## Archive for August, 2009

### Mathematical Research and the Internet

August 28, 2009

Terence Tao has given a talk on “Mathematical Research and the Internet” here is a link to a post about the talk which includes an advance copy of the talk:

http://terrytao.wordpress.com/2009/08/27/mathematical-research-and-the-internet/

Let me update this. The above now has an updated version of the talk and there are more talks up see:

http://terrytao.wordpress.com/2009/08/31/more-mahler-lectures/

and also:

August 27, 2009

Let us return to the problem. We have 8 points and we want to form two groups of three and one group of two so that the intersection of the convex hulls of the groups is not empty.

We take 7 of the points and apply Tverberg’s theorem we get three groups only one can be only one point so we have the following possibilities (4,2,1), (3,2,2), (3,3,1).
For (3,2,2) we add the remaining point to one of the 2 groups and we get a solution to the problem.
If we have groups of the form (3,3,1) we add a point to the group with one point and we get a solution.

The only set of groups left is the set (4,2,1). Here we use Caratheodory’s theorem. We use it to get three points out of the four containing the single point then we move the remaining point to the group containing the single point and then add the remaining point to one of the groups of two and we have a solution to the final case. In the next of this series of posts I want to look at possible generalizations.’

August 26, 2009

In the third of this series of posts I want to present Tverberg’s theorem. It is a generalization of Radon’s theorem: If m is greater than (r-1)*(d+1) +1 then any m points in d dimensional space can be divided into r sets that are not empty such that the intersection of the convex hulls of the sets is not empty.

Also let me introduce Caratheodory’s theorem if in d dimensional space x is in the convex hull of n points and n is greater than d+1 there is a subset of d+1 of the n points such that the convex hull of the d+1 points contains x. More information about these theorems is here:

http://gilkalai.wordpress.com/2008/11/24/sarkarias-proof-of-tverbergs-theorem-1//

In the next of these series of posts I want to start using these tools on the original problem.

August 25, 2009

This is a follow up to a post entitled Radon related Problem. I hope to continue this series of posts.

Let me introduce Radon’s theorem: It is the following if there are more than d+2 points in a d dimensional space the points may be divided into two sets whose convex hulls intersect in at least one point. When the problem mentioned in the previous post is referred to as Radon related it is this theorem which is meant. Let me give a reference for this theorem:

If we try to go beyond a division into 2 sets of points to 3 and more there is a generalization called Tverberg’s theorem. I will talk about it in the next of these series of posts.

August 23, 2009

In this post and possibly in future posts I want to look at the following problem. Suppose we have eight points in the plane can we divide these points into three sets two of three points one of two points such that the intersection of the convex hull of each of the three sets is nonzero. I want to look at a possible proof and some generalizations and connections.

August 10, 2009

There is a thread on “What mathematicians need to know about blogging” at

http://golem.ph.utexas.edu/category/2009/08/what_do_mathematicians_need_to.html

In it John Baez asks the question for a forthcoming article.

### Polymath4

August 9, 2009

Polymath4 is up and running. Here is a url for the research thread:

Let me update this. There is now a new research thread:

Let me update further. There is now another new research thread:

### Polymath4

August 8, 2009

In a few days it looks like Polymath4 will be launched. The problem to be solved is to find a prime with k digits using an algorithm which takes a polynomial in k amount of time. Polymath 3 has yet to start it involves the Hirsch conjecture.

### Two new wiki pages

August 3, 2009

There are two new wiki pages related to possible future polymath projects.
One on the Hirsch conjecture:

http://michaelnielsen.org/polymath1/index.php?title=The_polynomial_Hirsch_conjecture

The other on
Boshernitzan’s problem: