For the long days of ICM 2014 lectures, and long flights to and from Seoul, some mathematical gymnastics is needed. And this is precisely what Omer Angel taught us in his recent visit. Combining gymnastic with a demonstration of parallel transport and deep insights on human physiology!
You want to move from this position
to this position
without simply rotating your hand.
Here are two video-demonstrations by some HUJI top people
Click on the picture to see the video. (The drill is a sequence of five moves and the video skipped the first.)
Some other mathematical gymnastics is demonstrated in the post the ultimate riddle. And for a futurist mathematical application to sport (soccer) see this post.
I just saw in the Notices of the AMS a paper by Geelen, Gerards, and Whittle where they announce and give a high level description of their recent proof of Rota’s conjecture. The 1970 conjecture asserts that for every finite field, the class of matroids representable over the field can be described by a finite list of forbidden minors. This was proved by William Tutte in 1938 for binary matroids (namely those representable over the field of two elements). For binary matroids Tutte found a single forbidden minor. The ternary case was settled by by Bixby and by Seymour in the late 70s (four forbidden minors). Geelen, Gerards and Kapoor proved recently that there are seven forbidden minors over a field of four elements. The notices paper gives an excellent self-contained introduction to the conjecture.
This is a project that started in 1999 and it will probably take a couple more years to complete writing the proof. It relies on ideas from the Robertson-Seymour forbidden minor theorem for graphs. Congratulations to Jim, Bert, and Geoff!
Well, looking around I saw that this was announced in August 22’s post in a very nice group blog devoted by matroids- Matroid Union, with contributions by Dillon Mayhew, Stefan van Zwam, Peter Nelson, and Irene Pivotto. August 22? you may ask, yes! August 22, 2013. I missed the news by almost a year. It was reported also here and here and here, and here, and here, and here!
This is a good opportunity to mention two additional conjectures by Gian-Carlo Rota. But let me ask you, dear readers, before that a little question.
Rota’s unimodality conjecture and June Huh’s work
Rota’s unimodality conjecture predicts that the coefficients of the characteristic polynomial of a matroid form a log-concave sequence. This implies that the coefficients are unimodal. A special case of the conjecture is an earlier famous conjecture (by Read) asserting that the coefficients of the chromatic polynomial of a graph are unimodal (and log-concave). This conjecture about matroids was made also around the same time by Heron and Welsh.
Huh’s path to mathematics was quite amazing. He wanted to be a science-writer and accomponied Hironaka on whom he planned to write. Hironaka introduced him to mathematics in general and to algebraic geometry and this led June to study mathematics and a few years later to use deep connections between algebraic geometry and combinatorics to prove the conjecture.
David Kazhdan, a very famous mathematician from my department with a super-human understanding of mathematics (and more) is recovering from a terrible bike accident. Here is an article about him from “Maariv.” (In Hebrew)
Amnon Shashua, a computer science professor at the Hebrew University founded Mobileye fifteen years ago. Here is one of many articles about Mobileye. Mobileye helps eliminate car accidents and her sister company Orcam that Amnon also founded develops aids for the visually impaired.
Nathan Keller, now at Bar-Ilan University, is a former Ph D student of mine working in probabilistic combinatorics and he has a parallel impressive academic career in the area of cryptology. Here is an article about Nathan from Arutz 7 (in Hebrew). (The picture above shows Nathan with Eli Biham and Elad Barkan after their 2003 success in cracking the popular GSM cellular phone network encryption code.)