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1500 questions
60
votes
2 answers
Does this geometry theorem have a name?
Start with a circle and draw two tangent circles inside. The (black) inner tangent lines to the smaller circles intersect the large circle. The (red) lines through these intersection points are parallel to the (green) outer tangents to the small…
Simon
- 509
60
votes
8 answers
What does it mean to suspect that two conjectures are logically equivalent?
Here's a familiar conversation:
Me: Do you think Conjecture A and Conjecture B are equivalent?
Friend: Yes, because I think they're both true.
Me: [eye roll] You know what I mean...
Does there exist a rigorous notion of what I mean? Perhaps…
Dustin G. Mixon
- 7,563
60
votes
9 answers
Publishing conjectures
One has written a paper, the main contribution of which is a few conjectures. Several known theorems turned out to be special cases of the conjectures, however no new case of the conjectures was proven in the paper. In fact, no new theorem was…
asv
- 21,102
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60
votes
4 answers
Degree of sum of algebraic numbers
This is an elementary question (coming from an undergraduate student) about algebraic numbers, to which I don't have a complete answer.
Let $a$ and $b$ be algebraic numbers, with respective degrees $m$ and $n$. Suppose $m$ and $n$ are coprime. Does…
François Brunault
- 20,484
60
votes
10 answers
"Surprising" examples of Markov chains
I am looking for examples of Markov Chains which are surprising in the following sense: a stochastic process $X_1,X_2,...$ which is "natural" but for which the Markov property is not obvious at first glance. For example, it could be that the natural…
Adam Smith
- 708
60
votes
3 answers
Can a positive binary quadratic form represent 14 consecutive numbers?
NEW CONJECTURE: There is no general upper bound.
Wadim Zudilin suggested that I make this a separate question. This follows
representability of consecutive integers by a binary quadratic form
where most of the people who gave answers are worn out…
Will Jagy
- 25,349
60
votes
1 answer
What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last Theorem?
Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was not even aware of the fact that there was…
M.G.
- 6,683
60
votes
6 answers
Synthetic vs. classical differential geometry
To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of Lee's series) but I don't have any background in…
ಠ_ಠ
- 5,933
60
votes
1 answer
Probability that a stick randomly broken in five places can form a tetrahedron
Edit (June 2015): Addressing this problem is a brief project report from the Illinois Geometry Lab (University of Illinois at Urbana-Champaign), dated May 2015, that appears here along with a foot-note saying: An expanded version of this report is…
Benjamin Dickman
- 7,741
60
votes
2 answers
Are spectra really the same as cohomology theories?
Let $E \to F$ be a morphism of cohomology theories defined on finite CW complexes. Then by Brown representability, $E, F$ are represented by spectra, and the map $E \to F$ comes from a map of spectra. However, it is possible that the map on…
Akhil Mathew
- 25,291
60
votes
8 answers
Sheaf cohomology and injective resolutions
In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come into play? It is very confusing and awkward to me…
user709
60
votes
8 answers
Is the ultraproduct concept fundamentally category-theoretic?
Once again, I would like to take advantage of the large number of knowledgable category theorists on this site for a question I have about category-theoretic aspects of a fundamental logic concept.
My question is whether the ultraproduct…
Joel David Hamkins
- 224,022
60
votes
2 answers
What is a good roadmap for learning Shimura curves?
I am interested in learning about Shimura curves. Unlike most of the people who post reference requests however (see this question for example), my problem is not sorting through an abundance of books but rather dealing with what appears to be an…
user1073
60
votes
4 answers
Flipping coins on a budget
A coin is flipped $n$ times and you win if it comes up heads at least $k$ times. The coin is unusual in that you're allowed to pick the probability $p_i$ that it comes up heads on the $i$th flip, subject only to the constraint that $\sum_i p_i \le…
Timothy Chow
- 78,129
59
votes
5 answers
Are there any "related rates" calculus problems that don't feel contrived?
I just finished teaching a freshman calculus course (at an American state university), and one standard topic in the curriculum is related rates. I taught my students to answer questions such as the following (taken, more or less, from the…
Frank Thorne
- 7,199