Questions tagged [20-questions]

In the earliest days of MathOverflow, there was a '20 questions' seminar (see <http://sbseminar.wordpress.com/category/20-questions/>) 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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3 answers
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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 ...
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21 votes
4 answers
3k views

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

Let $\{a_n\}$ be a sequence of complex numbers indexed by the positive integers. Does there always exist an analytic function $f$ such that $f(n) = \{a_n\}$ for $n=1,2,...$? If not, are there any ...
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9 votes
5 answers
2k 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 ...
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18 votes
3 answers
4k 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 $H^n(...
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11 votes
4 answers
4k 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 (...
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13 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 ...
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6 votes
2 answers
498 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 ...
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8 votes
6 answers
2k 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 ...
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73 votes
9 answers
13k 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 ...
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7 votes
2 answers
439 views

Characterizing the Radon transforms of log-concave functions

$f:\mathbf{R}^d\to \mathbf{R}_{\ge 0}$ is log-concave if $\log(f)$ is concave (and the domain of $\log(f)$ is convex). Theorem: For all $\sigma$ on the sphere $\Bbb S^{d-1}$ and $r\in \mathbf{R}$, $$ ...
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19 votes
4 answers
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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?
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9 votes
6 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 ...
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11 votes
4 answers
757 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?
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7 votes
3 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 ...
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26 votes
7 answers
4k 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 ...
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7 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 ...
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7 votes
2 answers
485 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 ...
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47 votes
5 answers
5k 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 unlikely,...
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