Combinatorics, Mathematics, Academics, Polemics, …
Helly’s Theorem, “Hypertrees”, and Strange Enumeration I (There were 3 follow up posts:)
Extremal Combinatorics I: Extremal Problems on Set Systems (There were 4 follow up posts II ; III; IV; VI)
Rationality, Economics and Games
Five Open Problems Regarding Convex Polytopes
Seven Problems Around Tverberg’s Theorem
F ≤ 4E
The AC0 Prime Number Conjecture
Coloring Simple Polytopes and Triangulations
Some old and new problems in combinatorics and geometry
Noise Stability and Threshold Circuits
Taxi and other Stories
Here is, with Peter’s kind permission, a rebloging of Peter’s post on the automorphism group of . Other very nice accounts are by the Secret blogging seminar; John Baez,; A paper by Howard, Millson, Snowden, and Vakil; and most famously the legendary Chapter 6 (!) from the book by Cameron and Van-Lint (I dont have an electronic version for it).
My TYI 25 question about it arose from Sonia Balagopalan’s lecture in our combinatorics seminar on the 16 vertex triangulation of 4-dimensional projective space. (Here is the link to her paper.)
Originally posted on Peter Cameron's Blog:
No account of the symmetric group can be complete without mentioning the remarkable fact that the symmetric group of degree n (finite or infinite) has an outer automorphism if and only if n=6.
Here are the definitions. An automorphism of a group G is a permutation p of the group which preserves products, that is, (xy)p=(xp)(yp) for all x,y (where, as a card-carrying algebraist, I write the function on the right of its argument). The automorphisms of G themselves form a group, and the inner automorphisms (the conjugation maps x?g-1xg) form a normal subgroup; the factor group is the outer automorphism group of G. Abusing terminology, we say that G has outer automorphisms if the outer automorphism group is not the trivial group, that is, not all automorphisms are inner.
Now the symmetric group S
View original 1,245 more words
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.)
Here is a lovely lecture starting with quantum computers, going through the Feldenkrais method, and ending with a mathematical puzzle.
Click on the picture for the video of the talk.
Here are Dorit’s four body-mind principles for learning:
1. Start within your comfort zone and make it even more conforting,
2. Not too easy not too hard, pick an interesting challenge within your reach,
3. Move away from your desired place and come back to it from different angles,
4. Play with it, connect it with other things you know, make it your own.
Alexander Chervov asked over Mathoverflow about Noteworthy results in and around 2010 and some interesting results were offered in the answers. If you would like to mention additional results you can comment on them here. The only requirement is to explain what the result says and give links if possible.
Alex Lubotzky and I are running together a year long course at HU on High Dimensional Expanders. High dimensional expanders are simplical (and more general) cell complexes which generalize expander graphs. The course will take place in Room 110 of the mathematics building on Tuesdays 10-12.
Topics will include:
- Some background on expander graphs and on simplicial complexes and homology
- The geometric definition of high dimensional expanders in the recent paper: Overlap properties of geometric expanders. by J. Fox, M. Gromov, V. Lafforgue, A. Naor, and J. Pach;
- A cohomological definition arising in Linial-Meshulam’s work about homology of random complexes; possible definitions based on high Laplacians,
- Ramanujan complexes;
- Potential applications to error correcting codes and quantum error correcting codes.
(I will add further relevant links, and a more detailed description later.)
Jeff and I worked on the problem for several years. Once he visited me with his family for two weeks. Before the visit I emailed him and asked: What should we work on in your visit?
Jeff asnwered: We should settle Borsuk’s problem!
I asked: What should we do in the second week?!
and Jeff asnwered: We should write the paper!
And so it was.