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2 votes

Convex polygon shadows: Shortest equivalent segments

The current best lower bound on the total length of segments in $S$ is 2.00002: A. Kawamura, S. Moriyama, Y. Otachi and J. Pach, A lower bound on opaque sets, Comput. Geom. 80 (2019), 13–22. https://...
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1 vote

Question on consecutive integers with similar prime factorizations

This is how Heath-Brown resolved the d(n)=d(n+1) problem ! The idea is due to Claudia Spiro who used it for d(n)=d(n+5040) but was unable to reduce the 5040. The same idea has now been used in many ...
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1 vote

Birkhoff conjecture about integrable billiards

I think has recently obtained partial results on this conjecture (On polynomially integrable Birkhoff billiards on surfaces of constant curvature, JEMS 2021)
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6 votes

Mark Hovey's open problems in the theory of model categories

On the other hand, the category of commutative monoids seems to be much more subtle. The conditions for the existence of a model structure on commutative monoids were worked out by Jacob Lurie around ...
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34 votes
Accepted

Mark Hovey's open problems in the theory of model categories

I am a former student of Mark Hovey's, and during grad school, I wrote a document giving an update on the status of the 13 problems (as of 2012 or 2013, I guess). I just briefly went through it a ...
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0 votes

Can we cover the unit square by these rectangles?

it is hard to prove this problem directly, but it is not hard to prove (as someone mentioned in some comments) : If (n-1) rectangles has been put into the 1x1 square, then all rectangles can be put ...
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