In the earliest days of MathOverflow, there was a '20 questions' seminar (see ) run by graduate students at Berkeley. Many questions from the seminar were cross-posted to MathOverflow. This tag now exists solely for the historical record.

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8
votes
3answers
631 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 ...
8
votes
3answers
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 ...
7
votes
5answers
890 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
3answers
1k 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
4answers
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
3answers
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
2answers
415 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
6answers
903 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 ...
24
votes
7answers
3k views

understanding Steenrod squares

There is a function on Z/2Z-cohomology called Steenrod squaring: Sqi:H^k(X,Z/2Z) --> Hk+i(X,Z/2Z). (Coefficient group suppressed from here on out.) Its notable axiom (besides things like ...
5
votes
2answers
289 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 ...
13
votes
5answers
3k views

Can you explicitly write R^2 as a disjoint union of two totally path disconnected sets?

An anonymous question from the 20-questions seminar: Can you explicitly write R^2 as a disjoint union of two totally path disconnected sets?
5
votes
5answers
1k views

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

Darsh asks at the 20-questions seminar: Let f:P^1 -> P^1 be rational function. Can you say when the sequence ...
10
votes
4answers
622 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?
3
votes
2answers
871 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 ...
19
votes
7answers
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
2answers
1k views

Critical points on a fiber bundle

Consider a (smooth) bundle Eā†’_B_, and a (smooth) function f: E ā†’ 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
2answers
412 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 ...
36
votes
5answers
3k 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 ...