Monthly Archives: June 2009

Test Your Intuition (6)

This is not as clear cut a question as the earlier ones, and if you do not know an answer then it will be difficult to figure one out just based on intuition. (But perhaps possible).

If you are intrigued by the question and would like to explore what an answer could be, I would be interested to know how you tried to find an answer.  Asked a colleague? Looked at a book (which?)? Looked online (where?)? 

Here is the question:

A differentiable complex function automatically has derivatives of every order. (In contrast to differentiable real functions that need not have even second derivatives at any point.) 

Can you describe this “miracle” as part of a more general phenomenon?

Praise For ‘Gina says’

Praise for: ” ‘Gina Says,’ Adventures in the Blogsphere String War

(Below the dividing line:  Greg Kuperberg, Scott Aaronson, Clifford Johnson, Peter Woit, Motty Perry, Caterina Calsamiglia, Yuval Peres, Eva Illouz, and (right from the comment section)  Luca Trevisan, Thomas Love, John Sidles, Jacques Distler, Marni D Sheppeard, (and from other journals/blogs) Hamish Johnston, Lance Fortnow. New: Lubos Motl, Eytan Sheshinski and Tselil Schramm.

Download the first part of the book (pdf file) (See also this post)

   Shmuel Weinberger (August 07): Very much enjoyed the story of Gina’s involvement in the blog world– i read it through on my flight back to America. It was a very interesting if occasionally difficult read. Probably the part that resonated most was the advice (i think it came from your father) that every subject is fascinating after you study it deeply.

Avi Wigderson: (August 07)  I expected no less from the author of the immortal translation of the classic book “Where is Pluto?

Oded Schramm (Dec 2007): Though it is somewhat uneven, there were some definite enjoyable highlights. (Feb. 08 ) What about a sequel? I’m really curious what’s happening with Gina these days?

ראה תמונה בגודל מלא

Itai Benjamini It was such a joy to listen to your unique voice (music) yet again. A typo: Section 20 “children’s teaching disabilities” should be “children learning disabilities”. (Feb 08 ):


Elchanan Mossel (Feb 08): I read it when I was sick and couldn’t do other things, and it cheered me up



ראה תמונה בגודל מלאOlle Haggstrom (March 08): I found it a real page-turner, and read the entire thing for four straight hours last night! Very interesting stuff, on several levels. And very original, of course. May I ask what is your relation to Gina? You seem to have remarkable insight into her mind…


Ken Binmore (June 08) Dear Gil, I like your book a lot. If you get it published, it could do with Continue reading

My Book: “Gina Says,” Adventures in the Blogosphere String War

I wrote a book. It is a sort of a popular science book and it is also about blogging and debating.

You can download the first part of the book : It is a 94 page pdf file.


 “Gina Says,”

Adventures in the

Blogosphere String War

selected and edited by Gil Kalai


Praise for “Gina Says” 

To download the second part.


Debates portrayed in books, are the worst sort of readings,                        Jonathan Swift.


In the summer of 2006 two books attacking string theory, a prominent theory in physics, appeared. One by Peter Woit called Not even wrong” and the other by Lee Smolin called “The trouble with Physics.” A fierce public debate, much of it on weblogs, ensued.

Gina is very curious about science blogs.  Can they be useful for learning about, or discussing science? What happens in these blogs and who participates in them? Gina is eager to learn the issues and to form her own opinion about the string theory controversy. She is equipped with some academic background, even in mathematics, and has some familiarity with academic life. Her knowledge of physics is derived mainly from popular accounts. Gina likes to debate and to argue and to be carried by her associations. She is fascinated by questions about rationality and philosophy, and was exposed to various other scientific controversies in the past.

This book uses the blog string theory debate to tell about blogs, science, and mathematics. Meandering over various topics Continue reading

A Little Story Regarding Borsuk’s Conjecture

Jeff Kahn

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.

You can download our paper here. Here is the proof itself. Continue reading

Borsuk’s Conjecture

Karol Borsuk conjectured in 1933 that every bounded set in R^d can be covered by d+1 sets of smaller diameter.  Jeff Kahn and I found a counterexample in 1993. It is based on the Frankl-Wilson theorem.

Let \cal G be the set of \pm 1 vectors of length n. Suppose that n=4p and p is a prime, as the conditions of Frankl-Wilson theorem require. Let {\cal G'} = \{(1/\sqrt n)x:x \in {\cal G}\}. All vectors in {\cal G}' are unit vectors.

Consider the set X=\{x \otimes x: x \in {\cal G}'\}. X is a subset of R^{n^2}.

Remark: If x=(x_1,x_2,\dots,x_n), regard x\otimes x as the n by n matrix with entries (x_ix_j).

It is easy to verify that:

Claim: <x \otimes x,y\otimes y> = <x,y>^2.

It follows that all vectors in X are unit vectors, and that the inner product between every two of them is nonnegative. The diameter of X is therefore \sqrt 2. (Here we use the fact that the square of the distance between two unit vectors x and y is 2 minus twice their inner product.)

Suppose that Y \subset X has a smaller diameter. Write Y=\{x \otimes x: x \in {\cal F}\} for some subset \cal F of \cal G. This means that Y (and hence also \cal F) does not contain two orthogonal vectors and therefore by the Frankl-Wilson theorem

|{\cal F}| \le U=4({{n} \choose {0}}+{{n}\choose {1}}+\dots+{{n}\choose{p-1}}).

It follows that the number of sets of smaller diameter needed to cover X is at least 2^n / U. This clearly refutes Borsuk’s conjecture for large enough n. Sababa.

Let me explain in a few more words Continue reading