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The Inscribed Square Problem

October 21, 2018


A remark on an open problem with an application of the Lebesgue Density Theorem

[ Toeplitz ]

Otto Toeplitz was a mathematician who made key contributions to functional analysis. He is famous for many things, including a kind of matrix named after him.

Today we discuss one of his conjectures that remains open.
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London Calling

October 18, 2018


For chess and science: a cautionary tale about decision models

Clarke Chronicler blog source

Marmaduke Wyvill was a British chess master and Member of Parliament in the 1800s. He was runner-up in what is considered the first major international chess tournament, London 1851, but never played in a comparable tournament again. He promoted chess and helped organize and sponsor the great London 1883 chess tournament. Here is a fount of information on the name and the man, including that he once proposed marriage to Florence Nightingale, who became a pioneer of statistics.

Today we use Wyvill’s London 1883 tournament to critique statistical models. Our critique extends to ask, how extensively are models cross-checked?

London is about to take center stage again in chess. The World Championship match between the current world champion, Magnus Carlsen of Norway, and his American challenger Fabiano Caruana will begin there on November 9. This is the first time since 1972 that an American will play for the title. The organizer is WorldChess (previously Agon Ltd.) in partnership with the World Chess Federation (FIDE). Read more…

Watching Over the Zeroes

October 10, 2018


How confident should we be that the Riemann Hypothesis is true?

Composite of src1, src2

Andrew Odlyzko and Herman te Riele, in a 1985 paper, refuted a once widely believed conjecture from 1885 that implies the Riemann Hypothesis. The belief began a U-turn in the 1940s, and by the late 1970s the community was convinced its refutation would come from algorithmic advances to bring the needed computation into a feasible range. Odlyzko and te Riele duly credit advances in algorithms—not mere computing power—for their refutation.

Today we consider reasons for and against a belief in the Riemann Hypothesis (RH), contrast them with the situation for {\mathsf{P = NP}}, and point out that there are numerous conjectures weaker than RH but wide open which an RH claimant might try to crack first.
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Reading Into Atiyah’s Proof

September 26, 2018


The Todd function method

MacTutor biography source

John Todd was a British geometer who worked at Cambridge for most of his life. Michael Atiyah took classes from him there. He was not Atiyah’s doctoral advisor—that was William Hodge—but he advised Roger Penrose, Atiyah’s longtime colleague at Oxford.

Today Ken and I want to add to the discussion of Atiyah’s proof of the Riemann Hypothesis (RH).
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Preview of the Atiyah Talk

September 23, 2018


Why the Riemann hypothesis is hard and some other observations.

ICM 2018 “Matchmaking” source

Michael Atiyah, as we previously posted, claims to have a proof that the Riemann Hypothesis (RH) is true. In the second half of a 9:00–10:30am session (3:00–4:30am Eastern time), he will unveil his claim.

Today we discuss what the RH is and why it is hard.
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The Specter of Simple(r) Proofs

September 21, 2018


When has a strikingly simple proof come first?

Cropped from London Times 2017 source

Michael Atiyah is giving a lecture next Monday morning at the Heidelberg Laureate Forum (HLF). It is titled, simply, “The Riemann Hypothesis.” An unsourced image of his abstract says what his title does not: that he is claiming not only a proof of Riemann but a “simple proof using a radically new approach.”

Today we discuss cases where theorems had radically simpler proofs than were first contemplated.
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Abelian Groups, Mostly

September 14, 2018


Simple probabilistic arguments that apply to monoids too

Famous Mathematicians source

Niels Abel is of course a famous mathematician from the 19th century. Many mathematical objects have been named after him, including a type of group. My favorites, besides groups, are: Abel’s binomial theorem, Abel’s functions, and Abel’s summation formula. Not to mention the prize named after him, for which we congratulate Robert Langlands.

Today we will talk about commutative groups and a simple result concerning them.
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