Rejoining many who have forgotten or overlooked results

 St. Andrews Mac Tutor biographies source

Henry Smith was a mathematician of the 19th century who worked mainly in number theory. He especially did important work on the representation of numbers by various quadratic forms. We have discussed how even in something seemingly settled, like Joseph Lagrange’s theorem that every natural number is representable as a sum of four squares, new questions are always around—especially when one considers complexity.

Today Ken and I want to discuss a private slip of forgetfulness, and how often others may have done the same.

Various senses of ‘many’ from proofs of the infinitude of primes

 2008 Bowen Lectures source

Hillel Furstenberg is entering his 50th years as a professor of mathematics at Hebrew University in Jerusalem. He shared the 2006-07 Wolf Prize with Stephen Smale. He has shown many connections between analysis and combinatorics. One was showing how ergodic theory can prove Endre Szemerédi’s theorem that every positive-density subset of the integers includes arbitrarily-long arithmetic progressions ${[a + bn]}$. This led to a new multidimensional formulation with Yitzhak Katznelson, and the two later proved positive-density versions of the Hales-Jewett theorem.

Today Ken and I wish to discuss proofs of the infinitude of primes, and what they begin to say about analysis and combinatorics.

How some predictions fared in 2020 and other years

 Zimbio source

Simon Donaldson, Maxim Kontsevich, Terence Tao, Richard Taylor, and Jacob Lurie (photo order) won the 2015 Breakthrough Prize in Mathematics. This did not happen since Thursday; it happened last June. When Tao was asked to explain the 2015 prize date at the start of his 11/12/14 appearance on The Colbert Report, he said,

“It’s about the future.”

Today Dick and I salute the prize-winners, and preview a new book about advances that were made in the years 2015–2019.

 chessprogramming wiki source

Larry Kaufman is a Grandmaster of chess, and has teamed in the development of two champion computer chess programs, Rybka and Komodo. I have known him from chess tournaments since the 1970s. He earned the title of International Master (IM) from the World Chess Federation in 1980, a year before I did. He earned his GM title in 2008 by dint of winning the World Senior Chess Championship, equal with GM Mihai Suba.

Today we salute Komodo for winning the 7th Thoresen Chess Engines Competition (TCEC), which some regard as the de-facto world computer chess championship.

When does it become hard to compute?

Thomas Muir coined the term “permanent” as a noun in his treatise on determinants in 1882. He took it from Augustin Cauchy’s distinction in 1815 between symmetric functions that alternate when rows of a matrix are interchanged versus ones that “stay permanent.” To emphasize that all terms of the permanent have positive sign, he modified the contemporary notation ${\left| A \right|}$ for the determinant of a matrix ${A}$ into

$\displaystyle \overset{+}{|} A \overset{+}{|}$

for the permanent. Perhaps we should be glad that this notation did not become permanent.

Today Ken and I wish to highlight some interesting results on computing the permanent modulo some integer value.

What to do when afraid to see if what you want is true

 Cropped from Canadian Bergler Society source

Edmund Bergler coined the term in 1947, the great writers Francis Fitzgerald—F. Scott to most—and Joseph Conrad among many others suffered from it, as did the great cartoonist Charles Schulz. The problem is writer’s block.

Today Ken and I want to write about something that I wonder if any of you have ever had.