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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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
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4
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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
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5
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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 ...
- 3,808
18
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3
answers
4k
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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
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4
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4k
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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 (...
- 10.5k
13
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3
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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
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2
answers
498
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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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6
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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
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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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2
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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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4
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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
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6
answers
1k
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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
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4
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757
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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
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3
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1k
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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
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7
answers
4k
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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
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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
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2
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485
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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
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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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