questions seminar at Berkeley. Is there a positive integer $N$, not of the form $10^k$, such that the digits of $N^2$ are all 0's and 1's?
It seems very ...