Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba: Flat polynomials exist

gilkalai.wordpress.com/2019/09

[...] For a second year, The Big Internet Math-Off and World’s Most Interesting Mathematician has been won by a member of the IMA who works in industry. Following Dr Nira Chamberlain (IMA President Designate), Dr Sophie Carr CMath CSci MIMA won this year’s competition which saw 16 mathematicians from across the world (including lecturers, researchers, writers, and business owners) pitch against each other explaining how fun, interesting and relevant maths is.

ima.org.uk/12382/worlds-most-i

There are legitimate reasons to self-cite (as the article makes clear) but when senior researchers have a majority of citations from their own papers, and then get government awards for being highly cited, there's likely a problem. Or more than one problem, both in the researcher's citation habits and in the criteria for the awards.

[..] WOW! The new paper arxiv.org/abs/1908.08483 improved bounds for the sunflower lemma gives the most dramatic progress on the sunflower conjecture since it was asked. Congratulations to Ryan Alweiss, Shachar Lovett, Kewen Wu, Jiapeng Zhang.

Smaller Is Better: Why Finite Number Systems Pack More Punch
Recent progress on the “sum product” problem recalls a celebrated mathematical result that revealed the power of miniature number systems.

quantamagazine.org/smaller-is-

Why modern integer factoring algorithms have the time bounds they do, and what would be needed to improve them: blog.computationalcomplexity.o

SIAM Conference on Applied Algebraic Geometry 2019

Duffin–Schaeffer conjecture solved: quantamagazine.org/new-proof-s, arxiv.org/abs/1907.04593

This is about finding rational approximations to irrational numbers, like $$\pi\approx 355/113$$. Given a criterion for how good an approximation you want, depending only on the denominator (for instance, allowing only prime denominators and seeking an approximation accurate to $$\pm 1/p^{3/2}$$ for denominator $$p$$) the new theorem tells you when almost all irrationals have a good-enough approximation.

[...] But then Alice draws just the right card from her customized deck, and suddenly Bob finds himself caught in the equivalent of a Turing machine, the famed abstract device that can simulate any computer algorithm. Thanks to the peculiarities of the rules of Magic, Bob can now only finish the game when he meets whatever condition Alice has programmed her in-game algorithm to accomplish—for example, to find a pair of twin primes greater than one million.
arstechnica.com/science/2019/0

Network coding yields lower bounds: rjlipton.wordpress.com/2019/04

Lipton and Regan report on a new paper by Afshani, Freksen, Kamma, and Larsen (arxiv.org/abs/1902.10935) on lower bounds for multiplication. If algorithmically opening and recombining network messages never improves fractional flow, then $$O(n\log n)$$ circuit size for multiplication is optimal. But the same lower bound holds for simpler bit-shifting operations, so it's not clear how it could extend from circuits to bignum algorithms.

Planar point sets determine many pairwise crossing segments: arxiv.org/abs/1904.08845

János Pach, Natan Rubin, and Gábor Tardos make significant progress on whether every $$n$$ points in the plane have a large matching where all edges cross each other. A 1994 paper by Paul Erdős and half a dozen others only managed to prove this for "large" meaning $$\Omega(\sqrt{n})$$. The new paper proves a much stronger bound, $$n/2^{O(\sqrt{\log n})}$$ (Ryan Williams' favorite function).

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