My name is Gil Kalai and I am a mathematician working mainly in the field of Combinatorics. Within combinatorics, I work mainly on geometric combinatorics and the study of convex polytopes and related objects, and on the analysis of Boolean functions and related matters. I am a professor at the Institute of Mathematics at the Hebrew University of Jerusalem and also have a long-term visiting position at the departments of Computer Science and Mathematics at Yale University, New Haven.
Gosset polytope– a hand drawing by Peter McMullen of the plane projection of the 8-dimensional 4-simplicial 4-simple Gosset polytope.
The Shlegel diagram (a certain 3-d projection) of the 120-cell – a regular 4-dimensional polytope with 120 dodecahedral facets.
2. Getting started:
I have decided to follow the (separate) suggestions of Terry Tao, Noam Nisan and Oded Schramm (followed by a final push by Avi Wigderson) and set up a blog; I plan to do it for about a year. It will be centered around combinatorics which is my main area of research and will touch on other matters.
Tao, Schramm and Nisan
3. A problem: Frankl’s Conjecture
Let me tell you about a problem in extremal combinatorics that I like: I will probably return to the context later. But meanwile have a look:
Let be a finite family of finite sets which is closed under union. In other words, if then also .
There exists an element which belongs to at least half the sets in .
(Sept 09: Frankl’s conjecture was proposed, among other problems, for a “polymath project.”)
4. Earlier blog experience:
I became acquainted with blogs quite recently; Greg Kuperberg told me about Dave Bacon’s blog “The Quantum Pontiff“. Then I asked my children if they knew what blogs were, and they looked at me (again) like I said something very silly; It turned out my youngest son, Lior, had already completed a blog.
Greg Kuperberg (left) with his mother Krystyna. (Both Greg’s parents, Krystyna and Wlodek, are well known mathematicians). Dave Bacon (right)
Since then, I took part in a few blog discussions. I am curious about the meaning and role of mathematical discussions in general, and on blogs in particular.
5. Are mathematical debates possible?
I even took part in little “mathematical debates”. Here is a little quote from one of them; never mind the context. There aren’t many debates and controversies in mathematics, so if mathematical debating will ever become more developed, mathematical polemics may, perhaps, look like this:
Comment by Gil Kalai 3/14/2006:
In all these cases you approximate a rank-one matrix to start with. I believe that you may be able to approximate a rank-one matrix up to a rank-one error. I do not believe that you will be able to approximate an arbitrary matrix up to a rank one matrix.
Comment by Dave Bacon, later that day:
I will never look at rank one matrices the same ;)
6. Guest Blogging on “what’s new”:
I gave two guest posts over Terry Tao’s blog
One was about the weak epsilon net problem.
and the other was about the entropy/influence conjecture.
7. Other topics.
I will occasionally try to discuss areas of mathematics that touch on combinatorics, and also topics that go beyond my expertise: applied mathematics, and in particular its applications to, and connection with computer science, economics, statistics, and even physics and philosophy; and issues related to academic life, especially in Israel.